# Chaotic Dynamics of an Airfoil with Higher-Order Plunge and Pitch Stiffnesses in Incompressible Flow

Authors: Karthikeyan Rajagopal; Yesgat Admassu; Riessom Weldegiorgis; Prakash Duraisamy; Anitha Karthikeyan
Journal: Complexity
 (2019)

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## Abstract

Dynamical properties of a two-dimensional airfoil model with higher-order strong nonlinearities are investigated. Firstly, a state-space model is derived considering the plunge and pitch stiffnesses as generalized functions. Then, a stiffness function having square, cubic, and fifth-power nonlinearities is considered for both plunging and pitching stiffnesses, and the dimensionless state equations are derived. Various dynamical properties of the proposed model are investigated using equilibrium points, eigenvalues, and Lyapunov exponents. To further analyze the dynamical behavior of the system, bifurcation plots are derived. It is interesting to note that the new airfoil model with higher-order nonlinearities shows multistability with changing airspeed, and there are infinitely countable number of coexisting attractors generally called as megastability. Both multistability and megastability features of the airfoil model were not captured earlier in the literatures. To be clear, it is the first time a megastable feature is exposed in a physical system. Finally, to analyze the multifrequency effects of the airfoil model, we have presented the bicoherence plots.


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## Body

## 1. Introduction

Many literatures have shown that the airfoil (aeroelastic) systems show more complex dynamical behaviors such as limit cycles and chaotic oscillations [1–4]. A persistent flutter in an aeroelastic structure such as an aircraft wing may create dangerous effects to the structure and may cause structural unstability [1, 2]. Hence, controlling such unwanted and persistent oscillations has attracted importance among researchers [2–4]. A dynamical model of an airfoil system with cubic nonlinearity considered for the pitching stiffness was proposed in [5, 6], and it is shown that the system exhibits chaotic oscillations when the airspeed crosses a critical limit. A rigid wing supported by a nonlinear spring shows limit cycles as discussed in [7]. The authors investigated piecewise nonlinearities in aeroelastic systems, and the authors address continuous nonlinearities such as those found in structural systems that exhibit spring hardening or softening effects.A nonlinear active control method is adopted to control the limit cycle oscillations of an aeroelastic system with quasi-steady aerodynamic models [8, 9]. However, the results are limited to the elevation condition, and the real case should also be considered the actual vibration state and hence, the dynamic state must be set within an internal dynamic state when the nonlinear controller is designed [10]. In [11], a two-dimensional airfoil system with pitch and plunge stiffnesses using subsonic aerodynamics theory and classical nonlinearities, namely, cubic, freeplay, and hysteresis is investigated. Several cases of aerodynamic nonlinearities arising from transonic flow and dynamic stall are discussed, and numerical simulations are conducted. Poincaré mapping method and Floquet theory are adopted to analyze the limit cycle oscillation flutter and chaotic motion of a two-dimensional airfoil system with combined freeplay and cubic pitch stiffnesses in supersonic and hypersonic flows [12]. It is shown that the Floquet theory can effectively predict the occurrence of limit cycles in the system.In [13, 14], the aeroelastic airfoil system with freeplay is investigated and the transonic flow characteristics are discussed. The effect of hardening and nonlinearities on aeroelastic system is analyzed in [15]. The chaotic behavior and prediction of it with various methods and its robustness are presented in [16]. The comparative study reveals the effectiveness of Runge–Kutta method over other methods. A two-degree-of-freedom model of airfoil system is derived, and analysis is carried out to study the consequences of cubic nonlinearities [17]. Drastic changes are observed while the system entered to supersonic flow. Using precise integration method the nonlinear effect on airfoil system is simulated in [18]. The results show the presence of intricate behaviors of the system. The investigation on limit cycle oscillation and other aeroelastic responses is described in [19, 20] for system with freeplay in pitch. Challenges and complications during control and design of vibration absorber are discussed elaborately for the aeroelastic model with nonlinearities in [21, 22].An airfoil model with multiple strong nonlinearities for both pitch and plunge stiffnesses was studied, and incremental harmonic balance method was used to analyze the periodic state of the airfoil flutter [26]. Similarly, to analyze such periodic oscillations in an airfoil system, Monte Carlo method was adopted in [27]. A nonlinear adaptive control technique is used to suppress the flutter and limit cycle oscillations assuming that one state is known and the other states are compensated [28]. A terminal sliding-mode control technique is used to suppress the limit cycle oscillations with an exclusive choice between the plunge displacement and the pitch angle [29]. Differential transformation method (DTM) to examine the nonlinear dynamic response of a typical aeroelastic system with cubic nonlinearities for pitch stiffness under realistic operating parameters was proposed in [30], and the dynamical properties are investigated with bifurcation plots and Lyapunov spectrum. A nonlinear sliding-mode controller was designed to suppress the chaotic oscillations of an airfoil system proposed in [10], and the stability of the controllers was derived using the Lyapunov stability theorem [31]. A nonlinear energy sink (NES) is used to suppress the aeroelasticity of an airfoil with a control surface considering the freeplay and cubic stiffnesses in pitch. The harmonic balance method is used to determine the limit cycle oscillations occurring in the airfoil-NES system [32].In [29], the authors mentioned that a constant deterioration of wing structure influences on stiffness behavior, which demands higher-order nonlinearity in the dynamic model. In [33], influence of higher-order stiffness on aeroelastic model was discussed but no special properties are analyzed. Motivated by the above discussions, we are interested in exploring the airfoil system considering both plunge and pitch stiffnesses to be higher-order nonlinearities. This paper reports some new complex behaviors of the airfoil system like multistability and megastability which have not been reported earlier in the literatures. The proposed investigation falls under category 1 and 2 as described in [34].


