Homoclinic Trajectories Research Articles

Design of a Cislunar space navigation constellation based on special long-period orbits

This paper proposes a method for designing a navigation constellation in the Cislunar space. Firstly, the dynamic characteristics of the Cislunar space were analyzed. Northern Halo orbits and southern Halo orbits around L1 and L2 libration point are designed. Then, based on the invariant manifold theory, the homoclinic low-energy transfer trajectories of L1 Halo orbits and the low-energy heteroclinic transfer trajectories between L1 and L2 Halo orbits are provided respectively. Afterthat, two types of special long-period orbits are attained for the constellation design. Based on the geometric dilution of precision and the minimum number of visible satellites, design requirements for two parts are given. Next, two navigation constellation configurations are proposed. Lastly, simulations are conducted on the performance of two constellation configurations with different number of satellites. Compared to constellations in one orbit, constellations that configured in two orbits require fewer satellites to meet the design requirements. In different lunar transfer orbits, the performance of the dual orbit constellation was validated.

Chaotic behaviors and coexisting homoclinic cycles in a class of 3D piecewise systems

Investigating homoclinic trajectories and chaotic behaviors, is conducive to better understanding the fourteenth problem listed by Smale in his response to Arnold. This paper analytically detects coexisting homoclinic cycles without symmetry required in a class of three-dimensional (3D) piecewise systems, which is significantly different from the smooth case where a pair of homoclinic cycles with respect to one equilibrium point are generally symmetric. Moreover, it is rigorously shown that such coexisting cycles gives rise to chaos by means of analyzing a Poincaré map. An example finally is presented to illustrate the proposed results.

Bifurcation analysis of ion-acoustic superperiodic waves in ultra-relativistic dense plasma

Abstract Bifurcation analysis of ion-acoustic solitary, periodic and superperiodic waves in ultra-relativistic quantum degenerate electron-positron-ion plasma is investigated based on Sagdeev’s pseudopotential approach. The phase plane plots for the superperiodic, periodic, and homoclinic trajectories for the dynamical system are developed. Rarefactive and compressive solitary waves are demonstrated to exist. The influence of plasma parameters on the characteristic properties of the periodic and superperiodic waves is studied. Also, the effect of different plasma parameters on the Sagdeev pseudopotential profile is investigated. The results of this study may be used to shed light on the fundamental properties of nonlinear waves in dense objects (e.g., white dwarfs and neutron stars). These results also may be beneficial in describing laboratory plasma (e.g., interaction experiments of intense laser-solid matter), and plasma applications such as microelectronic devices.

Nonlinear Jeans-Buneman instability in gravitating complex plasmas

The nonlinear evolutionary dynamics of the hybrid Jeans-Buneman instability (JBI) in charge-fluctuating viscous dust molecular clouds (DMCs) of infinite extension is semi-analytically studied. The cloud macroscopic state is initially considered to be in a quasi-hydrostatic homogeneous equilibrium configuration, modeled in a classical non-relativistic framework, on the Jeans spatiotemporal scales. The global quasi-neutrality, dust-charge fluctuation, viscous dissipation, and inter-species collision effects are presumed to coexist. Such complex plasma situations are practically realizable in the star-forming DMCs in the diffused interstellar media extensively. A local nonlinear perturbative analysis herein yields a unique conjugational pair of coupled extended Korteweg-de Vries (e-KdV) equations on the gravito-electrostatic potential fluctuations. A numerical platform illustrates that the electrostatic and gravitational fluctuations grow as a unique admixture of compressive and rarefactive solitary wave patterns. It is seen that, even for cm-sized heavier dust grains, the gravitational wave amplitudes supersede the electrostatic counterparts. It is demonstrated geometrically that the electrostatic (gravitational) phase portraits are cyclically regular (irregular), unbounded (quasi-bounded), open (quasi-closed), and astable (quasi-stable) with no limit cycle behavior evolving as non(quasi)-homoclinic trajectories. The fluctuation amplitude-growth features with plasma multi-parametric variation is illustratively explored alongside its non-trivial futuristic applicability in bounded astrostructure formation.

