# Bursting Phenomenon and Chaos Phase Control in Plant Dynamics

Authors: Makenne Yemeli Lola; Kengne Romanic; Pelap Francois Beceau
Journal: Complexity
 (2023)

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## Abstract

New complex features of a Pinus family plant subjected to wind load by proposing an analog electronic simulator with an activeRC realization that mimics the real-time dynamics of the system was examined in this paper. Findings reveal that the periodic or chaotic dynamics of the system depends on the favored initial conditions. We see the dynamical behavior of the plant showing transition from periodic to chaotic states. This is very beneficial because it allows for the observation that when the value of the wind amplitude ratio changes continuously, the plant behavior may change in a discontinuous way. It also helps to understand the impact of wind on the plant dynamics, in particular, and in forest in general, which is very crucial for understanding several plant communities. We show that plants exhibit multiple forms (periodic/chaotic) of repetitive spiking oscillations reflecting the phenomenon of bursting. For the periodic bursting oscillations, we observe that after each peak of excitation due to the wind load, the plant returns directly to its state of rest, whereas the chaotic bursting oscillations break the energy of the wind into decreasing value packets and its occurrence has the merit of reducing the effects of wind energy on the plant; this action ensures the stability of the plant even in chaotic cases. We also observe that the numerical study based on the isospike technique permits efficient discovery and separation between periodic and chaotic orbits visited during the temporal evolution. We also establish through the chaos phase-control techniques that various chaotic oscillations disappeared after applying the chaos phase control strategy in the system compared to the uncontrolled case. Moreover, the chaos phase-control strategy to the system is introduced to instantly inform how the correct choice of the phase minimizes the effects of the wind and thus eliminates chaotic behavior of the plant by simply driving it towards a variety of periodic orbits.


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## Body

## 1. Introduction

Wind damage is among the most significant factors destroying plants and increasing climate change. Their impact can vary from a reduction in productivity to the elimination of whole plants, resulting in the loss of biodiversity. Recently, research has shown that some plants often uproot or break when they are forced by waves or gusts of wind [1, 2]. Therefore, to improve the understanding of the complex plant-wind interaction and avoid some damages, the plant was first considered as a beam subjected to air flow [3]. But this model was quickly replaced by another one proposed by Miller [4], in which the dynamics of the plant is examined with a third-order nonlinear damping term proportional to the power of its velocity [5]. This approach, which captures our attention here, corresponds to the natural structure of some trees belonging to the Pinus family known as Pinus sibirica [6], Pinus pinaster [7], and Pinus lambertiana [8]. Some authors have demonstrated that plants exhibit complex and adaptive behavior and incorporate flexible ways to unify this behavior intelligibly, contrary to what most people would expect [9].During the last twenty years, great attention was placed on physical systems exhibiting complex behavior such as bursting phenomena [10, 11]. Bursting oscillations are known as waveforms consisting of alternating large (fast) and small (slow) amplitude excursions. The fast subsystem can exhibit an active phase with fast oscillations, and a silent phase stable at steady states while the slow subsystem is responsible for periodically switching the dynamics of the fast subsystem between these two states [12]. In other words, there exist two types of bifurcations in a bursting oscillation, specifically, one which transforms the system from spiking states to quiescent states, and the other which moves the system from rest states to peak states [13]. Thanks to the fast-slow analysis method developed by Metzger et al. [14], bursting mechanism in nonlinear dynamical systems is easily analyzed by dividing them into fast and slow subsystems. This method was recently used in the classification of physical system exhibiting bursting oscillations such as neural firing [15]. As far as theoretic analysis is concerned, investigations of the bursting phenomenon were also examined in some systems, where authors revealed the essential dynamical nature of bursting through some crucial bifurcations [16, 17]. Bursting phenomenon was used to describe duct flows and bursts in the atmosphere in turbulence phase [18, 19]. Moreover, bursting oscillations are frequently observed in signal transduction processes and are supplementary to certain biological phenomena, such as oscillatory reactions of enzymes [20, 21]. To the best of our knowledge, this phenomenon has never been checked in the complex plant-wind interactions. To treat chaos in dynamical systems and develop the strategy of its control, researchers considered the motion of a periodically forced steel beam deflected towards two magnets and Duffing-like proxy. The two systems have the exceptional technological advantage of bypassing noisy spectra [22–25]. Akgul et al. [26] introduced a new three-dimensional chaotic system which contains cubic, quadratic, and quartic nonlinearities. The researchers studied the basic dynamical characteristics of the new system, such as eigenvalue structures, equilibria, fractal dimension, Lyapunov exponent spectrum, and chaotic behaviors. Furthermore, their work investigates bifurcation analysis of their proposed chaotic system by means of a selected parameter, and this chaotic system has been investigated by both theoretical analysis and detailed numerical analysis. They observed that the amplitude values are important in chaotic systems for real environment applications because of material limitations and electronic components. Thus, for real environment application, the new chaotic system is rescaled and executed an electronic circuit implementation. Their obtained results show that this system has complex dynamics with some fascinating characteristics and deserves a further detailed study. They conclude their work showing that the new chaotic system can be advantageous in many engineering and scientific applications, such as cryptology, physics, random number generator, and control. Let us keep in mind that Duffing oscillators are used to model a number of dynamical systems. Volos et al. [27, 28] studied experimentally the case of chaotic synchronization of two identical nonlinear electronic circuits. Their work was in a very interesting research area because of its applications to the field of secure communications. The circuit they have used is a second-order, Duffing-type, nonlinear electronic circuit driven by a sinusoidal voltage source. In their circuit, the nonlinear element has a cubic characteristic. They have also studied, in the same way, the dynamical behavior of the system in the case of the bidirectional coupling via a linear resistor. Their simulation and experimental results show that chaotic synchronization is possible in the Duffing oscillator.Luo et al. [29–31] examine the analytical prediction of the bifurcation trees of periodic motions in a time-delayed, periodically forced, and hardening Duffing oscillator using a semianalytical method. This method, based on the differential equation discretization of the time-delayed and nonlinear dynamical system, helps to evaluate the bifurcation trees for the unstable and stable solutions of periodic motions to chaos. Thus, the Duffing oscillator is achieved analytically. In the other part, using the finite discrete Fourier series, harmonic frequency amplitude curves for stable and unstable solutions of period-1 to period-4 motions were developed for a better understanding of singularity, quantity levels, and catastrophes of harmonic amplitudes in the frequency domain. By means of the analytical prediction, numerical results of periodic motions in the hardening Duffing oscillator and time-delayed motions are completed. Through their numerical illustrations, the authors showed that the asymmetry and complexity of period-1 motions to chaos in nonlinear dynamical systems strongly depend on the quantity levels and distributions of harmonic amplitudes. They show in their works that as the quantity level of specific harmonic amplitudes increases, the effects of the corresponding harmonics on the periodic motions become very strong, and the certain asymmetry and complexity of periodic motion and chaos can be identified through harmonic amplitudes with higher quantity levels.Regarding the control of chaos in physical systems, some strategies were developed to extract nonchaotic behaviors from chaos by applying slight perturbations [32, 33]. Among them, the phase control strategy seems to be the most precise device which aimed to separate the behaviors of the system (periodic from chaotic) by using weak harmonic disturbances. Accordingly, the veracity of the chaos phase control strategy has been demonstrated in different systems exhibiting bursting phenomena [34, 35]. Indeed, this technique was successfully applied to control entanglement in quantum parametric oscillators [36] and to manipulate escaping dynamics in an open system [37]. In spite of the variety of potential applications of phase control strategy and an attention drawn on it in recent years, numerous aspects remain unclear. For example, when considering the double-well Duffing oscillators, the sensitivities and the stability regions to each individual perturbation are not known a priori [38, 39].Based on the aforementioned works, we are interested in the work of Bashkirtseva et al. [40] who used a nonautonomous system to explain the comprehensive bifurcations associated with the bursting oscillations and provide a complete theoretical classification (fast-slow) of dynamical bifurcation analysis. Motivated by the results of the above studies and due to the complex dynamic behaviors of plants under air flow, we have no doubt that our investigations will reveal interesting features on the dynamics of the plant-wind system. Therefore, in our future works, we aim to study the factional other derivatives under wind load and also look for the contribution of our studies in the agricultural domain using machine learning.The rest of this paper is structured as follows. Section2 describes the model and presents its analyses. In Section 3, we consider the wind load as an external force and examine its influence on the plant dynamics in preparation for the study of the bursting phenomenon. Here, attention is focused on the effects of forcing frequency on the occurrence of bursting oscillations. The role of these new types of oscillations is also examined. In Section 4, the chaos phase control technique in plants is developed to show the dynamical mechanisms of minimizing the chaotic behavior in the plant dynamics by using weak harmonic perturbations. Section 5 presents the conclusions of the whole work.