## 2. Two-Dimensional Airfoil System with Higher-Order Nonlinear Spring (ASHS)

### 2.1. Mathematical Model

The dynamical model of an airfoil with cubic pitching stiffness and viscous damping as shown in Figure1 was proposed in [5, 6]. ρ is the air density, m is the mass, b is the semichord length, ab is the distance of the elastic axis E from the midchord point, 0.5+ab is the distance of E from the aerodynamic focus F, xαb is the distance of the center of gravity from E, rab is the radius of gyration of the airfoil with respect to E, and ωh and ωα are the eigenfrequencies of the constrained one-degree-of-freedom system associated with the linear plunging and the pitching springs, respectively. The parameter values are considered as follows: a=−0.1, b=1m, xa=0.25, ra2=0.5, ωh=28.1Hz, and ωα=62.8Hz.Figure 1
Two-degree-of-freedom airfoil model.The bifurcation analysis of the proposed model [5, 6] was investigated using harmonic balance method. It is to be noted that the literatures have investigated the dynamical behavior of the airfoil system using cubic nonlinearity stiffness. Such approximations of the nonlinear stiffness have not been useful in identifying the more complex behavior of the system. Hence, we propose a modified dynamical equation of the airfoil system as(1)h¨+0.25α¨+0.1h˙+0.2h+0.1βα+fh=0,0.25h¨+0.5α¨+0.1α˙+0.5α−0.04βα+fα=0,where fα is the pitching stiffness and fh is the plunging stiffness. The state h represents the plunging displacement, and α represents the pitching angle. The parameter β=V/bωα2, where V is the airspeed and ωα is the eigenfrequency.With higher-order nonlinear stiffness in an aeroelastic system, limit cycle oscillations occur, which leads to a fatigue in the wing structure as the consequence of a long-term vibration with constant amplitude at an invariant frequency [29]. In [35], the 5th order nonlinearity is introduced and its effects are analyzed; the authors observed that the resonant frequency is shifted toward higher frequency and the bandwidth of higher-order stiffness is wider for frequency up-sweeps. It is very clear that the 5th order nonlinearity increases the complexity, and its effects need to be studied.In this paper, we consider higher-order pitching and plunging stiffnesses in order to investigate the complex behaviors which have not been reported earlier.Usingx=α,x˙=y,z=h,andz˙=w, we derive the dimensionless model as(2)dxdt=y,dydt=11.754x0.065β−0.5+0.1w+0.2z+fz−4fx−0.4y,dzdt=w,dwdt=−11.75x0.24β−0.5+0.2w+0.4z+2fz−fx−0.1y,where fz=5z2+10z3+40z5 and fx=5x2+20x3+40x5 are the higher-order stiffnesses. The parameter β is considered as the bifurcation parameter, and for a fixed value of the airspeed, β=7.5, and for the initial conditions 0.1,0,0.1,0, the phase portraits of the system are shown in Figure 2.Figure 2
2D phase portraits of the ASHS.
(a)
(b)
(c)
(d)