Semiclassical asymptotic distribution of resonances created by homoclinic and heteroclinic trajectories

Here we review a part of the joint work with Jean-François Bony, Thierry Ramond and Maher Zerzeri [Astérisque 405 (2018), pp. vii+314] on the semiclassical distribution of resonances for Schrödinger operators in a complex neighborhood of a fixed energy, assuming that the trapped set of the underlying classical dynamics on the energy surface consists of hyperbolic fixed points and associated homoclinic and heteroclinic trajectories. Based on a microlocal theory of solutions near a hyperbolic fixed point established by Bony, Fujiié, Ramond, and Zerzeri [J. Funct. Anal. 252 (2007), pp. 68–125], we compute the quantization condition of resonances and prove its accuracy. We will see in particular that the principal term of the imaginary part of resonances is − D 0 h -D_0 h , where D 0 D_0 is the minimum over all ‘cycles’ in the trapped set of what we call damping index defined in terms of the exponents at the hyperbolic fixed points therein. We will also see the subprincipal term of order h / | log ⁡ h | h/|\log h| . In this scale, resonances present various interesting distributions reflecting the geometry of the classical dynamics.

Mutual Synchronization of Antiferromagnetic Spintronic Oscillators

Introduction. Recent studies into the properties of spintronic oscillators have led to broadening their scope of practical application as devices for generating and processing signals. The practical implementation of spintronic oscillators is, however, significantly limited by their low power capacity, thus requiring synchronization between devices.Aim. Determination of conditions for the implementation of the synchronous regime of two antiferromagnetic spintronic oscillators coupled by a common current.Materials and methods. To simplify the numerical simulation of a system of coupled resistively antiferromagnetic oscillators, the method of multiple-time-scale analysis was used. This allowed a system of Kuramoto equations to be considered instead of the original system. To determine the locking band of the Kuramoto model, the homoclinic trajectory approximation method was applied.Results. A system of Kuramoto equation for the phases of partial oscillators under the influence of the inertial term and phase shift was obtained. Expressions describing the locking and synchronization band as functions of the system parameters (bias currents and sizes) were derived. The numerically simulated Kuramoto model was used to determine the bands of the synchronous and asynchronous regimes.Conclusion. The results of numerical simulations of the system of Kuramoto equations and the Adler equation for two coupled spintronic oscillators agree well with the theoretically calculated values of locking and synchronization ranges. The scheme for reducing the model of antiferromagnetic oscillators to a Kuramoto model can be further extended to the case of a larger number of coupled oscillators, which will simplify computational experiments and significantly reduce the time required for numerical simulations.

Chaotic pitch motion of an aerodynamically stabilized magnetic satellite in polar orbits

The paper is devoted to the chaotic attitude dynamics of magnetic satellites with stabilizing panels. The pitch motion under the gravitational and restoring aerodynamic torques and small perturbations, namely, the magnetic torque and the aerodynamic damping, is considered. On the example of a CubeSat having an aerodynamic instability, it is demonstrated that the unperturbed phase space evolves with orbital altitude both quantitatively and qualitatively, forming different sets of homoclinic and heteroclinic trajectories. The Melnikov method is used to find the combinations of system parameters resulting in regular and chaotic motions. The occurrence of chaos is verified by means of Poincaré sections.

Dust-acoustic solitary and periodic waves in magnetized self-gravito-electrostatic opposite polarity dusty plasmas

Dust-acoustic (DA) solitary and periodic waves investigations were performed in a magnetized self-gravitating dusty plasma consisting of negatively and positively charged dust grains in the presence of inertialess ions and electrons. The Korteweg–de Vries–Burger (KdVB) equation has been derived. The numerical investigations revealed the compressive or rarefactive DA solitons depending on the plasma parameters. The nonlinear homoclinic and periodic trajectories from the KdVB equation were obtained for the phase portrait profiles when employing the phase plane theory of dynamical systems. The periodic wave solution depends also on the system parameters. The present results are considered to be beneficial in understanding the nonlinear structures in experimental devices and different astrophysical environments such as the Earth’s mesosphere, cometary tails, and Jupiter’s magnetosphere.