## 2. Model Description

### 2.1. Mechanical Representation and Characteristic Equation

The model used in this paper is a tree trunk of massm swaying in the air with a stiffness constant k and the damping d due to the internal viscoelastic of the wood, the roots, and the soil mass as shown in Figure 1:Figure 1
Dynamical model of a tree trunk: (a) structural representation and (b) mechanical equivalent model.
(a)(b)In this structural illustration, the plant is considered an isolated system subjected to an external loading called wind. This approach, which is the most used, allows analyzing the different behaviors of the plant and appreciating the effects of the different wind parameters on the dynamics of the plant. The model of Figure1 was firstly used to demonstrate its applicability for prediction of the precritical [41] and postcritical [42] behaviors. The possibility that the behavior of the plant displays certain aspects strikes many people, whether or not they are plant specialists. The notion of cognition seems appropriate even if it has not yet been extended to plants because researchers have established that plant systems deserve recognition of the intelligence developed in their behavior [43]. It was also exploited to describe the complex dynamics of various complex chemical, biological, and physical systems [44–46]. Moreover, a number of searches revealed that this model is very useful to detect a feeble complex signal against a background of sturdy noise [47]. Hereafter, the structural representation in Figure 1(a) deals with the natural structure of some trees [48]. However, the complex motion of the tree swaying in the air flow is often simplified using a conceptual model of a tree stem (without branches) and being considered as an upside-down pendulum. Within this concept, Miller [4] uses the structural model of Figure 1(a) and models the tree as a Duffing oscillator with an additional cubic stiffness term bx3. Thus, the nonlinear differential equation that governs the dynamics of our model is put in the form calculated as follows:(1)mx¨t+dx˙t−kxt+bx3t=I0t,in which d, k, b, and I0t are, respectively, the damping coefficient, the linear stiffness, the nonlinear stiffness of the spring, and the effective wind force acting on the plant. The quantity xt designates the deflection of the stem. Equation (1) describes the electromechanical control of the beam dynamics in a state where the single mode representation of these dynamics is that of a Duffing oscillator [49] and modeled the ship rolling motions [19]. Consequently, the dimensionless form associated to equation (1) is(2)X¨T+2ξX˙T−XT+αX3T=FcosωT.In equation (2), the diverse parameters are defined as follows: η1=bLmax3, η=kLmax, F=β0/η,ω=ω1/ω0,X=x/Lmax,α=η1/η, and T=ω0t. Within these quantities, parameter Lmax represents the characteristic maximum deflection of the tree, ω0 stands for the natural frequency of pure oscillations of the trunk, while ω1 is the frequency of the air flow; therefore, ω defines the frequency ratio of the system. Parameter α is the nonlinear stiffness ratio in which η defines the stiffness of the spring, whereas η1 designates the stiffness coefficient of the structure. The factor ξ=d/2mω0 represents the damping ratio. The quantity F denotes the amplitude force ratio, wherein β0 is the forcing amplitude. For simulation and experimental purposes, equation (2) can also be rewritten in the form(3)X˙=Y,Y˙=X−2ξY−αX3+Fcosωτ.Let us remind that the occurrence of symmetry property in a given system is known as a moderating factor of the amount of work needed [50]. System (3) shows that the symmetry of the plant-wind system depends on the wind action. In fact, this system is symmetric if F=0 and remains asymmetric otherwise. The symmetry of the attractors of equation (3) was recently analyzed [51].


### 2.2. PSpice Representation of the System

To improve the understanding of some new physical phenomena dealing with the real experimental setup of the plant subjected to wind load, we build the analog simulator of our mechanical system described by equation (3) as illustrated on Figure 2. Such approach is not new in physics because electronic circuits have extraordinary fascinated researcher’s attention for simulating complex nonlinear chaotic systems. Firstly, various nonlinear electronic circuits are used to examine and understand the dynamics of several complex physical systems under real-time conditions [52]. Secondly, electronic circuits provide a very useful medium for both modeling and experimental studies of nonlinear chaotic systems [53]. In recent years, concerning the electronic circuit in chaotic system, Sambas et al. [54, 55] introduced the new 3D chaotic dynamical system with a peanut-shaped closed curve of equilibrium points and described a detailed dynamical analysis of the new chaotic model using bifurcation diagrams and entropy analysis. They showed the multistability and coexisting convergent attractors in the new nonlinear plant. In the same work, they utilized the properties of their chaotic system in designing a new cipher colour image mechanism. Their experimental results demonstrated the efficiency of the presented encryption mechanism whose outcomes suggest promising applications for our chaotic system in various cryptographic applications. Few years after, the same authors announced a novel three-dimensional chaotic system with line equilibrium and discussed its dynamical properties such as Lyapunov exponents, phase portraits, equilibrium points, bifurcation diagram, multistability, and coexisting attractors. They established new synchronization results based on integral sliding mode control which are also derived for the new chaotic system with line equilibrium. In addition to these ideas, an electronic circuit implementation of the new chaotic system with line equilibrium is reported, and a good qualitative agreement is exhibited between the MATLAB simulations of the theoretical model and the Multisim results. To conclude their work, they propose an image encryption algorithm based on pixel-level scrambling, bit-level scrambling, and pixel value diffusion. Their experimental results showed that the encryption algorithm not only shuffles the pixel positions of the image but also replaces the pixel values with different values, which can effectively resist various attacks such as brute force attack and differential attack.Figure 2
Circuit diagram of system (3) with the values of the components: R2=50kΩ, R1=R3=R4=R6=R7=10kΩ, R8=R9=R10=10kΩ, R5=300Ω, and TL084CN≡Vjj=1,2,3,4.In the present case, the choice of the analog electrical simulator could be justified by the availability of electronic devices and the pedagogic necessity of having an experimental platform for graduate students. It is also based on the exploration of new features for researchers and the possibility to repeat the same experience at will. The diagram of Figure2 is implemented using four operational amplifiers (TL084CN), two multipliers (AD633JN), ten resistors, and two capacitors. The specific oscillator with active RC realization instead of the inductors is appropriate for assuring the circuit implementation. This RC block also spares us from looking for the links between R, L, and C components.To establish that the voltagesX and Y across the capacitors C1 and C2 are separately fulfilled in the subsequent set of the first-order differential equations, we apply the Kirchhoff’s laws to Figure 2 and obtain the following equations:(4)dxdτ=x˙=1R1C1ω0y,dydτ=y˙=−1R2C2ω0y+1R3C2ω0x−1R4C2ω0x3+eR5C2ω0cosωω0τ,in which τ=ω0t and ω0=104rad/s. While comparing equations (3) and (4), the upcoming links between the PSpice and MATLAB parameters are established:(5)1R1C1ω0=1,1R3C2ω0=1,α=1R4C2ω0,ξ=12R2C2ω0,andF=eR5C2ω0,within equation (5), the first three relations help to determine, respectively, the values of the resistors R1, R3, and R4 for a fixed frequency ω0. Indeed, the rheostat is varied until each of these conditions is fulfilled, thus indicating the value of the resistance sought. By exploiting the same principle, the last two relations of equation (5) permit to evaluate the values of the resistances R2 and R5 for various fixed values of ξ and F. Here, PSpice simulations are conducted at the frequency 15.93Hz with the print step 0.95 and the final time 1s.


### 2.3. Experimental Setup of the System

Many scientists have established that an experimental investigation is commonly needed to supplement theoretical/numerical results in complex nonlinear systems for technological and engineering applications [56]. In agreement with the theoretical abovementioned results, the chaotic oscillator of Figure 1 can experience some very interesting new dynamical behaviors. Hereafter, we will check the accuracy of our theoretical results by performing experimental analyses of the behavior of the system. The experimental setup of our system is drawn in Figure 3.Figure 3
A snapshot of the experimental device.As a starting point, we softly adjust the different values of resistorsR2 and R5 which denote, respectively, the dissipation in the plant and the strength of the wind acting on this plant. The experimental results of the phase-space trajectories are obtained by plotting VC1,VC2 by using a dual trace oscilloscope in XY mode.