### 2.2. Existence of Attractor

It has been proved in the literatures that nonlinear dissipative systems can produce chaotic attractors. Hence, to show that the ASHS is dissipative, we have computed the corresponding volume contraction rateVc, using summation of Lyapunov exponents (i.e., Vc=L.E1+L.E2+L.E3+L.E4), and thus, if Vc<0, the system is dissipative, thus experiences or presents attractors. For Vc=0, phase space volume is conserved and the dynamical system is conservative. If Vc>0, the volume in phase space expands, and hence there exist only unstable cycles or possibly chaotic repellors.For ASHS,L.E1=0.2014,L.E2=0,L.E3=0.1852,andL.E4=0.1852. It can be observed that Vc=−0.169<0 for all state vectors; thus, the introduced system is dissipative.


### 2.3. Stability of Equilibrium Points

In order to obtain the equilibrium of our model, letx˙=y˙=z˙=w˙=0; then, the only real equilibrium point of ASHS is at the origin.The Jacobian matrix of the ASHS system evaluated at any equilibrium is given by(3)JX=0100a1a2a3a40001a5a6a7a8,a1=26β175−160x7−960x27−3200x47−87,a2=−835,a3=800z4+120z2+40z7+435,a4=235,a5=800x4+240x2+40x7−24β175+27,a6=235,a7=−80z−240z+1600z47−835,a8=−435.The eigenvalues associated with the above Jacobian matrix are obtained by solving the following characteristic equation (detMJ−λId=0), where Id is the identity matrix:(4)λ4+0.342λ3+1.39−0.148βλ2+0.16−9.14e−3βλ−0.0182β+0.229=0.Figure3 shows the stability of the equilibrium point for various values of β. It is to be noted that the system shows unstable oscillations when the airspeed exceeds the critical divergent speed β≥4.08015 which agrees with the results described in [5].Figure 3
Stability of the equilibrium point for various values ofβ.From the Routh–Hurwitz stability criterion, the stability conditions of all the principal minors need to be positive for the ASHS system to be stable. The principal minors are as folows:(5)Δ1=δ1>0,Δ2=δ1δ0δ3δ2>0,Δ3=>δ1δ00δ3δ2δ100δ3>0,that is, δ1>0,δ1δ2−δ3>0,andδ3>0, where δ1=0.342, δ2=1.39−0.148β,andδ3=0.16−9.14e−3β, where the conditions are satisfied; then, ASHS is stable, leading to the situation of point attractor; otherwise, the system is unstable, and the model can experience periodic or chaotic oscillations.


