# Dynamical Complexity and Multistability in a Novel Lunar Wake Plasma System

Authors: Bo. Yan; Punam K. Prasad; Sayan Mukherjee; Asit Saha; Santo Banerjee
Journal: Complexity
 (2020)

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## Abstract

Dynamical complexity and multistability of electrostatic waves are investigated in a four-component homogeneous and magnetized lunar wake plasma constituting of beam electrons, heavier ions (alpha particles, He++), protons, and suprathermal electrons. The unperturbed dynamical system of the considered lunar wake plasma supports nonlinear and supernonlinear trajectories which correspond to nonlinear and supernonlinear electrostatic waves. On the contrary, the perturbed dynamical system of lunar wake plasma shows different types of coexisting attractors including periodic, quasiperiodic, and chaotic, investigated by phase plots and Lyapunov exponents. To confirm chaotic and nonchaotic dynamics in the perturbed lunar wake plasma, 0−1 chaos test is performed. Furthermore, a weighted recurrence-based entropy is implemented to investigate the dynamical complexity of the system. Numerical results show existence of chaos with variation of complexity in the perturbed dynamics.


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## Body

## 1. Introduction

The Moon is nonconducting and has no atmosphere and intrinsic magnetic field, so the solar wind freely interacts with the Moon and forms a wake on the antisunward side of the Moon [1]. The magnetic field of solar wind enters the Moon easily compared with particles of solar wind. The variations in density across the boundary of lunar wake steer the solar wind plasma to replenish the void area by ambipolar diffusion [2, 3]. The presence of ion and electron beams with fluctuating temperature of solar wind plasma produces different kinds of waves. The wind satellite revealed ion beams [2] and different modes of nonlinear waves [4] in the tail region of lunar wake. A lunar orbiter SELENE revealed the existence of electrostatic waves which are generated due to the electrostatic instability driven by energetic solar wind particles in the lunar wake [5].The particles of astrophysical plasmas such as solar wind plasma were generally found to follow non-Maxwellian distribution containing suprathermal particles with high-energy tails [6]. The kappa distribution appropriately defines the influence of suprathermal particles [7]. Recently, Saini [8] and Devanandhan et al. [9] investigated arbitrary nonlinear wave structures in two-temperature plasmas with suprathermal electrons and found the effect of suprathermal electrons on amplitude of solitons.Some nonlinear systems can exhibit many solutions with specified parameters and distinct initial conditions [10]. This nonlinear behavior is termed as coexisting attractors or multistability. Multistability behaviors [11, 12] of the physical system act as important feature in the dynamics of nonlinear systems. Experimentally, multistability feature was firstly investigated in a Q-switched gas laser [13]; thereafter, various works [14, 15] were reported in different complex systems exhibiting multistability features. In this study, we show multistability features of the lunar wake plasma system for the first time.Recently, a new wave structure called supernonlinear wave was introduced in theoretical [16] and astrophysical plasmas [17]. The nonlinear Alfvén waves and solitons defined in the framework of derivative nonlinear Schrodinger equation [18] are found to support supernonlinear waves. Tamang and Saha [19] reported supernonlinear waves and chaotic motion in a non-Maxwellian plasma. Singh and Lakhina [20] investigated ion-acoustic supersolitons in multicomponent plasma. Streaming charged debris moving in space plasma may cause an external disturbance to the system. These disturbances can disrupt the motion of the system [21, 22]. To the best of our knowledge, the study on dynamical properties of nonlinear electrostatic waves in lunar wake plasma is not reported. So, in this work, we employ the concept of nonlinear dynamics to study dynamical properties nonlinear electrostatic structures in magnetized, collisionless, homogeneous plasma comprising of beam electrons, and heavier ions (alpha particles and He++), protons, and kappa distributed electrons.The article is organized as follows. In Section2, model equations for the lunar wake plasma system are considered. In Section 3, dynamics of the perturbed and unperturbed system are studied. It has been noticed that the novel system can produce coexisting attractors under the influence of an external forcing term. Variation of Lyapunov exponents shows the conservative nature of the system. To quantify chaos, Lyapunov exponents do not produce constructive information, since very small oscillations of the Lyapunov spectra are observed. To classify chaotic and nonchaotic regimes, 0−1 chaos test [23, 24] is then implemented. The analysis is given in Section 4. A dynamical complexity is also investigated by weighted recurrence entropy [25] in Section 5. Section 6 is the conclusion.