### 2.4. Transition to Chaos in the Plant-Wind System

Chaos theory investigates the behavior of dynamical systems which are very sensitive to initial conditions. Its applications are found in physics, sociology, meteorology, and engineering [35, 57]. The main purpose of this subsection is to ensure an exhaustive analysis of some new phenomena linked to equation (3), which governs the dynamics of the plant subjected to wind loads. Therefore, direct integration of this equation is carried out through the fourth-order Runge–Kutta algorithm. Numerical simulations are made by asserting variables and constants solicited with a time step of ∆t=0.5.10−5s. During this process, the system is integrated for a satisfactorily long period, so that the transient phase is avoided for each set of parameters. Consequently, we started by examining the effect of the wind on the dynamics of the tree trunk by the means of the bifurcation diagram and its corresponding Lyapunov exponent (Figure 4).Figure 4
Bifurcation diagram (a) and Lyapunov exponent (b) versus the amplitude force ratio. The other parameters areξ=0.1, ω=1.35, and α=1. It appears a perfect correspondence between the two curves.
(a)(b)Figure4 highlights various routes to chaos for a particular set of parameters defined as ξ=0.1, ω=1.35, and α=1. These curves reveal that the behavior of the plant depends on its initial states when the values of the amplitude ratio F are chosen between 24 and 27, for given initial conditions. The same graphs allow seeing the dynamical behavior of the plant showing transition from periodic to chaotic states. This figure is very beneficial because it allows observing that when the value of the wind amplitude ratio changes continuously, the plant behavior may change in a discontinuous way. Consequently, understanding the impact of wind on the plant dynamics in particular and in forest in general is crucial for understanding several plant communities. Furthermore, the disruption caused by the wind can range from slight disturbances with local changes to a very large-scale tumult that initiates a whole change in the succession of the plant ecosystem.Next, we carry out the PSpice simulations of the network dynamics and observe its experimental behavior exploiting Figure3. The results of these investigations are superimposed in Figure 5. Note that to achieve our experiment, we exploit the PSpice version 9.2 for the implementation of equation (4).Figure 5
Different phase portraits of the system. (a) MATLAB curves obtained with parameters of Figure4, (b) PSpice results, and (c) the experimental simulations of the system. The curves of Figure 5 (c) are obtained from the circuit of Figure 2. These curves display the multiperiodic states (left hand side-lhs) and the chaotic behaviors (right hand side-rhs) of the system. The values of the parameters are defined as: ξ=0.1, ω=1.35, F=26, α=1R2=50kΩ, R1=R3=R4=R6=R7=10kΩ, R8=R9=R10=10kΩ,and R5=300Ω.
(a)(b)(c)Figure5 exhibits the different phase portraits for the quasiperiodic and chaotic behaviors of the system. The dynamical response of the plant is obtained when the amplitude ratio varies. Indeed, Figure 5(a) presents the numerical results obtained using the MATLAB version R2015a environment. These curves are followed by the PSpice behavior of the system (Figure 5(b)). We observe that the results established in both cases (Figures 5(a) and 5(b)) are in perfect agreement with the experimental analyses (Figure 5(c)).


## 2.1. Mechanical Representation and Characteristic Equation

The model used in this paper is a tree trunk of massm swaying in the air with a stiffness constant k and the damping d due to the internal viscoelastic of the wood, the roots, and the soil mass as shown in Figure 1:Figure 1
Dynamical model of a tree trunk: (a) structural representation and (b) mechanical equivalent model.
(a)(b)In this structural illustration, the plant is considered an isolated system subjected to an external loading called wind. This approach, which is the most used, allows analyzing the different behaviors of the plant and appreciating the effects of the different wind parameters on the dynamics of the plant. The model of Figure1 was firstly used to demonstrate its applicability for prediction of the precritical [41] and postcritical [42] behaviors. The possibility that the behavior of the plant displays certain aspects strikes many people, whether or not they are plant specialists. The notion of cognition seems appropriate even if it has not yet been extended to plants because researchers have established that plant systems deserve recognition of the intelligence developed in their behavior [43]. It was also exploited to describe the complex dynamics of various complex chemical, biological, and physical systems [44–46]. Moreover, a number of searches revealed that this model is very useful to detect a feeble complex signal against a background of sturdy noise [47]. Hereafter, the structural representation in Figure 1(a) deals with the natural structure of some trees [48]. However, the complex motion of the tree swaying in the air flow is often simplified using a conceptual model of a tree stem (without branches) and being considered as an upside-down pendulum. Within this concept, Miller [4] uses the structural model of Figure 1(a) and models the tree as a Duffing oscillator with an additional cubic stiffness term bx3. Thus, the nonlinear differential equation that governs the dynamics of our model is put in the form calculated as follows:(1)mx¨t+dx˙t−kxt+bx3t=I0t,in which d, k, b, and I0t are, respectively, the damping coefficient, the linear stiffness, the nonlinear stiffness of the spring, and the effective wind force acting on the plant. The quantity xt designates the deflection of the stem. Equation (1) describes the electromechanical control of the beam dynamics in a state where the single mode representation of these dynamics is that of a Duffing oscillator [49] and modeled the ship rolling motions [19]. Consequently, the dimensionless form associated to equation (1) is(2)X¨T+2ξX˙T−XT+αX3T=FcosωT.In equation (2), the diverse parameters are defined as follows: η1=bLmax3, η=kLmax, F=β0/η,ω=ω1/ω0,X=x/Lmax,α=η1/η, and T=ω0t. Within these quantities, parameter Lmax represents the characteristic maximum deflection of the tree, ω0 stands for the natural frequency of pure oscillations of the trunk, while ω1 is the frequency of the air flow; therefore, ω defines the frequency ratio of the system. Parameter α is the nonlinear stiffness ratio in which η defines the stiffness of the spring, whereas η1 designates the stiffness coefficient of the structure. The factor ξ=d/2mω0 represents the damping ratio. The quantity F denotes the amplitude force ratio, wherein β0 is the forcing amplitude. For simulation and experimental purposes, equation (2) can also be rewritten in the form(3)X˙=Y,Y˙=X−2ξY−αX3+Fcosωτ.Let us remind that the occurrence of symmetry property in a given system is known as a moderating factor of the amount of work needed [50]. System (3) shows that the symmetry of the plant-wind system depends on the wind action. In fact, this system is symmetric if F=0 and remains asymmetric otherwise. The symmetry of the attractors of equation (3) was recently analyzed [51].


## 2.2. PSpice Representation of the System

To improve the understanding of some new physical phenomena dealing with the real experimental setup of the plant subjected to wind load, we build the analog simulator of our mechanical system described by equation (3) as illustrated on Figure 2. Such approach is not new in physics because electronic circuits have extraordinary fascinated researcher’s attention for simulating complex nonlinear chaotic systems. Firstly, various nonlinear electronic circuits are used to examine and understand the dynamics of several complex physical systems under real-time conditions [52]. Secondly, electronic circuits provide a very useful medium for both modeling and experimental studies of nonlinear chaotic systems [53]. In recent years, concerning the electronic circuit in chaotic system, Sambas et al. [54, 55] introduced the new 3D chaotic dynamical system with a peanut-shaped closed curve of equilibrium points and described a detailed dynamical analysis of the new chaotic model using bifurcation diagrams and entropy analysis. They showed the multistability and coexisting convergent attractors in the new nonlinear plant. In the same work, they utilized the properties of their chaotic system in designing a new cipher colour image mechanism. Their experimental results demonstrated the efficiency of the presented encryption mechanism whose outcomes suggest promising applications for our chaotic system in various cryptographic applications. Few years after, the same authors announced a novel three-dimensional chaotic system with line equilibrium and discussed its dynamical properties such as Lyapunov exponents, phase portraits, equilibrium points, bifurcation diagram, multistability, and coexisting attractors. They established new synchronization results based on integral sliding mode control which are also derived for the new chaotic system with line equilibrium. In addition to these ideas, an electronic circuit implementation of the new chaotic system with line equilibrium is reported, and a good qualitative agreement is exhibited between the MATLAB simulations of the theoretical model and the Multisim results. To conclude their work, they propose an image encryption algorithm based on pixel-level scrambling, bit-level scrambling, and pixel value diffusion. Their experimental results showed that the encryption algorithm not only shuffles the pixel positions of the image but also replaces the pixel values with different values, which can effectively resist various attacks such as brute force attack and differential attack.Figure 2
Circuit diagram of system (3) with the values of the components: R2=50kΩ, R1=R3=R4=R6=R7=10kΩ, R8=R9=R10=10kΩ, R5=300Ω, and TL084CN≡Vjj=1,2,3,4.In the present case, the choice of the analog electrical simulator could be justified by the availability of electronic devices and the pedagogic necessity of having an experimental platform for graduate students. It is also based on the exploration of new features for researchers and the possibility to repeat the same experience at will. The diagram of Figure2 is implemented using four operational amplifiers (TL084CN), two multipliers (AD633JN), ten resistors, and two capacitors. The specific oscillator with active RC realization instead of the inductors is appropriate for assuring the circuit implementation. This RC block also spares us from looking for the links between R, L, and C components.To establish that the voltagesX and Y across the capacitors C1 and C2 are separately fulfilled in the subsequent set of the first-order differential equations, we apply the Kirchhoff’s laws to Figure 2 and obtain the following equations:(4)dxdτ=x˙=1R1C1ω0y,dydτ=y˙=−1R2C2ω0y+1R3C2ω0x−1R4C2ω0x3+eR5C2ω0cosωω0τ,in which τ=ω0t and ω0=104rad/s. While comparing equations (3) and (4), the upcoming links between the PSpice and MATLAB parameters are established:(5)1R1C1ω0=1,1R3C2ω0=1,α=1R4C2ω0,ξ=12R2C2ω0,andF=eR5C2ω0,within equation (5), the first three relations help to determine, respectively, the values of the resistors R1, R3, and R4 for a fixed frequency ω0. Indeed, the rheostat is varied until each of these conditions is fulfilled, thus indicating the value of the resistance sought. By exploiting the same principle, the last two relations of equation (5) permit to evaluate the values of the resistances R2 and R5 for various fixed values of ξ and F. Here, PSpice simulations are conducted at the frequency 15.93Hz with the print step 0.95 and the final time 1s.