## 2.1. Mathematical Model

The dynamical model of an airfoil with cubic pitching stiffness and viscous damping as shown in Figure1 was proposed in [5, 6]. ρ is the air density, m is the mass, b is the semichord length, ab is the distance of the elastic axis E from the midchord point, 0.5+ab is the distance of E from the aerodynamic focus F, xαb is the distance of the center of gravity from E, rab is the radius of gyration of the airfoil with respect to E, and ωh and ωα are the eigenfrequencies of the constrained one-degree-of-freedom system associated with the linear plunging and the pitching springs, respectively. The parameter values are considered as follows: a=−0.1, b=1m, xa=0.25, ra2=0.5, ωh=28.1Hz, and ωα=62.8Hz.Figure 1
Two-degree-of-freedom airfoil model.The bifurcation analysis of the proposed model [5, 6] was investigated using harmonic balance method. It is to be noted that the literatures have investigated the dynamical behavior of the airfoil system using cubic nonlinearity stiffness. Such approximations of the nonlinear stiffness have not been useful in identifying the more complex behavior of the system. Hence, we propose a modified dynamical equation of the airfoil system as(1)h¨+0.25α¨+0.1h˙+0.2h+0.1βα+fh=0,0.25h¨+0.5α¨+0.1α˙+0.5α−0.04βα+fα=0,where fα is the pitching stiffness and fh is the plunging stiffness. The state h represents the plunging displacement, and α represents the pitching angle. The parameter β=V/bωα2, where V is the airspeed and ωα is the eigenfrequency.With higher-order nonlinear stiffness in an aeroelastic system, limit cycle oscillations occur, which leads to a fatigue in the wing structure as the consequence of a long-term vibration with constant amplitude at an invariant frequency [29]. In [35], the 5th order nonlinearity is introduced and its effects are analyzed; the authors observed that the resonant frequency is shifted toward higher frequency and the bandwidth of higher-order stiffness is wider for frequency up-sweeps. It is very clear that the 5th order nonlinearity increases the complexity, and its effects need to be studied.In this paper, we consider higher-order pitching and plunging stiffnesses in order to investigate the complex behaviors which have not been reported earlier.Usingx=α,x˙=y,z=h,andz˙=w, we derive the dimensionless model as(2)dxdt=y,dydt=11.754x0.065β−0.5+0.1w+0.2z+fz−4fx−0.4y,dzdt=w,dwdt=−11.75x0.24β−0.5+0.2w+0.4z+2fz−fx−0.1y,where fz=5z2+10z3+40z5 and fx=5x2+20x3+40x5 are the higher-order stiffnesses. The parameter β is considered as the bifurcation parameter, and for a fixed value of the airspeed, β=7.5, and for the initial conditions 0.1,0,0.1,0, the phase portraits of the system are shown in Figure 2.Figure 2
2D phase portraits of the ASHS.
(a)
(b)
(c)
(d)


## 2.2. Existence of Attractor

It has been proved in the literatures that nonlinear dissipative systems can produce chaotic attractors. Hence, to show that the ASHS is dissipative, we have computed the corresponding volume contraction rateVc, using summation of Lyapunov exponents (i.e., Vc=L.E1+L.E2+L.E3+L.E4), and thus, if Vc<0, the system is dissipative, thus experiences or presents attractors. For Vc=0, phase space volume is conserved and the dynamical system is conservative. If Vc>0, the volume in phase space expands, and hence there exist only unstable cycles or possibly chaotic repellors.For ASHS,L.E1=0.2014,L.E2=0,L.E3=0.1852,andL.E4=0.1852. It can be observed that Vc=−0.169<0 for all state vectors; thus, the introduced system is dissipative.


