Dynamical Complexity and Multistability in a Novel Lunar Wake Plasma System

Department of Information Engineering, Shaoyang University, Shaoyang 422000, China Department of Mathematics, Sikkim Manipal Institute of Technology, Sikkim Manipal University, Majitar, Rangpo, East-Sikkim 737136, India Department of Mathematics, Sivanath Sastri College, Kolkata, India Institute for Mathematical Research, Universiti Putra Malaysia, Serdang, Malaysia Malaysia-Italy Centre of Excellence for Mathematical Science, Universiti Putra Malaysia, Serdang, Malaysia


Introduction
e 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]. e magnetic field of solar wind enters the Moon easily compared with particles of solar wind. e variations in density across the boundary of lunar wake steer the solar wind plasma to replenish the void area by ambipolar diffusion [2,3]. e presence of ion and electron beams with fluctuating temperature of solar wind plasma produces different kinds of waves. e 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]. e particles of astrophysical plasmas such as solar wind plasma were generally found to follow non-Maxwellian distribution containing suprathermal particles with highenergy tails [6]. e 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]. is 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]. e 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. ese 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. e article is organized as follows. In Section 2, 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. e 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.

Model Equations
A homogeneous four-component magnetized lunar wake plasma constituting of protons (N p0 , T p ), electron beams (N b0 , T b ), heavier ions, such as alpha particles, He ++ (N i0 , T i ), and suprathermal elections (N e0 , T e ), where N j0 and T j 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 (V b0 ) are assumed to be propagating along the ambient magnetic field (B 0 ). e suprathermal electrons of the lunar wake plasma are assumed to follow the κ-distribution [26]: where κ represents spectral index, Γ(κ) stands for gamma function, and θ denotes modified electron thermal velocity given by where T e and m e are electron temperature and mass, respectively. e kappa distribution tends to Maxwellian distribution, for κ ⟶ ∞.
e suprathermal electron number density is given by [26] (3) e normalized fluid equations for lunar wake plasma propagating parallel to B 0 are given by where μ pj � (m p /m j ) is the ratio of mass of proton to mass of jth species, n 0 � n e0 + n b0 � n p0 + Z i n i0 is equilibrium number density, and Z j denotes the electronic charge of the jth species with Z p � 1, Z b � − 1, and Z i � 2. In equations and n j by n 0 . Furthermore, for one-dimensional case, we consider the adiabatic index, c j � 3 for all species.

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 f 0 cos(ωξ). In presence of the source f 0 cos(ωξ), the dynamical system (12) can be expressed in the following form: where f 0 is the strength and ω is the frequency of the external force.
In Figure 2, 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, n b0 � 0.01, n e0 � 0.99, n i0 � 0.05, n p0 � 0.9, σ b � 0.0025, 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  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.

Characterization of Chaos
In this section, we investigate chaos by 0 − 1 test method. In 0 − 1 test method, only one component, say x(n) N k�1 (N being the length of the component), of a system is considered [23,24]. Using the following transformation: 4 Complexity e component x(n) N k�1 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 f 0 ∈ [0, 0.012] and initial conditions I k � (0, − k), k ∈ [0.00161, 0.00327]. Some of the plots are shown in  Figure 4 shows chaotic as well as nonchaotic dynamics with the variation of f 0 and k.
In the next, we compute fluctuation of K c with the variations of f 0 ∈ [0, 0.012] and k ∈ [0.00161, 0.00327], where K c is defined by where M c (n) is defined as  Figure 5(c). In Figure 5(c), most of the region shows K c ≈ 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 (f 0 , k).
In the following section, we have investigated dynamical complexity using weighted recurrence plot (WRP) [25].

Analysis of Dynamical Complexity
For a given n-dimensional phase space P � x i ∈ R n , weighted recurrence w(i, j) is defined by where N is the length of the trajectory of the phase space.
As ‖x i − x j ‖ indicates dispersion between x i and x j , w(i, j) can measure exponential divergence between the trajectories. e corresponding matrix [w(i, j)] N×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 f 0 � 0.001 and 0.012 (fixed k � 0.0032), respectively. From Figure 6(a), it can be seen that range of variation as well as its pattern in w(i, j) are very less as compared to same in Figure 6(b). It indicates that the corresponding phase space of system (13) at f 0 � 0.012 is more complex than the same at f 0 � 0.001 for k � 0.001. Furthermore, similar investigation is carried out for k � 0.00161, 0.00261 with fixed f 0 � 0.012. e 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 [w(i, j)] can be seen in Figure 6  8 Complexity complex structure in the corresponding phase space compared to the other cases. However, the above mentioned analysis is not enough to understand the complexity for the whole range. is is why we utilize a complexity measure-weighted recurrence entropy measure to investigate how complexity varies with the variations of f 0 and k. e weight recurrence entropy (S w ) is defined as where p(s k ) denotes probability of s k ∈ S � s k : s k � (1/N) M j�1 ω kj , 1 ≤ k ≤ M} (M being number of events). In our case, "events" means s k s.
Using (18) We further investigate complexity of system (13) over the region (f 0 , k) ∈ [0, 0.012] × [0.00161, 0.00327]. e 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 f 0 and k, but higher complexity can be seen in the same system for small regions of f 0 and k. erefore, chaos with high complexity in system (13) Figure 6: (a) and (b) represent [w(i, j)] matrix plots for system (13) with f 0 � 0.001 and 0.012, respectively. In order to calculate [w(i, 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 f 0 � 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. Complexity 9

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. ese 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. e 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. e corresponding results assure that the perturbed system (13) has high complexity in some region inside the parametric space.

Data Availability
No data were used to support this study.  Figure 2. In each calculation, we consider last 10,000 points on the trajectories.

Conflicts of Interest
e authors declare that they have no conflicts of interest.