## 2.3. Experimental Setup of the System

Many scientists have established that an experimental investigation is commonly needed to supplement theoretical/numerical results in complex nonlinear systems for technological and engineering applications [56]. In agreement with the theoretical abovementioned results, the chaotic oscillator of Figure 1 can experience some very interesting new dynamical behaviors. Hereafter, we will check the accuracy of our theoretical results by performing experimental analyses of the behavior of the system. The experimental setup of our system is drawn in Figure 3.Figure 3
A snapshot of the experimental device.As a starting point, we softly adjust the different values of resistorsR2 and R5 which denote, respectively, the dissipation in the plant and the strength of the wind acting on this plant. The experimental results of the phase-space trajectories are obtained by plotting VC1,VC2 by using a dual trace oscilloscope in XY mode.


## 2.4. Transition to Chaos in the Plant-Wind System

Chaos theory investigates the behavior of dynamical systems which are very sensitive to initial conditions. Its applications are found in physics, sociology, meteorology, and engineering [35, 57]. The main purpose of this subsection is to ensure an exhaustive analysis of some new phenomena linked to equation (3), which governs the dynamics of the plant subjected to wind loads. Therefore, direct integration of this equation is carried out through the fourth-order Runge–Kutta algorithm. Numerical simulations are made by asserting variables and constants solicited with a time step of ∆t=0.5.10−5s. During this process, the system is integrated for a satisfactorily long period, so that the transient phase is avoided for each set of parameters. Consequently, we started by examining the effect of the wind on the dynamics of the tree trunk by the means of the bifurcation diagram and its corresponding Lyapunov exponent (Figure 4).Figure 4
Bifurcation diagram (a) and Lyapunov exponent (b) versus the amplitude force ratio. The other parameters areξ=0.1, ω=1.35, and α=1. It appears a perfect correspondence between the two curves.
(a)(b)Figure4 highlights various routes to chaos for a particular set of parameters defined as ξ=0.1, ω=1.35, and α=1. These curves reveal that the behavior of the plant depends on its initial states when the values of the amplitude ratio F are chosen between 24 and 27, for given initial conditions. The same graphs allow seeing the dynamical behavior of the plant showing transition from periodic to chaotic states. This figure is very beneficial because it allows observing that when the value of the wind amplitude ratio changes continuously, the plant behavior may change in a discontinuous way. Consequently, understanding the impact of wind on the plant dynamics in particular and in forest in general is crucial for understanding several plant communities. Furthermore, the disruption caused by the wind can range from slight disturbances with local changes to a very large-scale tumult that initiates a whole change in the succession of the plant ecosystem.Next, we carry out the PSpice simulations of the network dynamics and observe its experimental behavior exploiting Figure3. The results of these investigations are superimposed in Figure 5. Note that to achieve our experiment, we exploit the PSpice version 9.2 for the implementation of equation (4).Figure 5
Different phase portraits of the system. (a) MATLAB curves obtained with parameters of Figure4, (b) PSpice results, and (c) the experimental simulations of the system. The curves of Figure 5 (c) are obtained from the circuit of Figure 2. These curves display the multiperiodic states (left hand side-lhs) and the chaotic behaviors (right hand side-rhs) of the system. The values of the parameters are defined as: ξ=0.1, ω=1.35, F=26, α=1R2=50kΩ, R1=R3=R4=R6=R7=10kΩ, R8=R9=R10=10kΩ,and R5=300Ω.
(a)(b)(c)Figure5 exhibits the different phase portraits for the quasiperiodic and chaotic behaviors of the system. The dynamical response of the plant is obtained when the amplitude ratio varies. Indeed, Figure 5(a) presents the numerical results obtained using the MATLAB version R2015a environment. These curves are followed by the PSpice behavior of the system (Figure 5(b)). We observe that the results established in both cases (Figures 5(a) and 5(b)) are in perfect agreement with the experimental analyses (Figure 5(c)).


## 3. Bursting Phenomenon in the System

### 3.1. Generalities on Bursting Phenomenon in Dynamical System

Before checking its impact in the plant-wind interactions, let us stress that bursting is a nonlinear phenomenon initially used to describe a type of neuron motion in endocrine cells and biological neurons. Bursting phenomenon appears in all dynamical systems when the oscillations exchange between the stationary states and the repetitive spiking states [58, 59]. Bursting oscillations can also be viewed as a dynamic behavior which connects two states (spiking and quiescent). In this particular case, authors have shown that connections between these two states come from the fact that there exist two types of bifurcations in one bursting oscillation, precisely one that transforms the system from quiescent states to spiking states and other that transforms the same system from spiking states to quiescent states [60, 61]. In this phenomenon, when all state variables of a given system behave as large amplitude oscillations, then the system is in its spiking state. When these state variables manifest themselves as microamplitude oscillations, the system is in its quiescent state [62]. Consequently, the bursting patterns before depending on the patterns of the bifurcations that connect the two states depend firstly on the patterns of the quiescent states and spiking states. For the generation of bursting oscillations in a given complex system, special methods such as fast-slow burster analysis have been developed, but the system must incorporate both fast and slow processes [63, 64]. Let us keep in mind that the coupling effect of these fast/slow time scales generally results in a physical system that exhibits periodic motion characterized by a combination of relatively large amplitude and nearly harmonic small amplitude oscillations, which can be very dangerous for the system. For this purpose, other authors have shown that the bursting oscillation can be easily controlled by the amplitude and frequency of the slow excitation [65]. In the next subsections, we present our results concerning the dynamics of the plant under the effects of the wind.


### 3.2. Numerical Analyses of the Bursting Oscillations in the System

Previous research [51] has established that the higher the damping ratio, the less the plant oscillates under the action of the wind. Hereafter, the numerical integration of equation (3) is made through the fourth-order Runge–Kutta algorithm for different values of the damping ratio ξ, and the results are displayed in Figures 6–8. These curves show that the plant exhibits multiple forms of repetitive spiking oscillations separated by stationary states, reflecting the phenomenon of bursting when the value of certain parameters of the system changes.Figure 6
Periodic busting oscillations in the plant-wind system for parametersF=27, ξ=2.5, and α=0.675: (a) ω=0.025; (b) ω=0.05; (c) ω=0.1; (d) zooming of (a).
(a)(b)(c)(d)Figure 7
Quasiperiodic busting oscillations in the system forF=27, ξ=0.5, and α=0.675: (a) ω=0.025; (b) ω=0.05; (c) ω=0.1; (d) enlargement of (a).
(a)(b)(c)(d)Figure 8
Chaotic busting oscillations in the system for the parametersF=27, ξ=0.1, and α=0.675: (a) ω=0.025; (b) ω=0.05; (c) ω=0.1; (d) the zoom view of (a).
(a)(b)(c)(d)Indeed, Figure6 presents the periodic bursting oscillations of the plant under air flow for several values of the frequency ratio. It appears from Figure 6(d) that after each peak of excitation due to the wind load, the plant returns directly to its state of rest. These periodic bursting oscillations reflect the exchange between the stable state of the plant’s motion and the activity of the wind. They help strengthen the stable movement of the plant.Figure7 displays the quasiperiodic bursting oscillations of the plant-wind system for different values of the frequency ratio. In these graphs, Figure 7(d) shows that after each excitation peak due to the wind load, the plant returns to its stationary state after a few slight oscillations. This trend increases with the increasing frequency ratio. They help strengthen the stable movement of the plant since the appearance of these small oscillations subdivides the energy of the wind into small packets, having the merit of minimizing the effects of wind energy on the plant and consolidating its stability.Furthermore, Figures8 exhibits the chaotic bursting oscillations of the plant subjected to the wind load for various values of the frequency ratio. Within these plots, the curve of Figure 8(d) allows observing that after each peak of excitation linking with the activity of the wind activity, the plant returns to its stationary state after numerous oscillations of decreasing amplitudes. This tendency increases with the growth in wind frequency. This type of oscillation breaks the energy of the wind into decreasing value packets, and its occurrence has the merit of reducing the effects of wind energy on the plant, an action which ensures the stability of the plant even in chaotic cases. Such chaotic bursting was observed in a neuron cell.Figures6–8 exhibit the periodic, quasiperiodic, and chaotic oscillations of the plant under air flow. They show that the plant exhibits multiple forms of repetitive spiking oscillations separated by stationary states reflecting bursting phenomena when the value of certain parameters of the system changes. In each case (of Figures 6–8), we observe that when the frequency ratio doubles, the number of spikes of bursting oscillations in the dynamics of the plant increases twice, and the time interval between two adjacent peaks is reduced by almost half. This possibility of doubling the number of peaks by doubling the frequency of the wind makes it possible to transform a given movement of the plant into a succession of spikes separated by stationary oscillations. Whatever the nature of the oscillations (periodic, quasiperiodic, or chaotic), the initial motion of the plant is transformed into a set of peaks separated by stationary states, thereby reducing the initial energy that could have destroyed the plant in the chaotic case.Figures6–8 also allow noticing that for high values of the damping ratio (ξ), periodic and quasiperiodic bursting patterns can be observed in the dynamics of plant (Figures 6 and 7), while the chaotic bursting patterns can be created otherwise (Figure 8). In other words, the higher the damping ratio, the less the plant oscillates under the action of the wind. These results help observing that the bursting oscillations seen in autonomous dynamical systems could also appear in nonautonomous dynamical systems.All these curves exhibit the impact of the damping of the plant and the characteristics (amplitude and frequency) of the wind on the appearance of bursting oscillations in the plant dynamics. One could also note that the damping ratio of a given plant is responsible for the type of bursting oscillations appearing in its dynamics, while the wind frequency helps to control these bursting oscillations. The occurrence of these bursting oscillations in our system strengthens its stability or at least minimizes the effects of chaos on the dynamics of the plant.Owing to the complex nature of the bursting oscillations, we also examine the phenomenon of amplitude jump. In the plant-wind interaction, this phenomenon is traduced by the fact that the motion of the plant enters a relaxation regime for given values of the damping ratio. In other words, the system periodically switches from oscillations of large amplitudes to those of small amplitudes (Figure9), thus ensuring its stability.Figure 9
Numerical results of the amplitude jump in the plant-wind system for the parametersα=0.675 and ω=0.1. (a) Periodic state for F=25 and ξ=0.5. (b) Chaotic behavior for F=27 and ξ=0.1.
(a)(b)