## 2.3. Stability of Equilibrium Points

In order to obtain the equilibrium of our model, letx˙=y˙=z˙=w˙=0; then, the only real equilibrium point of ASHS is at the origin.The Jacobian matrix of the ASHS system evaluated at any equilibrium is given by(3)JX=0100a1a2a3a40001a5a6a7a8,a1=26β175−160x7−960x27−3200x47−87,a2=−835,a3=800z4+120z2+40z7+435,a4=235,a5=800x4+240x2+40x7−24β175+27,a6=235,a7=−80z−240z+1600z47−835,a8=−435.The eigenvalues associated with the above Jacobian matrix are obtained by solving the following characteristic equation (detMJ−λId=0), where Id is the identity matrix:(4)λ4+0.342λ3+1.39−0.148βλ2+0.16−9.14e−3βλ−0.0182β+0.229=0.Figure3 shows the stability of the equilibrium point for various values of β. It is to be noted that the system shows unstable oscillations when the airspeed exceeds the critical divergent speed β≥4.08015 which agrees with the results described in [5].Figure 3
Stability of the equilibrium point for various values ofβ.From the Routh–Hurwitz stability criterion, the stability conditions of all the principal minors need to be positive for the ASHS system to be stable. The principal minors are as folows:(5)Δ1=δ1>0,Δ2=δ1δ0δ3δ2>0,Δ3=>δ1δ00δ3δ2δ100δ3>0,that is, δ1>0,δ1δ2−δ3>0,andδ3>0, where δ1=0.342, δ2=1.39−0.148β,andδ3=0.16−9.14e−3β, where the conditions are satisfied; then, ASHS is stable, leading to the situation of point attractor; otherwise, the system is unstable, and the model can experience periodic or chaotic oscillations.


## 3. Numerical Simulation

### 3.1. Bifurcation and Multistability

The bifurcation plots are derived and investigated to study the impact of the parameters on the system behavior. The parameterβ is considered as the bifurcation parameter with the other parameters fixed at their respective chaotic values. The initial condition for the first iteration is taken as 0.1,0,0.1,0. Multistability in physical systems is already discussed in the literatures [36, 37], and it is shown that such coexisting oscillations are dangerous and can affect the structural stability of a system.To show the existence of multistability, we use a robust way to plot the bifurcation plots where the initial conditions are changed in every iteration to the end values of the state variables wherein the parameter is increased or decreased in tiny steps. It is to be noted that the airfoil system shows multistability as shown in Figure4 which was not reported earlier in the literatures.Figure 4
Maximum of ASHS with forward (red) and backward (blue) continuation.Figure4 (blue) shows the forward continuation where the parameter β is increased from minimum to maximum, and Figure 4 (red) shows the backward continuation where the parameter β is decreased from maximum to minimum, and the local maxima of the state variables are plotted. The corresponding Lyapunov exponents (LEs) are calculated using Wolf’s algorithm [38] for a finite time of 40,000 s. It should be noted that we used the same forward and backward continuation to generate the Lyapunov spectrum for β. Figures 5(a) and 5(b) show the LEs for forward and backward continuation.Figure 5
Lyapunov exponents for (a) forward continuation and (b) backward continuation.
(a)
(b)The ASHS system takes a period-doubling route to chaos as shown in Figure6. Also, we could see the period-doubling route to chaos for β≥6 and an inverse period-doubling exit from chaos for 4.7≤β≤5.3. Such a phenomenon of period doubling and inverse period doubling occurring in a bifurcation diagram is termed as antimonotonicity [39].Figure 6
Phase portraits in theX-Y plane for various values of β: (a) point attractor; (b) period-1 limit cycle; (c) period-2 limit cycle; (d) chaotic attractor.
(a)
(b)
(c)
(d)Different two-dimensional projections of the ASHS attractor are presented in Figure6. We can easily note that there is no linear dependency between the state variables of the ASHS and also such dependencies between state vectors can be nonlinear and can involve several of the variables.From Figures4 and 5, it is evident that the ASHS shows coexisting attractors. We have plotted the 2D phase portraits of the coexisting attractors for different values of initial conditions and parameter β. It can be seen that a period-1 limit cycle (red) coexists with a chaotic attractor (blue) (Figure 7).Figure 7
Coexisting attractors for different values ofβ: for β = 5 and initial conditions for (a, b) blue [−0.012, −0.055, 0.009, −0.125]; red [−0.13, 0, 0.09, 0]; for β = 7 and initial conditions for (c, d) blue [−0.09, 0.003, 0.13, 0.22]; red [−0.17, 0.02, 0.13,−0.02].
(a)
(b)
(c)
(d)