## 2. Model Equations

A homogeneous four-component magnetized lunar wake plasma constituting of protonsNp0,Tp, electron beams Nb0,Tb, heavier ions, such as alpha particles, He++Ni0,Ti, and suprathermal elections Ne0,Te, where Nj0 and Tj denote number densities at equilibrium state and temperature of jth species, where j=b,e,i, and p for beam electrons and suprathermal electrons, ions, and positrons, respectively. Here, nonlinear electrostatic waves and drift velocity of beam electron Vb0 are assumed to be propagating along the ambient magnetic field B0.The suprathermal electrons of the lunar wake plasma are assumed to follow theκ-distribution [26]:(1)fev=ne0π1/2θΓkκΓk−1/21+v2κθ2−κ,κ>32,where κ represents spectral index, Γκ stands for gamma function, and θ denotes modified electron thermal velocity given by(2)θ2=2−3κTeme,where Te and me are electron temperature and mass, respectively. The kappa distribution tends to Maxwellian distribution, for κ⟶∞.The suprathermal electron number density is given by [26](3)ne=1−ϕκ−3/2−κ+1/2.The normalized fluid equations for lunar wake plasma propagating parallel toB0 are given by(4)∂nj∂t+∂njvj∂x=0,(5)∂vj∂t+vj∂vj∂x+Zjμpj∂ϕ∂x+3μpjσjnjnj02∂nj∂x=0,(6)∂2ϕ∂x2=ne+nb−np−Zini,where μpj=mp/mj is the ratio of mass of proton to mass of jth species, n0=ne0+nb0=np0+Zini0 is equilibrium number density, and Zj denotes the electronic charge of the jth species with Zp=1, Zb=−1, and Zi=2. In equations (3)–(6), velocity vj is normalized by Ca=Te/mp, time by ωpp−1=1/4πn0e2/mp, x by λD=Te/4πn0e2, ϕ by Te/e, and nj by n0. Furthermore, for one-dimensional case, we consider the adiabatic index, γj=3 for all species.