### 3.3. Generation of Bursting Oscillations in the System: Roles of Wind Parameters

In this subsection, we examine the conditions for the appearance of bursting oscillations in the plant dynamics. To achieve this aim, the damping ratio and the frequency ratio of the system are exploited. Subsequently, forξ>0, our system admits three equilibrium points in which the equilibrium point P0 is a saddle, while P± are sinks. Accordingly, the plant-wind system under consideration is a bistable system, and its phase portraits around these equilibria are drawn in Figure 10 for ξ∈2;+∞.Figure 10
Bistability of equilibrium attractors in the plant-wind system. In each curve, there are two stable focuses separated by a saddle obtained forc=0 with (a) ξ=0.5 and (b) ξ=2.5.
(a)(b)It emerges from these curves that the two equilibriaP0 and P−, (resp. P+) meet and vanish by a fold bifurcation when c decreases (or increases, resp.) through the point −23α/9α (or 23α/9α, resp.). This condition allows P+ (P−, resp.) to be the only attractor of the plant-wind system in the phase plane. Thus, the system becomes stable by losing its bistable state.To complete our findings, we start by checking deeply the stability of the system by plotting the deflection of the plant with respect toc (Figure 11).Figure 11
Deflection of the plant with respect toc for different values of the damping ratio ξ showing the equilibrium points of the plant-wind system for parameters of Figure 7 with (a) ξ=0.5 and (b) ξ=2.5. In these curves, LP1 and LP2 are fold bifurcation points of equilibria with cLP1=−23α/9α and cLP2=23α/9α, respectively.
(a)(b)From the equilibrium points of equation (3), it appears there are two main cases dealing with the values of the displacement of the plant: For x=0, equation (3) is rewritten in the form fP0λ=λ2+2ξλ−1, and the mathematical solution helps us to conclude that P0 is an unstable equilibrium point, while for x=±α/α, it takes the form fP±λ=λ2+2ξλ+2,and an analog analysis shows that the system is stable around the points P±. Our findings also reveal that the system possesses stable focuses (SF±) for ξ∈0,2 dealing with eigenvalues having negative real parts (Figure 11(a)) and stable nodes (SN±) for ξ∈2;+∞ for that corresponds to solutions with negative eigenvalues (Figure 11(b)).Nevertheless, when the eigenvalues are real and of opposite signs, we have a saddle (Sa) which is always unstable. Therefore, the phase portrait is plotted in X−Y plane. From Figure 12, we first observe that to the right of cLP2, the plant-wind system is stable and the only attractor is SF+. Then, we notice that when external excitation 27cos0.025T increases through this bifurcation point, the orbit of equation (3) changes to SF+ by fold bifurcation LP2, and remains there. When the amplitude force ratio reaches the value F=27, the external excitation begins to decrease. Likewise, when this external forcing decreases through cLP1, the stable focus SF+ vanishes by means of the fold bifurcation LP1 and the orbit jumps towards the lower stable focus SF−. As shown in Figure 12(a), when the trajectory passes through the fold bifurcation LP2, the system undergoes a transition from the stable focus SF− to SF+. In a similar manner, when this trajectory passes through the bifurcation point LP1, the plant now jumps back to the stable focus SF−. Bursting oscillations are then created in the system since the plant periodically switches between the stable focuses SF+ and SF−.Figure 12
(a) Chaotic phase portrait inX−Y plane obtained with the parameters of Figure 8(a) together with the equilibrium points of the plant-wind system defined in Figure 11(a) and (b) transformed phase diagram of bursting trajectory for ξ=0.5.
(a)(b)In fact, the external amplitude ratio plays a significant role in the generation of bursting oscillations in the plant dynamics, a role that is not highlighted in the strategy developed above. To get a clear idea of this role, we construct the transformed phase diagram of the system (Figure12(b)). This diagram describes the process of evolution of equation (3) with respect to the external amplitude ratio, superimposed on the bifurcation diagram of the mathematical modeling of the plant dynamics in order to detect the associated bifurcation mechanisms. Finally, according to the slow modulation of the external amplitude ratio, the plant-wind system periodically switches amongst two stable focuses (SF±) and bursting oscillations are naturally created in the plant dynamics.When parameters of Figure6(a) are used, periodic bursting oscillations obtained in Figure 13 displayed a dynamic of the plant similar to the chaotic case described beyond. This means that the plant switches between two equilibrium attractors by the fold bifurcations which form these two bursting patterns. However, in the latter case (Figure 13) the attractors are nodes, while in the former case (Figure 12), the attractors are focuses.Figure 13
(a) Overlay of the phase portrait in the planeX−Y obtained for parameters of Figure 6(a) with the equilibrium points of the system defined in Figure 11(b) and (b) transformed phase diagram of bursting trajectory for ξ=2.5.
(a)(b)On the contrary, the phase portrait inX,Y plane for parameters of Figure 7(a), corresponding to the quasiperiodic bursting oscillations of the system is drawn in Figure 14(a).Figure 14
Phase portraits of the periodic (a) and chaotic (b) bursting oscillations of the system obtained in the MATLAB environment for parameters of Figures7(a) and 8(a), respectively.
(a)(b)Physically, the aforementioned results reveal that the plant behaves as a dynamical system that evolves at multiple scales over time. These results show that, during a given period of time, the stable state of a plant can pass very quickly from a periodic to a chaotic state. When this reaction time is very short, these transitions modify the properties of the plant, and its state of relaxation considered as fixed and stable suddenly turns into an unstable state, leading to breakage or uprooting of the plant. This is most often observed in plantations and forests during or after wind load and, therefore, reflects the bifurcations in the dynamic system considered.


### 3.4. PSpice Results on Bursting Phenomenon in the Plant Dynamics

Based on the electronic circuit of Figure2, PSpice simulations of the bursting phenomenon in the plant dynamics lead to the results depicted in Figure 15. These plots exhibit the time-history of the variable X (lhs curves) and the corresponding phase portraits in X,Y plane (rhs curves) for parameters of Figure 5.Figure 15
PSpice implementation of the time-history of the variableX (lhs) and phase portraits (rhs) for (a) the periodic and (b) chaotic bursting oscillations of the system obtained with parameters of Figure 5.
(a)(b)The obtained graphs are in perfect agreement with the MATLAB curves of Figure14 for the phase portraits and with those of Figures 7(a) and 8(a) for the time-history of the variable. These results show that, for different values of the frequency ratio, bursting oscillations are observed in the plant dynamics. These last results confirm the fact that a plant subjected to wind loads (whether large or not) could be very sensitive to variations in wind characteristics.


### 3.5. Experimental Observations of the Bursting Oscillations in the Plant Dynamics

The setup for the experimental simulations of the plant dynamics subjected to wind load is given in Figure3. It appears from the above numerical results that a plant exhibits various types of behaviors (periodic, quasiperiodic, and chaotic). To observe the complete experimental behavior of the plant dynamics under the wind load, the resistors R2 and R5 are gradually varied. Figure 16 exhibits the results, displaying the existence of periodic and chaotic bursting oscillations in the system and revealing the same scenarios as perceived using theoretical methods.Figure 16
Numerical time domain and phase portraits (lhs) and the corresponding experimental results obtained with a dual trace oscilloscope inX−Y mode (rhs). In the experimental setup, the quantities VX and VY are replaced by the capacitor voltages VC1 and VC2, respectively. The scales are x=2V/div and y=5V/div.
(a)(b)Nevertheless, between numerical and experimental results, an insignificant discrepancy may be allowed due to the accuracy of the values of the electronic components.At the end of this investigation on bursting phenomenon in the plant-wind system, some links can be made between the established results and the existing knowledge in other areas of science. Indeed, bursting oscillations were observed in neuronal dynamics [66, 67], and neurologists could solicit our system for analyzing neurons dynamics without doing so on living beings because it was shown that the human brain is an endlessly captivating organ (structural complexity, information processing, motor control, and memory formation). On the contrary, our study on bursting oscillations can allow making analyses on heart behavior without doing it on humans because Corrigan et al. [68] showed that plant displacement can be interpreted as synapses motion and the introduction of hybrid plant in technology can allow the implementation of supervised and unsupervised machine learning algorithms that make use of the feature of synapses. They have also demonstrated that synapses play an important role in biology, as, for instance, in the case of plant signaling.