### 3.2. Megastability

It was Sprott et al. who introduced the term “megastability” which is defined as the coexistence of a countable infinity of attractors in a system. He proposed a system which is a periodically-forced oscillator with a spatially-periodic damping term [40]. The system looks like a cross-sectioned cabbage with multiple layers of periodic, quasiperiodic, and chaotic attractors. A new oscillator with infinite coexisting asymmetric attractors with the megastability property was proposed in [41], in which the attractors are a combination of self-excited and hidden attractors. A two-dimensional chaotic oscillator producing a whirlpool of attractors was proposed in [42]. Similarly, Tang et al. proposed a chaotic system with coexisting attractors which forms a carpet-like structure [43]. To show the controllability of such megastable oscillators, the authors in [44] have proposed a fuzzy-based control algorithm to suppress chaotic oscillations. Most of these oscillators use a periodic forcing term and it was in [45], the forcing term was modified to a quasiperiodic function and was proved that the quasiperiodic forcing can also produce megastable oscillators. It is to be noted that, in the entire literatures on megastable oscillators, there were no discussions on such megastability in a real physical system. The proposed ASHS model discussed in this paper shows such megastability as shown in Figure 8 for β=7 and different initial conditions.Figure 8
Multiple coexisting attractors forβ = 7 with different initial conditions P,0,0.1,0 where P takes the values of 0.5,1,2,3,4,5,6,7,8,9,10 for initial-1 to initial-11, respectively.


### 3.3. Bicoherence

Bispectral analysis or bicoherence is a powerful tool in signal processing which offers a way to analyze the nonlinear coupling between frequencies and helps us in areas where linear power spectral analysis provides insufficient information [46]. Bicoherence analysis was used to investigate the nonlinearities in the aeroelastic systems [47]. The nonlinear aspects of the aerodynamic loading are determined from estimates of higher-order spectral moments, namely, the auto- and cross-bispectrum through which the quadratic nonlinear interaction between two frequency components are calculated and are used to detect a quadratic coupling or interaction among different frequency components of a signal [47].Bicoherence is the squared normalized version of bispectral density. Bicoherence gives a measure of phase coupling between signals at three different frequencies. Bicoherence is mostly used in fault diagnosis because of its ability to trace and analyze multifrequency components. It is most effective in analyzing systems with nonlinear coupling between frequencies and is useful in detecting and quantifying the presence of nonlinearity, thus indicating the severity of the fault in the machine [48, 49]. Bicoherence is also considered as a tool to analyze the coupling effects between states of a dynamical system at different frequencies [50–52].The power spectrum of a discrete time seriesxn is given by(6)Pxxk=Exkx∗k,where k is the discrete frequency variable. The bispectrum can be defined by(7)Bxk,l=Exkxlx∗k+l.The bicoherence is the normalized bispectrum given by(8)b2k,l=Exkxlx∗k+l2Exkxl2Exk+l2.The cross bicoherence can be calculated by using the following definition:(9)bxy2k,l=Exkxly∗k+l2Exkxl2Eyk+l2.The bicoherence at any frequency pairk+l can be interpreted as the fraction of power at frequency k+l which is phase coupled to the component at k+l. We have used the Welch periodogram method to estimate the bispectrum of the airfoil system (ASHS) and then the bicoherence, but the lengths of data required to obtain consistent estimates are longer than those required for power spectrum estimation; hence, we sampled the time series data generated from the ASHS state equations at 1KHz and have used 30,000 samples for the bicoherence analysis.We have presented the bicoherence plots of the ASHS for different values of the parameterβ as shown in Figure 9. We could see that the coupling between states are much weaker for β=4.1 but becomes stronger (yellow spots) for β=4.6 and β=5 and forms multiple islands of small bandwidths for β=7 indicating the strength of the coupling effects of the frequencies. We have used a fixed initial condition of 0.1,0,0.1,0. The bicoherence spectrum of surface elevations at the first measured location (Figure 9) far from the focal location indicates that many wave modes were involved in the wave-wave interactions. The bicoherence (β=5) at b20.16,0.16=1.4 denotes a self wave-wave interactions, while b20.16,0.12=1.4 denotes a nonlinear coupling between two different frequencies.Figure 9
Bicoherence plots of the ASHS.
(a)
(b)
(c)
(d)