## 3. Dynamical Systems

### 3.1. Unperturbed Dynamical System

To analyze the dynamical properties of electrostatic waves in lunar wake plasma, we take transformationξ=x−Vt, where V signifies wave speed. Employing ξ and imposing the conditions ϕ⟶0 and dϕ/dξ=0 as ξ⟶±∞ in equations (4)–(5), we obtain(7)np=np023σpV+3σp2−2ϕ1/2−V−3σp2−2ϕ1/2,(8)ni=ni023σiVμpi+3σi2−2Ziϕ1/2−Vμpi−3σi2−2Ziϕ1/2,(9)nb=nb023σiV−Vb0μpb+3σb2+2ϕ1/2−V−Vb0μpb−3σb2+2ϕ1/2.Again using equations (3), (7), (8), and (9) in equation (6), we obtain(10)d2ϕdξ2=Aϕ+Bϕ2+Cϕ3+Dϕ4,where(11)A=2κ−12κ−3−np023σp−1V+3σp+1V−3σp−ni03σi−2V/μpi+3σi+2V/μpi−3σi+nb023σb1V−Vb0/μpb+3σb−1V−Vb0/μpb−3σb,B=2κ−12κ+122κ−32−np023σp−12V+3σp3+12V−3σp3−ni03σi−2V/μpi+3σi3+2V/μpi−3σi3+nb023σb−12V−Vb0/μpb+3σb3+12V−Vb0/μpb−3σb3,C=2κ−12κ+12κ+362κ−33−np023σp−12V+3σp5+12V−3σp5−ni03σi−4V/μpi+3σi5+4V/μpi−3σi5+nb023σb12V−Vb0/μpb+3σb5−12V−Vb0/μpb−3σb5,D=2κ−12κ+12κ+32κ+5242κ−34−np023σp−58V+3σp7+58V−3σp7−ni03σi−10V/μpi+3σi7+10V/μpi−3σi7+nb023σb−58V−Vb0/μpb+3σb7+58V−Vb0/μpb−3σb7.Equation (10) is represented as the following dynamical system:(12)dϕdξ=y,dydξ=Aϕ+Bϕ2+Cϕ3+Dϕ4.Here, equation (12) represents a dynamical system with physical parameters κ, nb0, ne0, ni0, np0, σb, σi, σp, Vbo, μpb, μpi, and V.Considering typical parameteric values of lunar wake [3, 26], we set nb0=0.01, ne0=0.99, ni0=0.05, np0≈0.9, σb=0.0025, σi=0.4, σp=0.2, and Vbo=17.14.In Figure1, we present probable phase plots of system (12) for nonlinear electrostatic waves in lunar wake plasma. Based on the values of parameters κ, np0, ni0, ne0, nb0, σp, σi, σb, Vbo, μpb, μpi, and V, we have four distinct types of phase plots. Each trajectory in a phase plane corresponds to a traveling wave solution. The phase plots presented in Figure 1 constitute different families of phase trajectories, such as superhomoclinic SH3,1, superperiodic SP3,1, periodic P1,0, and homoclinic H1,0 trajectories which correspond to supersolitary, superperiodic, periodic, and solitary wave solutions of system (12), respectively. Considering different speeds (V) of the nonlinear wave with κ=5, nb0=0.01, ne0=0.99, ni0=0.05, np0=0.9, σb=0.0025, σi=0.4, σp=0.2, Vbo=17.14, μpb=1836, and μpi=0.25187, all qualitatively distinct phase plots are depicted in Figures 1(a)–1(d). If we consider V=1.1 with specified values of other parameters, there exist only two fixed points at ϕ0,0 and ϕ1,0, as shown in Figure 1(a), where ϕ0=0 and ϕ1<0. The two fixed points ϕ0,0 and ϕ1,0 are the center and saddle point, respectively. The homoclinic trajectory H1,0 at ϕ1,0 and a periodic trajectory P1,0 at ϕ0,0 correspond to solitary and periodic wave solutions in lunar wake plasma. The phase portrait in Figure 1(b) is presented for V=1.225 with specified values of other parameters. In this case, we obtain a pair of saddle points and centers which occur at ϕ0,0, ϕ3,0, ϕ1,0, and ϕ2,0, respectively, where ϕ1<0 and ϕ2,ϕ3>0. It shows signatures of superperiodic and supersolitary wave structures due to the presence of SH3,1 and SP3,1 trajectories. For V=1.24, the existence of four fixed points can still be seen and there exist a pair of P1,0 and H1,0 trajectories but there is no superperodicity as depicted in Figure 1(c). Figure 1(d) is obtained for V=1.3. In this case, one saddle point at ϕ0,0 and a center at ϕ1,0 occur, where ϕ1<0. There also exist a class of P1,0 and H1,0 trajectories. Thus, the existence of supernonlinear waves (superperiodic and supersoliton) is confirmed in lunar wake for the first time.Figure 1
Probable phase plots of the dynamical system (12) for κ=5, nb0=0.01, ne0=0.99, ni0=0.05, np0=0.9, σb=0.0025, σi=0.4, σp=0.2, Vbo=17.14, μpb=1836, and μpi=0.25187 with (a) V=1.1, (b) V=1.225, (c) V=1.24, and (d) V=1.3.
(a)
(b)
(c)
(d)Figure 2
Coexisting attractors of system (13) for κ=5, nb0=0.01, ne0=0.99, ni0=0.05, np0=0.9, σb=0.0025, σi=0.4, σp=0.2, Vbo=17.14, μpb=1836, μpi=0.25187, V=1.225, and f0=0.01 and (a) ω=0.08 with intial conditions 0,−0.00161 (green curve) and 0,−0.00327 (red curve), (b) ω=2.09 with intial conditions 0,0.011 (brown curve), 0.013,0 (black curve), and 0.049,0 (ocean green curve), (c) enlarged view of coexisting attractors enveloping attractors shown in Figure 2(c) with initial conditions −0.049,−0.001897 (magenta curve) and −0.049,0.0021 (blue curve), and (d) for ω=1.08 with initial conditions 0,−0.0169 (red curve) and 0.21,0 (blue curve).