## 3.1. Generalities on Bursting Phenomenon in Dynamical System

Before checking its impact in the plant-wind interactions, let us stress that bursting is a nonlinear phenomenon initially used to describe a type of neuron motion in endocrine cells and biological neurons. Bursting phenomenon appears in all dynamical systems when the oscillations exchange between the stationary states and the repetitive spiking states [58, 59]. Bursting oscillations can also be viewed as a dynamic behavior which connects two states (spiking and quiescent). In this particular case, authors have shown that connections between these two states come from the fact that there exist two types of bifurcations in one bursting oscillation, precisely one that transforms the system from quiescent states to spiking states and other that transforms the same system from spiking states to quiescent states [60, 61]. In this phenomenon, when all state variables of a given system behave as large amplitude oscillations, then the system is in its spiking state. When these state variables manifest themselves as microamplitude oscillations, the system is in its quiescent state [62]. Consequently, the bursting patterns before depending on the patterns of the bifurcations that connect the two states depend firstly on the patterns of the quiescent states and spiking states. For the generation of bursting oscillations in a given complex system, special methods such as fast-slow burster analysis have been developed, but the system must incorporate both fast and slow processes [63, 64]. Let us keep in mind that the coupling effect of these fast/slow time scales generally results in a physical system that exhibits periodic motion characterized by a combination of relatively large amplitude and nearly harmonic small amplitude oscillations, which can be very dangerous for the system. For this purpose, other authors have shown that the bursting oscillation can be easily controlled by the amplitude and frequency of the slow excitation [65]. In the next subsections, we present our results concerning the dynamics of the plant under the effects of the wind.


## 3.2. Numerical Analyses of the Bursting Oscillations in the System

Previous research [51] has established that the higher the damping ratio, the less the plant oscillates under the action of the wind. Hereafter, the numerical integration of equation (3) is made through the fourth-order Runge–Kutta algorithm for different values of the damping ratio ξ, and the results are displayed in Figures 6–8. These curves show that the plant exhibits multiple forms of repetitive spiking oscillations separated by stationary states, reflecting the phenomenon of bursting when the value of certain parameters of the system changes.Figure 6
Periodic busting oscillations in the plant-wind system for parametersF=27, ξ=2.5, and α=0.675: (a) ω=0.025; (b) ω=0.05; (c) ω=0.1; (d) zooming of (a).
(a)(b)(c)(d)Figure 7
Quasiperiodic busting oscillations in the system forF=27, ξ=0.5, and α=0.675: (a) ω=0.025; (b) ω=0.05; (c) ω=0.1; (d) enlargement of (a).
(a)(b)(c)(d)Figure 8
Chaotic busting oscillations in the system for the parametersF=27, ξ=0.1, and α=0.675: (a) ω=0.025; (b) ω=0.05; (c) ω=0.1; (d) the zoom view of (a).
(a)(b)(c)(d)Indeed, Figure6 presents the periodic bursting oscillations of the plant under air flow for several values of the frequency ratio. It appears from Figure 6(d) that after each peak of excitation due to the wind load, the plant returns directly to its state of rest. These periodic bursting oscillations reflect the exchange between the stable state of the plant’s motion and the activity of the wind. They help strengthen the stable movement of the plant.Figure7 displays the quasiperiodic bursting oscillations of the plant-wind system for different values of the frequency ratio. In these graphs, Figure 7(d) shows that after each excitation peak due to the wind load, the plant returns to its stationary state after a few slight oscillations. This trend increases with the increasing frequency ratio. They help strengthen the stable movement of the plant since the appearance of these small oscillations subdivides the energy of the wind into small packets, having the merit of minimizing the effects of wind energy on the plant and consolidating its stability.Furthermore, Figures8 exhibits the chaotic bursting oscillations of the plant subjected to the wind load for various values of the frequency ratio. Within these plots, the curve of Figure 8(d) allows observing that after each peak of excitation linking with the activity of the wind activity, the plant returns to its stationary state after numerous oscillations of decreasing amplitudes. This tendency increases with the growth in wind frequency. This type of oscillation breaks the energy of the wind into decreasing value packets, and its occurrence has the merit of reducing the effects of wind energy on the plant, an action which ensures the stability of the plant even in chaotic cases. Such chaotic bursting was observed in a neuron cell.Figures6–8 exhibit the periodic, quasiperiodic, and chaotic oscillations of the plant under air flow. They show that the plant exhibits multiple forms of repetitive spiking oscillations separated by stationary states reflecting bursting phenomena when the value of certain parameters of the system changes. In each case (of Figures 6–8), we observe that when the frequency ratio doubles, the number of spikes of bursting oscillations in the dynamics of the plant increases twice, and the time interval between two adjacent peaks is reduced by almost half. This possibility of doubling the number of peaks by doubling the frequency of the wind makes it possible to transform a given movement of the plant into a succession of spikes separated by stationary oscillations. Whatever the nature of the oscillations (periodic, quasiperiodic, or chaotic), the initial motion of the plant is transformed into a set of peaks separated by stationary states, thereby reducing the initial energy that could have destroyed the plant in the chaotic case.Figures6–8 also allow noticing that for high values of the damping ratio (ξ), periodic and quasiperiodic bursting patterns can be observed in the dynamics of plant (Figures 6 and 7), while the chaotic bursting patterns can be created otherwise (Figure 8). In other words, the higher the damping ratio, the less the plant oscillates under the action of the wind. These results help observing that the bursting oscillations seen in autonomous dynamical systems could also appear in nonautonomous dynamical systems.All these curves exhibit the impact of the damping of the plant and the characteristics (amplitude and frequency) of the wind on the appearance of bursting oscillations in the plant dynamics. One could also note that the damping ratio of a given plant is responsible for the type of bursting oscillations appearing in its dynamics, while the wind frequency helps to control these bursting oscillations. The occurrence of these bursting oscillations in our system strengthens its stability or at least minimizes the effects of chaos on the dynamics of the plant.Owing to the complex nature of the bursting oscillations, we also examine the phenomenon of amplitude jump. In the plant-wind interaction, this phenomenon is traduced by the fact that the motion of the plant enters a relaxation regime for given values of the damping ratio. In other words, the system periodically switches from oscillations of large amplitudes to those of small amplitudes (Figure9), thus ensuring its stability.Figure 9
Numerical results of the amplitude jump in the plant-wind system for the parametersα=0.675 and ω=0.1. (a) Periodic state for F=25 and ξ=0.5. (b) Chaotic behavior for F=27 and ξ=0.1.
(a)(b)