## 3.1. Bifurcation and Multistability

The bifurcation plots are derived and investigated to study the impact of the parameters on the system behavior. The parameterβ is considered as the bifurcation parameter with the other parameters fixed at their respective chaotic values. The initial condition for the first iteration is taken as 0.1,0,0.1,0. Multistability in physical systems is already discussed in the literatures [36, 37], and it is shown that such coexisting oscillations are dangerous and can affect the structural stability of a system.To show the existence of multistability, we use a robust way to plot the bifurcation plots where the initial conditions are changed in every iteration to the end values of the state variables wherein the parameter is increased or decreased in tiny steps. It is to be noted that the airfoil system shows multistability as shown in Figure4 which was not reported earlier in the literatures.Figure 4
Maximum of ASHS with forward (red) and backward (blue) continuation.Figure4 (blue) shows the forward continuation where the parameter β is increased from minimum to maximum, and Figure 4 (red) shows the backward continuation where the parameter β is decreased from maximum to minimum, and the local maxima of the state variables are plotted. The corresponding Lyapunov exponents (LEs) are calculated using Wolf’s algorithm [38] for a finite time of 40,000 s. It should be noted that we used the same forward and backward continuation to generate the Lyapunov spectrum for β. Figures 5(a) and 5(b) show the LEs for forward and backward continuation.Figure 5
Lyapunov exponents for (a) forward continuation and (b) backward continuation.
(a)
(b)The ASHS system takes a period-doubling route to chaos as shown in Figure6. Also, we could see the period-doubling route to chaos for β≥6 and an inverse period-doubling exit from chaos for 4.7≤β≤5.3. Such a phenomenon of period doubling and inverse period doubling occurring in a bifurcation diagram is termed as antimonotonicity [39].Figure 6
Phase portraits in theX-Y plane for various values of β: (a) point attractor; (b) period-1 limit cycle; (c) period-2 limit cycle; (d) chaotic attractor.
(a)
(b)
(c)
(d)Different two-dimensional projections of the ASHS attractor are presented in Figure6. We can easily note that there is no linear dependency between the state variables of the ASHS and also such dependencies between state vectors can be nonlinear and can involve several of the variables.From Figures4 and 5, it is evident that the ASHS shows coexisting attractors. We have plotted the 2D phase portraits of the coexisting attractors for different values of initial conditions and parameter β. It can be seen that a period-1 limit cycle (red) coexists with a chaotic attractor (blue) (Figure 7).Figure 7
Coexisting attractors for different values ofβ: for β = 5 and initial conditions for (a, b) blue [−0.012, −0.055, 0.009, −0.125]; red [−0.13, 0, 0.09, 0]; for β = 7 and initial conditions for (c, d) blue [−0.09, 0.003, 0.13, 0.22]; red [−0.17, 0.02, 0.13,−0.02].
(a)
(b)
(c)
(d)


## 3.2. Megastability

It was Sprott et al. who introduced the term “megastability” which is defined as the coexistence of a countable infinity of attractors in a system. He proposed a system which is a periodically-forced oscillator with a spatially-periodic damping term [40]. The system looks like a cross-sectioned cabbage with multiple layers of periodic, quasiperiodic, and chaotic attractors. A new oscillator with infinite coexisting asymmetric attractors with the megastability property was proposed in [41], in which the attractors are a combination of self-excited and hidden attractors. A two-dimensional chaotic oscillator producing a whirlpool of attractors was proposed in [42]. Similarly, Tang et al. proposed a chaotic system with coexisting attractors which forms a carpet-like structure [43]. To show the controllability of such megastable oscillators, the authors in [44] have proposed a fuzzy-based control algorithm to suppress chaotic oscillations. Most of these oscillators use a periodic forcing term and it was in [45], the forcing term was modified to a quasiperiodic function and was proved that the quasiperiodic forcing can also produce megastable oscillators. It is to be noted that, in the entire literatures on megastable oscillators, there were no discussions on such megastability in a real physical system. The proposed ASHS model discussed in this paper shows such megastability as shown in Figure 8 for β=7 and different initial conditions.Figure 8
Multiple coexisting attractors forβ = 7 with different initial conditions P,0,0.1,0 where P takes the values of 0.5,1,2,3,4,5,6,7,8,9,10 for initial-1 to initial-11, respectively.