### 3.2. Perturbed Dynamical System

Recently, effect of the Gaussian-shaped source term on nonlinear plasma waves is investigated [27]. But, the nonlinear source term as an external forcing can be of different types [28, 29]. In this work, we consider a source term or perturbation as f0cosωξ. In presence of the source f0cosωξ, the dynamical system (12) can be expressed in the following form:(13)dϕdξ=y,dydξ=Aϕ+Bϕ2+Cϕ3+Dϕ4+f0cosωξ,where f0 is the strength and ω is the frequency of the external force.In Figure2, we depict possible phase plots of attractors corresponding to system (13) for nonlinear electrostatic structures of lunar wake. We display multistability for different values of ω by varying the initial condition with κ=5, nb0=0.01, ne0=0.99, ni0=0.05, np0=0.9, σb=0.0025, σi=0.4, σp=0.2, Vbo=17.14, μpb=1836, μpi=0.25187, V=1.225, and f0=0.01. For ω=0.08, we obtain Figure 2(a) which shows chaotic and quasiperiodic attractors in ϕy plane with initial conditions 0,−0.00161 (green curve) and 0,−0.00327 (red curve), respectively. In Figure 2(c), we set ω=2.09 with specified values of other physical parameters and detect the presence of three kinds of attractors which are quasiperiodic, chaotic, and periodic-2 attractors. Quasiperiodic attractors are obtained for initial conditions −0.049,0.0021 (blue curve), 0,0.011 (brown curve), and 0.049,0 (ocean green curve). Chaotic and periodic-2 attractors are obtained for initial conditions −0.049,−0.001897 (magenta curve) and 0.013,0 (black curve), respectively. Here, Figure 2(b) is a part of attractors shown in Figure 2(c). For ω=1.08, we show chaotic and periodic-1 attractors with initial conditions 0,−0.0169 (red curve) and 0.21,0 (blue curve), respectively, in Figure 2(d). Thus, multistability behaviors are confirmed in lunar wake plasma in presence of external periodic force.Lyapunov exponent is an effective tool to check the chaotic motion of any system. For a system to be chaotic, there must be at least one positive Lyapunov exponent. In Figure3, Lyapunov exponents are plotted against extent of the external periodic force f0 with specified values of other physical parameters as in Figure 2. Figures 3(a)–3(c) show the Lyapunov exponents corresponding to the chaotic phase trajectories shown in Figures 2(a)–2(d), respectively. From Figure 3, it can be also observed that the fluctuations of Lyapunov exponents are very small (near to 0) in all the cases. So, chaos in (13) cannot be confirmed strongly by the study of the Lyapunov exponent. A test of chaos is thus also performed which is given in the following section.Figure 3
Lyapunov exponents of system (13) of chaotic attractors corresponding to (a) Figure 2(a), (b) Figure 2(c), and (c) Figure 2(d).
(a)
(b)
(c)