## 3.3. Generation of Bursting Oscillations in the System: Roles of Wind Parameters

In this subsection, we examine the conditions for the appearance of bursting oscillations in the plant dynamics. To achieve this aim, the damping ratio and the frequency ratio of the system are exploited. Subsequently, forξ>0, our system admits three equilibrium points in which the equilibrium point P0 is a saddle, while P± are sinks. Accordingly, the plant-wind system under consideration is a bistable system, and its phase portraits around these equilibria are drawn in Figure 10 for ξ∈2;+∞.Figure 10
Bistability of equilibrium attractors in the plant-wind system. In each curve, there are two stable focuses separated by a saddle obtained forc=0 with (a) ξ=0.5 and (b) ξ=2.5.
(a)(b)It emerges from these curves that the two equilibriaP0 and P−, (resp. P+) meet and vanish by a fold bifurcation when c decreases (or increases, resp.) through the point −23α/9α (or 23α/9α, resp.). This condition allows P+ (P−, resp.) to be the only attractor of the plant-wind system in the phase plane. Thus, the system becomes stable by losing its bistable state.To complete our findings, we start by checking deeply the stability of the system by plotting the deflection of the plant with respect toc (Figure 11).Figure 11
Deflection of the plant with respect toc for different values of the damping ratio ξ showing the equilibrium points of the plant-wind system for parameters of Figure 7 with (a) ξ=0.5 and (b) ξ=2.5. In these curves, LP1 and LP2 are fold bifurcation points of equilibria with cLP1=−23α/9α and cLP2=23α/9α, respectively.
(a)(b)From the equilibrium points of equation (3), it appears there are two main cases dealing with the values of the displacement of the plant: For x=0, equation (3) is rewritten in the form fP0λ=λ2+2ξλ−1, and the mathematical solution helps us to conclude that P0 is an unstable equilibrium point, while for x=±α/α, it takes the form fP±λ=λ2+2ξλ+2,and an analog analysis shows that the system is stable around the points P±. Our findings also reveal that the system possesses stable focuses (SF±) for ξ∈0,2 dealing with eigenvalues having negative real parts (Figure 11(a)) and stable nodes (SN±) for ξ∈2;+∞ for that corresponds to solutions with negative eigenvalues (Figure 11(b)).Nevertheless, when the eigenvalues are real and of opposite signs, we have a saddle (Sa) which is always unstable. Therefore, the phase portrait is plotted in X−Y plane. From Figure 12, we first observe that to the right of cLP2, the plant-wind system is stable and the only attractor is SF+. Then, we notice that when external excitation 27cos0.025T increases through this bifurcation point, the orbit of equation (3) changes to SF+ by fold bifurcation LP2, and remains there. When the amplitude force ratio reaches the value F=27, the external excitation begins to decrease. Likewise, when this external forcing decreases through cLP1, the stable focus SF+ vanishes by means of the fold bifurcation LP1 and the orbit jumps towards the lower stable focus SF−. As shown in Figure 12(a), when the trajectory passes through the fold bifurcation LP2, the system undergoes a transition from the stable focus SF− to SF+. In a similar manner, when this trajectory passes through the bifurcation point LP1, the plant now jumps back to the stable focus SF−. Bursting oscillations are then created in the system since the plant periodically switches between the stable focuses SF+ and SF−.Figure 12
(a) Chaotic phase portrait inX−Y plane obtained with the parameters of Figure 8(a) together with the equilibrium points of the plant-wind system defined in Figure 11(a) and (b) transformed phase diagram of bursting trajectory for ξ=0.5.
(a)(b)In fact, the external amplitude ratio plays a significant role in the generation of bursting oscillations in the plant dynamics, a role that is not highlighted in the strategy developed above. To get a clear idea of this role, we construct the transformed phase diagram of the system (Figure12(b)). This diagram describes the process of evolution of equation (3) with respect to the external amplitude ratio, superimposed on the bifurcation diagram of the mathematical modeling of the plant dynamics in order to detect the associated bifurcation mechanisms. Finally, according to the slow modulation of the external amplitude ratio, the plant-wind system periodically switches amongst two stable focuses (SF±) and bursting oscillations are naturally created in the plant dynamics.When parameters of Figure6(a) are used, periodic bursting oscillations obtained in Figure 13 displayed a dynamic of the plant similar to the chaotic case described beyond. This means that the plant switches between two equilibrium attractors by the fold bifurcations which form these two bursting patterns. However, in the latter case (Figure 13) the attractors are nodes, while in the former case (Figure 12), the attractors are focuses.Figure 13
(a) Overlay of the phase portrait in the planeX−Y obtained for parameters of Figure 6(a) with the equilibrium points of the system defined in Figure 11(b) and (b) transformed phase diagram of bursting trajectory for ξ=2.5.
(a)(b)On the contrary, the phase portrait inX,Y plane for parameters of Figure 7(a), corresponding to the quasiperiodic bursting oscillations of the system is drawn in Figure 14(a).Figure 14
Phase portraits of the periodic (a) and chaotic (b) bursting oscillations of the system obtained in the MATLAB environment for parameters of Figures7(a) and 8(a), respectively.
(a)(b)Physically, the aforementioned results reveal that the plant behaves as a dynamical system that evolves at multiple scales over time. These results show that, during a given period of time, the stable state of a plant can pass very quickly from a periodic to a chaotic state. When this reaction time is very short, these transitions modify the properties of the plant, and its state of relaxation considered as fixed and stable suddenly turns into an unstable state, leading to breakage or uprooting of the plant. This is most often observed in plantations and forests during or after wind load and, therefore, reflects the bifurcations in the dynamic system considered.


## 3.4. PSpice Results on Bursting Phenomenon in the Plant Dynamics

Based on the electronic circuit of Figure2, PSpice simulations of the bursting phenomenon in the plant dynamics lead to the results depicted in Figure 15. These plots exhibit the time-history of the variable X (lhs curves) and the corresponding phase portraits in X,Y plane (rhs curves) for parameters of Figure 5.Figure 15
PSpice implementation of the time-history of the variableX (lhs) and phase portraits (rhs) for (a) the periodic and (b) chaotic bursting oscillations of the system obtained with parameters of Figure 5.
(a)(b)The obtained graphs are in perfect agreement with the MATLAB curves of Figure14 for the phase portraits and with those of Figures 7(a) and 8(a) for the time-history of the variable. These results show that, for different values of the frequency ratio, bursting oscillations are observed in the plant dynamics. These last results confirm the fact that a plant subjected to wind loads (whether large or not) could be very sensitive to variations in wind characteristics.


## 3.5. Experimental Observations of the Bursting Oscillations in the Plant Dynamics

The setup for the experimental simulations of the plant dynamics subjected to wind load is given in Figure3. It appears from the above numerical results that a plant exhibits various types of behaviors (periodic, quasiperiodic, and chaotic). To observe the complete experimental behavior of the plant dynamics under the wind load, the resistors R2 and R5 are gradually varied. Figure 16 exhibits the results, displaying the existence of periodic and chaotic bursting oscillations in the system and revealing the same scenarios as perceived using theoretical methods.Figure 16
Numerical time domain and phase portraits (lhs) and the corresponding experimental results obtained with a dual trace oscilloscope inX−Y mode (rhs). In the experimental setup, the quantities VX and VY are replaced by the capacitor voltages VC1 and VC2, respectively. The scales are x=2V/div and y=5V/div.
(a)(b)Nevertheless, between numerical and experimental results, an insignificant discrepancy may be allowed due to the accuracy of the values of the electronic components.At the end of this investigation on bursting phenomenon in the plant-wind system, some links can be made between the established results and the existing knowledge in other areas of science. Indeed, bursting oscillations were observed in neuronal dynamics [66, 67], and neurologists could solicit our system for analyzing neurons dynamics without doing so on living beings because it was shown that the human brain is an endlessly captivating organ (structural complexity, information processing, motor control, and memory formation). On the contrary, our study on bursting oscillations can allow making analyses on heart behavior without doing it on humans because Corrigan et al. [68] showed that plant displacement can be interpreted as synapses motion and the introduction of hybrid plant in technology can allow the implementation of supervised and unsupervised machine learning algorithms that make use of the feature of synapses. They have also demonstrated that synapses play an important role in biology, as, for instance, in the case of plant signaling.


## 4. Phase Control Strategy in the Plant Dynamics

### 4.1. Presentation of the Method

From an environmental and economical point of view, to increase the plant protection, minimizing the hazards of wind damage is required. The first objective of chaos phase-control techniques is to separate the different behaviors of the system (periodic from chaotic) by using weak harmonic perturbations. The control method called for in this work is a method taken from recent literature on the optimal control of physical systems. It is based on the control of system-specific parameters influenced by small external disturbances. Therefore, the method does not link on a control law from the states of the system, but simply allows examining the consequences of external disturbances on parameters of the system during its evolution [69, 70]. Since chaotic behaviors are not desired in the plant dynamics, this technique seems to be an excellent tool for selecting the appropriate perturbations to reach preferred states. Because the effects of additive perturbations differ from one state to another, we characterize different types of controlling perturbations in an analog implementation of a plant-wind system, allowing the opportunity to differentiate stable and unstable states. However, we note that the numerical study based on the isospike technique permits efficient discovery and separation between periodic and chaotic orbits visited during the temporal evolution as explained in [68, 69]. Based on these works, we exploit the idea developed by Bi et al. [49] who consider a nonautonomous system, show the comprehensive bifurcations associated with bursting, and provide a complete theoretical classification of the dynamical bifurcation analysis. A more physically illuminating approach comes from multiscale analysis, where a dynamical system model is decomposed into elastic and dissipative terms. The phase control mechanism in this model is not well understood. In both cases, one has the elastic term in the form ft=1+ε0cosωt+φ and the dissipative term defined by ft=ξ0+ξ1cosωt+φ.