## 3.3. Bicoherence

Bispectral analysis or bicoherence is a powerful tool in signal processing which offers a way to analyze the nonlinear coupling between frequencies and helps us in areas where linear power spectral analysis provides insufficient information [46]. Bicoherence analysis was used to investigate the nonlinearities in the aeroelastic systems [47]. The nonlinear aspects of the aerodynamic loading are determined from estimates of higher-order spectral moments, namely, the auto- and cross-bispectrum through which the quadratic nonlinear interaction between two frequency components are calculated and are used to detect a quadratic coupling or interaction among different frequency components of a signal [47].Bicoherence is the squared normalized version of bispectral density. Bicoherence gives a measure of phase coupling between signals at three different frequencies. Bicoherence is mostly used in fault diagnosis because of its ability to trace and analyze multifrequency components. It is most effective in analyzing systems with nonlinear coupling between frequencies and is useful in detecting and quantifying the presence of nonlinearity, thus indicating the severity of the fault in the machine [48, 49]. Bicoherence is also considered as a tool to analyze the coupling effects between states of a dynamical system at different frequencies [50–52].The power spectrum of a discrete time seriesxn is given by(6)Pxxk=Exkx∗k,where k is the discrete frequency variable. The bispectrum can be defined by(7)Bxk,l=Exkxlx∗k+l.The bicoherence is the normalized bispectrum given by(8)b2k,l=Exkxlx∗k+l2Exkxl2Exk+l2.The cross bicoherence can be calculated by using the following definition:(9)bxy2k,l=Exkxly∗k+l2Exkxl2Eyk+l2.The bicoherence at any frequency pairk+l can be interpreted as the fraction of power at frequency k+l which is phase coupled to the component at k+l. We have used the Welch periodogram method to estimate the bispectrum of the airfoil system (ASHS) and then the bicoherence, but the lengths of data required to obtain consistent estimates are longer than those required for power spectrum estimation; hence, we sampled the time series data generated from the ASHS state equations at 1KHz and have used 30,000 samples for the bicoherence analysis.We have presented the bicoherence plots of the ASHS for different values of the parameterβ as shown in Figure 9. We could see that the coupling between states are much weaker for β=4.1 but becomes stronger (yellow spots) for β=4.6 and β=5 and forms multiple islands of small bandwidths for β=7 indicating the strength of the coupling effects of the frequencies. We have used a fixed initial condition of 0.1,0,0.1,0. The bicoherence spectrum of surface elevations at the first measured location (Figure 9) far from the focal location indicates that many wave modes were involved in the wave-wave interactions. The bicoherence (β=5) at b20.16,0.16=1.4 denotes a self wave-wave interactions, while b20.16,0.12=1.4 denotes a nonlinear coupling between two different frequencies.Figure 9
Bicoherence plots of the ASHS.
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## 4. Conclusion

We have modified the dynamics of the well-known airfoil system by introducing higher-order nonlinearity in plunging and pitching stiffnesses. Chaotic motions exist in an airfoil system when the airspeed exceeds the critical divergent speed. The dynamical analysis of the proposed model shows unique characters of multistability and infinitely coexisting attractors known as megastability. Such features of an airfoil system were not captured earlier in the literatures. Bicoherence plots are investigated to know the impact of multifrequency terms and coupled nonlinearities on the system.



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*Source: 5234382-2019-10-30.xml*