## 3.1. Unperturbed Dynamical System

To analyze the dynamical properties of electrostatic waves in lunar wake plasma, we take transformationξ=x−Vt, where V signifies wave speed. Employing ξ and imposing the conditions ϕ⟶0 and dϕ/dξ=0 as ξ⟶±∞ in equations (4)–(5), we obtain(7)np=np023σpV+3σp2−2ϕ1/2−V−3σp2−2ϕ1/2,(8)ni=ni023σiVμpi+3σi2−2Ziϕ1/2−Vμpi−3σi2−2Ziϕ1/2,(9)nb=nb023σiV−Vb0μpb+3σb2+2ϕ1/2−V−Vb0μpb−3σb2+2ϕ1/2.Again using equations (3), (7), (8), and (9) in equation (6), we obtain(10)d2ϕdξ2=Aϕ+Bϕ2+Cϕ3+Dϕ4,where(11)A=2κ−12κ−3−np023σp−1V+3σp+1V−3σp−ni03σi−2V/μpi+3σi+2V/μpi−3σi+nb023σb1V−Vb0/μpb+3σb−1V−Vb0/μpb−3σb,B=2κ−12κ+122κ−32−np023σp−12V+3σp3+12V−3σp3−ni03σi−2V/μpi+3σi3+2V/μpi−3σi3+nb023σb−12V−Vb0/μpb+3σb3+12V−Vb0/μpb−3σb3,C=2κ−12κ+12κ+362κ−33−np023σp−12V+3σp5+12V−3σp5−ni03σi−4V/μpi+3σi5+4V/μpi−3σi5+nb023σb12V−Vb0/μpb+3σb5−12V−Vb0/μpb−3σb5,D=2κ−12κ+12κ+32κ+5242κ−34−np023σp−58V+3σp7+58V−3σp7−ni03σi−10V/μpi+3σi7+10V/μpi−3σi7+nb023σb−58V−Vb0/μpb+3σb7+58V−Vb0/μpb−3σb7.Equation (10) is represented as the following dynamical system:(12)dϕdξ=y,dydξ=Aϕ+Bϕ2+Cϕ3+Dϕ4.Here, equation (12) represents a dynamical system with physical parameters κ, nb0, ne0, ni0, np0, σb, σi, σp, Vbo, μpb, μpi, and V.Considering typical parameteric values of lunar wake [3, 26], we set nb0=0.01, ne0=0.99, ni0=0.05, np0≈0.9, σb=0.0025, σi=0.4, σp=0.2, and Vbo=17.14.In Figure1, we present probable phase plots of system (12) for nonlinear electrostatic waves in lunar wake plasma. Based on the values of parameters κ, np0, ni0, ne0, nb0, σp, σi, σb, Vbo, μpb, μpi, and V, we have four distinct types of phase plots. Each trajectory in a phase plane corresponds to a traveling wave solution. The phase plots presented in Figure 1 constitute different families of phase trajectories, such as superhomoclinic SH3,1, superperiodic SP3,1, periodic P1,0, and homoclinic H1,0 trajectories which correspond to supersolitary, superperiodic, periodic, and solitary wave solutions of system (12), respectively. Considering different speeds (V) of the nonlinear wave with κ=5, nb0=0.01, ne0=0.99, ni0=0.05, np0=0.9, σb=0.0025, σi=0.4, σp=0.2, Vbo=17.14, μpb=1836, and μpi=0.25187, all qualitatively distinct phase plots are depicted in Figures 1(a)–1(d). If we consider V=1.1 with specified values of other parameters, there exist only two fixed points at ϕ0,0 and ϕ1,0, as shown in Figure 1(a), where ϕ0=0 and ϕ1<0. The two fixed points ϕ0,0 and ϕ1,0 are the center and saddle point, respectively. The homoclinic trajectory H1,0 at ϕ1,0 and a periodic trajectory P1,0 at ϕ0,0 correspond to solitary and periodic wave solutions in lunar wake plasma. The phase portrait in Figure 1(b) is presented for V=1.225 with specified values of other parameters. In this case, we obtain a pair of saddle points and centers which occur at ϕ0,0, ϕ3,0, ϕ1,0, and ϕ2,0, respectively, where ϕ1<0 and ϕ2,ϕ3>0. It shows signatures of superperiodic and supersolitary wave structures due to the presence of SH3,1 and SP3,1 trajectories. For V=1.24, the existence of four fixed points can still be seen and there exist a pair of P1,0 and H1,0 trajectories but there is no superperodicity as depicted in Figure 1(c). Figure 1(d) is obtained for V=1.3. In this case, one saddle point at ϕ0,0 and a center at ϕ1,0 occur, where ϕ1<0. There also exist a class of P1,0 and H1,0 trajectories. Thus, the existence of supernonlinear waves (superperiodic and supersoliton) is confirmed in lunar wake for the first time.Figure 1
Probable phase plots of the dynamical system (12) for κ=5, nb0=0.01, ne0=0.99, ni0=0.05, np0=0.9, σb=0.0025, σi=0.4, σp=0.2, Vbo=17.14, μpb=1836, and μpi=0.25187 with (a) V=1.1, (b) V=1.225, (c) V=1.24, and (d) V=1.3.
(a)
(b)
(c)
(d)Figure 2
Coexisting attractors of system (13) for κ=5, nb0=0.01, ne0=0.99, ni0=0.05, np0=0.9, σb=0.0025, σi=0.4, σp=0.2, Vbo=17.14, μpb=1836, μpi=0.25187, V=1.225, and f0=0.01 and (a) ω=0.08 with intial conditions 0,−0.00161 (green curve) and 0,−0.00327 (red curve), (b) ω=2.09 with intial conditions 0,0.011 (brown curve), 0.013,0 (black curve), and 0.049,0 (ocean green curve), (c) enlarged view of coexisting attractors enveloping attractors shown in Figure 2(c) with initial conditions −0.049,−0.001897 (magenta curve) and −0.049,0.0021 (blue curve), and (d) for ω=1.08 with initial conditions 0,−0.0169 (red curve) and 0.21,0 (blue curve).