### 4.2. Application to the Plant Dynamics

In this subsection, we use numerical simulations to propose a method of controlling the dynamical behavior of a plant subjected to wind load. To achieve our goal, we reconstruct numerically the stability phases predicted by equation (3). The method consists of using a real-time analog circuit to construct the curves that can discriminate periodic behaviors from chaos. Thus, we plot the phase of the system according to the stability spaces of the perturbation strength when periodic perturbations are applied to the terms governing the oscillator (plant). We intend to verify and show that our system is more sensitive to external disturbances applied to it by separating periodic from chaotic behaviors.To achieve this aim, we construct the two dimensional Lyapunov stability diagrams linked to the plant-wind system. They are based on plotting the number of local maxima per period for the periodic oscillations. Thus, to produce such diagrams, we ought to cover with a mesh ofP×P equidistant points, a parameter window of interest. Moreover, through 103 equal steps h=0.05, the temporal evolution for each point is obtained by numerically integrating the oscillators using the standard fourth-order Runge–Kutta algorithm. Results are displayed in Figure 17, where from x,y,t=2,1,3, parameters are horizontally scanned from left to right, respectively. When determining the Lyapunov exponents, it became easier to compute the number of local extrema per period. For this purpose, we continue with the integration of an additional 103 time steps for recording up to 500 minima and maxima.Figure 17
Phase-control strategy predicted by equation (3) for parameters F =27 , ω=1.44, and φ=2πt/Tsω. Lyapunov stability diagrams in ε0,φ and ξ0,φ planes, respectively, for a control applied to the elastic term (a, b) and the dissipative term (c, d). The left panel (Figures 17 (a) and 17(c)) consists of the results before the phase control, while the right panel (Figures 17(b) and 17(d)) exhibits the corresponding results after applying the chaos phase control strategy. We can observe that only a correct choice of the phase makes it possible to ensure chaos control for all values of other parameters.
(a)(b)(c)(d)The aim of this technique is to check whether pulses are repeated in the system or not. The maxima are represented using a palette of many colors. The left panel (Figures17(a) and 17(c)) consists of the results before the phase control, while the right panel (Figures 17(b) and 17(d)) exhibits the corresponding results after applying the chaos phase control strategy. Moreover, the curves in Figures 17(a) and 17(b) show the standard Lyapunov stability diagram for the plant with parametric perturbation applied to the elastic term in ε0,φ plane. In those diagrams, gray shadings represent the negative exponents (periodic oscillations), whereas colors are the positive exponents (chaos). From these plots, one can observe that periodic solutions begin in regular parabolic structures with regions in which the phase is fruitless up to ε0 around 0.4 and 0.7, respectively. Consequently, for larger values of ε0, up to 1, the organization of chaotic or periodic solutions becomes more and more complex owed to parametric perturbation. Figure 17(b) displays fine structures, where periodic solutions are entrained for small ranges (0.1) of the phase. On the contrary, it appears from Figure 17(d) that, when perturbations are applied to the dissipative term, the periodic solutions appear in the larger domains (more than 0.2), but their value is greater with respect to parametric values.According to the phase control diagram of Figure17, one could note that various chaotic oscillations have disappeared after applying the chaos phase control strategy (Figures 17(b) and 17(d)) in the system compared to the uncontrolled case (Figures 17(a) and 17(c)). In this work, phase control strategy is view as an optimal strategy to suppress nonautonomous chaos in the plant dynamics. The results of Figure 17 allow showing that a correct choice of the phase minimizes the effects of the wind and, thus, eliminates chaotic behavior in the dynamics of the plant, by simply leading it towards a variety of periodic orbits. This work shows that the use of the chaos phase control strategy is not only reserved for electronic systems but can be extended to all dynamical systems. It allows us to characterize the dynamical behavior of the said system according to a pair of parameters.


## 4.1. Presentation of the Method

From an environmental and economical point of view, to increase the plant protection, minimizing the hazards of wind damage is required. The first objective of chaos phase-control techniques is to separate the different behaviors of the system (periodic from chaotic) by using weak harmonic perturbations. The control method called for in this work is a method taken from recent literature on the optimal control of physical systems. It is based on the control of system-specific parameters influenced by small external disturbances. Therefore, the method does not link on a control law from the states of the system, but simply allows examining the consequences of external disturbances on parameters of the system during its evolution [69, 70]. Since chaotic behaviors are not desired in the plant dynamics, this technique seems to be an excellent tool for selecting the appropriate perturbations to reach preferred states. Because the effects of additive perturbations differ from one state to another, we characterize different types of controlling perturbations in an analog implementation of a plant-wind system, allowing the opportunity to differentiate stable and unstable states. However, we note that the numerical study based on the isospike technique permits efficient discovery and separation between periodic and chaotic orbits visited during the temporal evolution as explained in [68, 69]. Based on these works, we exploit the idea developed by Bi et al. [49] who consider a nonautonomous system, show the comprehensive bifurcations associated with bursting, and provide a complete theoretical classification of the dynamical bifurcation analysis. A more physically illuminating approach comes from multiscale analysis, where a dynamical system model is decomposed into elastic and dissipative terms. The phase control mechanism in this model is not well understood. In both cases, one has the elastic term in the form ft=1+ε0cosωt+φ and the dissipative term defined by ft=ξ0+ξ1cosωt+φ.


## 4.2. Application to the Plant Dynamics

In this subsection, we use numerical simulations to propose a method of controlling the dynamical behavior of a plant subjected to wind load. To achieve our goal, we reconstruct numerically the stability phases predicted by equation (3). The method consists of using a real-time analog circuit to construct the curves that can discriminate periodic behaviors from chaos. Thus, we plot the phase of the system according to the stability spaces of the perturbation strength when periodic perturbations are applied to the terms governing the oscillator (plant). We intend to verify and show that our system is more sensitive to external disturbances applied to it by separating periodic from chaotic behaviors.To achieve this aim, we construct the two dimensional Lyapunov stability diagrams linked to the plant-wind system. They are based on plotting the number of local maxima per period for the periodic oscillations. Thus, to produce such diagrams, we ought to cover with a mesh ofP×P equidistant points, a parameter window of interest. Moreover, through 103 equal steps h=0.05, the temporal evolution for each point is obtained by numerically integrating the oscillators using the standard fourth-order Runge–Kutta algorithm. Results are displayed in Figure 17, where from x,y,t=2,1,3, parameters are horizontally scanned from left to right, respectively. When determining the Lyapunov exponents, it became easier to compute the number of local extrema per period. For this purpose, we continue with the integration of an additional 103 time steps for recording up to 500 minima and maxima.Figure 17
Phase-control strategy predicted by equation (3) for parameters F =27 , ω=1.44, and φ=2πt/Tsω. Lyapunov stability diagrams in ε0,φ and ξ0,φ planes, respectively, for a control applied to the elastic term (a, b) and the dissipative term (c, d). The left panel (Figures 17 (a) and 17(c)) consists of the results before the phase control, while the right panel (Figures 17(b) and 17(d)) exhibits the corresponding results after applying the chaos phase control strategy. We can observe that only a correct choice of the phase makes it possible to ensure chaos control for all values of other parameters.
(a)(b)(c)(d)The aim of this technique is to check whether pulses are repeated in the system or not. The maxima are represented using a palette of many colors. The left panel (Figures17(a) and 17(c)) consists of the results before the phase control, while the right panel (Figures 17(b) and 17(d)) exhibits the corresponding results after applying the chaos phase control strategy. Moreover, the curves in Figures 17(a) and 17(b) show the standard Lyapunov stability diagram for the plant with parametric perturbation applied to the elastic term in ε0,φ plane. In those diagrams, gray shadings represent the negative exponents (periodic oscillations), whereas colors are the positive exponents (chaos). From these plots, one can observe that periodic solutions begin in regular parabolic structures with regions in which the phase is fruitless up to ε0 around 0.4 and 0.7, respectively. Consequently, for larger values of ε0, up to 1, the organization of chaotic or periodic solutions becomes more and more complex owed to parametric perturbation. Figure 17(b) displays fine structures, where periodic solutions are entrained for small ranges (0.1) of the phase. On the contrary, it appears from Figure 17(d) that, when perturbations are applied to the dissipative term, the periodic solutions appear in the larger domains (more than 0.2), but their value is greater with respect to parametric values.According to the phase control diagram of Figure17, one could note that various chaotic oscillations have disappeared after applying the chaos phase control strategy (Figures 17(b) and 17(d)) in the system compared to the uncontrolled case (Figures 17(a) and 17(c)). In this work, phase control strategy is view as an optimal strategy to suppress nonautonomous chaos in the plant dynamics. The results of Figure 17 allow showing that a correct choice of the phase minimizes the effects of the wind and, thus, eliminates chaotic behavior in the dynamics of the plant, by simply leading it towards a variety of periodic orbits. This work shows that the use of the chaos phase control strategy is not only reserved for electronic systems but can be extended to all dynamical systems. It allows us to characterize the dynamical behavior of the said system according to a pair of parameters.


## 5. Conclusion

In this paper, we have formulated a new mathematical model to investigate some new features of a Pinus family plant subjected to wind load. To achieve our goal, we have constructed an electrical analog simulator with an activeRC realization that mimic the system motion. We have showed that the periodic or chaotic behavior of the system strongly depends on the initial conditions. It has appeared that when the amplitude force ratio changes continuously, the plant behavior may change in a discontinuous way. Our findings have revealed that plant exhibits several shapes of repetitive spiking oscillations linking with the magnitude of the damping ratio and reflecting the phenomenon of bursting. Furthermore, we have established that the occurrence of such oscillations in the system allows its chaotic behavior to be less robust. Furthermore, the existence of the chaotic bursting oscillations in the plant dynamics has been shown for the first time. It was also demonstrated that the time interval between two adjacent spikes decreases as the forcing frequency increases. Experimental investigations of this phenomenon have led to results that are in perfect agreement with the numerical outcomes established in the MATLAB and PSpice environments. On the contrary, we have applied the chaos phase control strategy based on the control of the system’s parameters influenced by small disturbances external to the system and observed that various chaotic oscillations disappear compared to the uncontrolled case. In addition, we have showed that a correct choice of the phase minimizes the wind effects and, thus, eliminates chaotic behavior of the plant by simply driving it towards a variety of periodic orbits.



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*Source: 3206434-2023-02-17.xml*