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, 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.
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 ), protons, and kappa distributed electrons.
The 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, 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 protons , electron beams , heavier ions, such as alpha particles, , and suprathermal elections , where and denote number densities at equilibrium state and temperature of species, where , and p for beam electrons and suprathermal electrons, ions, and positrons, respectively. Here, nonlinear electrostatic waves and drift velocity of beam electron are assumed to be propagating along the ambient magnetic field .
The suprathermal electrons of the lunar wake plasma are assumed to follow the κ-distribution [26]:where κ represents spectral index, stands for gamma function, and θ denotes modified electron thermal velocity given bywhere and are electron temperature and mass, respectively. The kappa distribution tends to Maxwellian distribution, for .
The suprathermal electron number density is given by [26]
The normalized fluid equations for lunar wake plasma propagating parallel to are given bywhere is the ratio of mass of proton to mass of species, is equilibrium number density, and denotes the electronic charge of the jth species with , , and . In equations (3)–(6), velocity is normalized by , time by , x by , ϕ by , and by . Furthermore, for one-dimensional case, we consider the adiabatic index, 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 transformation , where V signifies wave speed. Employing ξ and imposing the conditions and as in equations (4)–(5), we obtain
Again using equations (3), (7), (8), and (9) in equation (6), we obtainwhere
Equation (10) is represented as the following dynamical system:
Here, equation (12) represents a dynamical system with physical parameters κ, , , , , , , , , , , and V.
Considering typical parameteric values of lunar wake [3, 26], we set , , , , , , , and .
In Figure 1, we present probable phase plots of system (12) for nonlinear electrostatic waves in lunar wake plasma. Based on the values of parameters κ, , , , , , , , , , , 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 , superperiodic , periodic , and homoclinic trajectories which correspond to supersolitary, superperiodic, periodic, and solitary wave solutions of system (12), respectively. Considering different speeds (V) of the nonlinear wave with , , , , , , , , , , and , all qualitatively distinct phase plots are depicted in Figures 1(a)–1(d). If we consider with specified values of other parameters, there exist only two fixed points at and , as shown in Figure 1(a), where and . The two fixed points and are the center and saddle point, respectively. The homoclinic trajectory at and a periodic trajectory at correspond to solitary and periodic wave solutions in lunar wake plasma. The phase portrait in Figure 1(b) is presented for with specified values of other parameters. In this case, we obtain a pair of saddle points and centers which occur at , , , and , respectively, where and . It shows signatures of superperiodic and supersolitary wave structures due to the presence of and trajectories. For , the existence of four fixed points can still be seen and there exist a pair of and trajectories but there is no superperodicity as depicted in Figure 1(c). Figure 1(d) is obtained for . In this case, one saddle point at and a center at occur, where . There also exist a class of and trajectories. Thus, the existence of supernonlinear waves (superperiodic and supersoliton) is confirmed in lunar wake for the first time.
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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 . In presence of the source , the dynamical system (12) can be expressed in the following form:where 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 , , , , , , , , , , , , and . For , we obtain Figure 2(a) which shows chaotic and quasiperiodic attractors in plane with initial conditions (green curve) and (red curve), respectively. In Figure 2(c), we set 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 (blue curve), (brown curve), and (ocean green curve). Chaotic and periodic-2 attractors are obtained for initial conditions (magenta curve) and (black curve), respectively. Here, Figure 2(b) is a part of attractors shown in Figure 2(c). For , we show chaotic and periodic-1 attractors with initial conditions (red curve) and (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 Figure 3, Lyapunov exponents are plotted against extent of the external periodic force 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.
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4. Characterization of Chaos
In this section, we investigate chaos by test method. In test method, only one component, say (N being the length of the component), of a system is considered [23, 24]. Using the following transformation:
The component 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 -plots. So, we investigate nature of the -plots for system (13) with the variation of and initial conditions . Some of the plots are shown in Figure 4. It can be observed from Figures 4(a) and 4(b) that the corresponding clouds are of regular and Brownian motion-like structures, respectively. It indicates nonchaotic and chaotic dynamics in system (13) for the respective with . On the contrary, both Figures 4(c) and 4(d) show Brownian-like structure in the corresponding plots with and . Thus, Figure 4 shows chaotic as well as nonchaotic dynamics with the variation of and k.
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In the next, we compute fluctuation of with the variations of and , where is defined bywhere is defined as
The values of and 1 correspond nonchaotic and chaotic dynamics of the system. Figures 5(a) and 5(b) show fluctuation of over (fixed ) and (fixed ), respectively. From Figure 5(a), it can be observed that the for , except . It assures that chaos in system (13) can only be seen at when . Furthermore, can be seen for . It confirms chaotic dynamics in perturbed system (13) over the region . On the contrary, Figure 5(b) shows for almost all values of with fixed . It confirms chaos in (13) over . We also investigate fluctuation of with the variation of . The corresponding contour is given in Figure 5(c). In Figure 5(c), most of the region shows , except for few closed regions. It establishes chaotic dynamics of system (13) over , except few values of .
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In the following section, we have investigated dynamical complexity using weighted recurrence plot (WRP) [25].
5. Analysis of Dynamical Complexity
For a given n-dimensional phase space , weighted recurrence is defined bywhere N is the length of the trajectory of the phase space. As indicates dispersion between and , can measure exponential divergence between the trajectories. The corresponding matrix 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 (fixed ), respectively. From Figure 6(a), it can be seen that range of variation as well as its pattern in are very less as compared to same in Figure 6(b). It indicates that the corresponding phase space of system (13) at is more complex than the same at for . Furthermore, similar investigation is carried out for with fixed . 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 can be seen in Figure 6(d), which indicates higher complex structure in the corresponding phase space compared to the other cases.
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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 of and k. The weight recurrence entropy is defined aswhere denotes probability of (M being number of events). In our case, “events” means s.
Using (18), we have computed fluctuation of over and . 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 (fixed ). On the contrary, almost parallel trend exists in Figure 7(b). It assures that, variation in s does not fluctuate abruptly. So, complexity does not differ significantly in system (13) with increasing (fixed ).
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We further investigate complexity of system (13) over the region . 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 , but higher complexity can be seen in the same system for small regions of . Therefore, chaos with high complexity in system (13) for the interval 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 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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Dr. Asit Saha is grateful to SMIT and SMU for research support funded by TMA Pai University Research Fund-Minor Grant (6100/SMIT/R&D/Project/05/2018).