## 3.2. Perturbed Dynamical System

Recently, effect of the Gaussian-shaped source term on nonlinear plasma waves is investigated [27]. But, the nonlinear source term as an external forcing can be of different types [28, 29]. In this work, we consider a source term or perturbation as f0cosωξ. In presence of the source f0cosωξ, the dynamical system (12) can be expressed in the following form:(13)dϕdξ=y,dydξ=Aϕ+Bϕ2+Cϕ3+Dϕ4+f0cosωξ,where f0 is the strength and ω is the frequency of the external force.In Figure2, we depict possible phase plots of attractors corresponding to system (13) for nonlinear electrostatic structures of lunar wake. We display multistability for different values of ω by varying the initial condition with κ=5, nb0=0.01, ne0=0.99, ni0=0.05, np0=0.9, σb=0.0025, σi=0.4, σp=0.2, Vbo=17.14, μpb=1836, μpi=0.25187, V=1.225, and f0=0.01. For ω=0.08, we obtain Figure 2(a) which shows chaotic and quasiperiodic attractors in ϕy plane with initial conditions 0,−0.00161 (green curve) and 0,−0.00327 (red curve), respectively. In Figure 2(c), we set ω=2.09 with specified values of other physical parameters and detect the presence of three kinds of attractors which are quasiperiodic, chaotic, and periodic-2 attractors. Quasiperiodic attractors are obtained for initial conditions −0.049,0.0021 (blue curve), 0,0.011 (brown curve), and 0.049,0 (ocean green curve). Chaotic and periodic-2 attractors are obtained for initial conditions −0.049,−0.001897 (magenta curve) and 0.013,0 (black curve), respectively. Here, Figure 2(b) is a part of attractors shown in Figure 2(c). For ω=1.08, we show chaotic and periodic-1 attractors with initial conditions 0,−0.0169 (red curve) and 0.21,0 (blue curve), respectively, in Figure 2(d). Thus, multistability behaviors are confirmed in lunar wake plasma in presence of external periodic force.Lyapunov exponent is an effective tool to check the chaotic motion of any system. For a system to be chaotic, there must be at least one positive Lyapunov exponent. In Figure3, Lyapunov exponents are plotted against extent of the external periodic force f0 with specified values of other physical parameters as in Figure 2. Figures 3(a)–3(c) show the Lyapunov exponents corresponding to the chaotic phase trajectories shown in Figures 2(a)–2(d), respectively. From Figure 3, it can be also observed that the fluctuations of Lyapunov exponents are very small (near to 0) in all the cases. So, chaos in (13) cannot be confirmed strongly by the study of the Lyapunov exponent. A test of chaos is thus also performed which is given in the following section.Figure 3
Lyapunov exponents of system (13) of chaotic attractors corresponding to (a) Figure 2(a), (b) Figure 2(c), and (c) Figure 2(d).
(a)
(b)
(c)


## 4. Characterization of Chaos

In this section, we investigate chaos by0−1 test method. In 0−1 test method, only one component, say xnk=1N (N being the length of the component), of a system is considered [23, 24]. Using the following transformation:(14)pn,c=∑j=1nxjcosjc,qn,c=∑j=1nxjsinjc,c∈0,π.The componentxnk=1N is decomposed into two components p and q. In [23, 24], it has been established that the chaotic and nonchaotic behavior can be recognized by the respective regular and Brownian motion-like structure in the corresponding p,q-plots. So, we investigate nature of the p,q-plots for system (13) with the variation of f0∈0,0.012 and initial conditions Ik=0,−k,k∈0.00161,0.00327. Some of the plots are shown in Figure 4. It can be observed from Figures 4(a) and 4(b) that the corresponding p,q clouds are of regular and Brownian motion-like structures, respectively. It indicates nonchaotic and chaotic dynamics in system (13) for the respective f0=0.001,0.012 with k=0.00161. On the contrary, both Figures 4(c) and 4(d) show Brownian-like structure in the corresponding plots with k=0.00161,0.0261 and f0=0.012. Thus, Figure 4 shows chaotic as well as nonchaotic dynamics with the variation of f0 and k.Figure 4
(a) and (b) representp,q-plots for system (13) with f0=0.001 and 0.012, respectively. In order to calculate pandq, we choose y-component of system (13) with the initial condition 0,−0.0032. Same plots are represented in (c) and (d) with respect to the different initial conditions 0,−0.00161and0,−0.00261 for fixed f0=0.012. In both the cases, the values of parameters are considered same as chosen in Figure 2. In each calculation, the value of (c) is considered as 2π/3.
(a)
(b)
(c)
(d)In the next, we compute fluctuation ofKc with the variations of f0∈0,0.012 and k∈0.00161,0.00327, where Kc is defined by(15)Kc=limn⟶∞logMcnlogn,where Mcn is defined as(16)Mcn=limN⟶∞1N∑j=1Npcj+n−pcj2+qcj+n−qcj2.The values ofKc≈0 and 1 correspond nonchaotic and chaotic dynamics of the system. Figures 5(a) and 5(b) show fluctuation of Kc over f0∈0,0.012 (fixed k=0.0032) and k∈0.00161,0.00327 (fixed f0=0.012), respectively. From Figure 5(a), it can be observed that the Kc≈1 for f0∈0,0.0097, except f0=0.004,0.005,0.0055. It assures that chaos in system (13) can only be seen at f0=0.004,0.005,0.0055 when f0∈0,0.0097. Furthermore, Kc≈1 can be seen for f0∈0.098,0.12. It confirms chaotic dynamics in perturbed system (13) over the region f0∈0.098,0.12. On the contrary, Figure 5(b) shows Kc≈1 for almost all values of k∈0.00161,0.00327 with fixed f0=0.012. It confirms chaos in (13) over k∈0.00161,0.00327. We also investigate fluctuation of Kc with the variation of f0,k∈0,0.012×0.00161,0.00327. The corresponding contour is given in Figure 5(c). In Figure 5(c), most of the region shows Kc≈1, except for few closed regions. It establishes chaotic dynamics of system (13) over 0,0.012×0.00161,0.00327, except few values of f0,k.Figure 5
(a) representsKc vs. f0 graph for system (13) with f0∈0,0.013 and k=0.00161. (b) represents Kc vs. k for the same system with k∈0.00161,0.00327 and f0=0.012. (c) represents Kck,f0 matrix plot with k∈0.00161,0.00327,f0∈0,0.013. The associated color bar indicates values of Kc over the region 0.00161,0.00327×0,0.013. In order to calculate Kc, we have taken n≪ncut=N/10.
(a)
(b)
(c)In the following section, we have investigated dynamical complexity using weighted recurrence plot (WRP) [25].


## 5. Analysis of Dynamical Complexity

For a givenn-dimensional phase space P=xi∈Rn, weighted recurrence wi,j is defined by(17)wi,j=e−xi−xj,i,j=1,2,…,N,where N is the length of the trajectory of the phase space. As xi−xj indicates dispersion between xi and xj, wi,j can measure exponential divergence between the trajectories. The corresponding matrix wi,jN×N can thus recognize disorder in the phase space. Figures 6(a) and 6(b) represent some of the weighted matrix plots for system (13) with f0=0.001and0.012 (fixed k=0.0032), respectively. From Figure 6(a), it can be seen that range of variation as well as its pattern in wi,j are very less as compared to same in Figure 6(b). It indicates that the corresponding phase space of system (13) at f0=0.012 is more complex than the same at f0=0.001 for k=0.001. Furthermore, similar investigation is carried out for k=0.00161,0.00261 with fixed f0=0.012. The corresponding weighted matrix plots are shown in Figures 6(c) and 6(d), respectively. As variation in the weights is almost similar between Figures 6(b) and 6(c), same kind of disorder can be observed in the respective phase spaces. On the contrary, completely different as well as various patterns in wi,j can be seen in Figure 6(d), which indicates higher complex structure in the corresponding phase space compared to the other cases.Figure 6
(a) and (b) representwi,j matrix plots for system (13) with f0=0.001 and 0.012, respectively. In order to calculate wi,j matrix, we solve system (13) with the initial condition 0,−0.0032. Same plots are represented in (c) and (d) with respect to the different initial conditions 0,−0.00161,0,−0.00261 for fixed f0=0.012. In both the cases, the values of parameters are considered same, as chosen in Figure 2. In each calculation, we consider last 10,000 points on the trajectories.
(a)
(b)
(c)
(d)However, the above mentioned analysis is not enough to understand the complexity for the whole range. This is why we utilize a complexity measure-weighted recurrence entropy measure to investigate how complexity varies with the variations off0 and k. The weight recurrence entropy Sw is defined as(18)Sw=−∑sk∈Spsklogpsk,where psk denotes probability of sk∈S=sk:sk=1/N∑j=1Mωkj,1≤k≤M (M being number of events). In our case, “events” means sks.Using (18), we have computed fluctuation of Sw over f0∈0,0.012 and k∈0.00161,0.00327. Corresponding oscillations are given in Figures 7(a) and 7(b), respectively. An increasing trend can be seen in Figure 7(a). It indicates increasing pattern in the complexity with the increasing f0∈0,0.012 (fixed k=0.0032). On the contrary, almost parallel trend exists in Figure 7(b). It assures that, variation in Sws does not fluctuate abruptly. So, complexity does not differ significantly in system (13) with increasing k∈−0.00161,−0.00327 (fixed f0=0.012).Figure 7
(a) and (b) representwi,j matrix plots for system (13) with f0=0.001 and 0.012, respectively. In order to calculate wi,j matrix, we solve system (13) with the initial condition 0,−0.0032. Same plots are represented in (c) and (d) with respect to the different initial conditions 0,−0.00161and0,−0.00261, for fixed f0=0.012. In both the cases, the values of parameters are considered same, as chosen in Figure 2. In each calculation, we consider last 10,000 points on the trajectories.
(a)
(b)
(c)We further investigate complexity of system (13) over the region f0,k∈0,0.012×0.00161,0.00327. The corresponding contour is shown in Figure 7(c). In Figure 7, it can be observed that higher complexity bounded regions are very fewer compared with its complement. However, some discrete increasing as well as decreasing patterns can be seen in the whole contour.So, the analysis on the novel system reveals that the chaotic dynamics can be observed in system (13) for large regions of f0andk, but higher complexity can be seen in the same system for small regions of f0andk. Therefore, chaos with high complexity in system (13) for the interval f0,k∈0,0.012×0.00161,0.00327 can be observed.


## 6. Conclusions

Phase portrait analysis of a novel dynamical system corresponding to lunar wake has been performed in plasma constituting of beam electrons, heavier ions (alpha particles, He++), protons, and suprathermal electrons. Typical values of physical parameters of lunar wake [3, 26] have been applied in the unperturbed system to investigate qualitatively different phase portraits comprising of superperiodic, superhomoclinic, periodic, and homoclinic trajectories. These trajectories correspond to different types of nonlinear and supernonlinear wave solutions. For an external periodic perturbation due to the nonlinear source term, multistability features have been confirmed in a lunar wake plasma system. The existence of multistability in such a plasma model is never been reported. We have also investigated that the system does not confirm chaos with the observations of Lyapunov exponents as the Lyapunov exponents are close to zero with conservative characteristics. To quantify the existence of chaos, we have constructed the 0−1 test. Furthermore, a detailed dynamical complexity analysis has been implemented by using weighted recurrence. The corresponding results assure that the perturbed system (13) has high complexity in some region inside the parametric space.



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*Source: 5428548-2020-03-16.xml*