One-dimensional dynamical systems

The survey is devoted to the topological dynamics of maps defined on one-dimensional continua such as a closed interval, a circle, finite graphs (for instance, finite trees), or dendrites (locally connected continua without subsets homeomorphic to a circle). Connections between the periodic behaviour of trajectories, the existence of a horseshoe and homoclinic trajectories, and the positivity of topological entropy are investigated. Necessary and sufficient conditions for entropy chaos in continuous maps of an interval, a circle, or a finite graph, and sufficient conditions for entropy chaos in continuous maps of dendrites are presented. Reasons for similarities and differences between the properties of maps defined on the continua under consideration are analyzed. Extensions of Sharkovsky’s theorem to certain discontinuous maps of a line or an interval and continuous maps on a plane are considered. Bibliography: 207 titles.

Nonlinear excitations and dynamic features of dust ion-acoustic waves in a magnetized electron–positron–ion plasma

Abstract A wide class of nonlinear excitations and the dynamics of wave groups of finite amplitude ion-acoustic waves are investigated in multicomponent magnetized plasma system comprising warm ions, and superthermal electrons as well as positrons in presence of negatively charged impurities or dust particles. Employing the reductive perturbation technique (RPT), the Korteweg–de-Vries (KdV) equation, and extended KdV equation are derived. The presence of excess superthermal electrons as well as positrons and other plasma parameters are shown to influence the characteristics of both compressive and rarefactive solitons as well as double layers (DLs). Also, we extend our investigation by deriving the nonlinear Schrödinger equation from the extended KdV equation employing a suitable transformation to study the wave group dynamics for long waves. The analytical and numerical simulation results demonstrate that nonlinear wave predicts solitons, “table-top” solitons, DLs, bipolar structure, rogue waves, and breather structures. Moreover, implementing the concept of dynamical systems, phase portraits of nonlinear periodic, homoclinic trajectories, and supernonlinear periodic trajectories are presented through numerical simulation.

Curvature effects and radial homoclinic snaking

Abstract In this work, we revisit some general results on the dynamics of circular fronts between homogeneous states and the formation of localized structures in two dimensions (2D). We show how the bifurcation diagram of axisymmetric structures localized in radius fits within the framework of collapsed homoclinic snaking. In 2D, owing to curvature effects, the collapse of the snaking structure follows a different scaling that is determined by the so-called nucleation radius. Moreover, in the case of fronts between two symmetry-related states, the precise point in parameter space to which radial snaking collapses is not a ‘Maxwell’ point but is determined by the curvature-driven dynamics only. In this case, the snaking collapses to a ‘zero surface tension’ point. Near this point, the breaking of symmetry between the homogeneous states tilts the snaking diagram. A different scaling law is found for the collapse of the snaking curve in each case. Curvature effects on axisymmetric localized states with internal structure are also discussed, as are cellular structures separated from a homogeneous state by a circular front. While some of these results are well understood in terms of curvature-driven dynamics and front interactions, a proper mathematical description in terms of homoclinic trajectories in a radial spatial dynamics description is lacking.

Properties of Dynamical Systems on Dendrites and Graphs

Dynamical systems generated by continuous maps on compact metric spaces can have various properties, e.g. the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, the existence of an omega-limit set containing two minimal sets and other. In [Kočan et al., 2014] we consider six such properties and survey the relations among them for the cases of graph maps, dendrite maps and maps on compact metric spaces. In this paper, we consider fourteen such properties, provide new results and survey all the relations among the properties for the case of graph maps and all known relations for the case of dendrite maps. We formulate some open problems at the end of the paper.

Čech cohomology, homoclinic trajectories and robustness of non-saddle sets

In this paper we study flows $\varphi:M\times\mathbb{R}\longrightarrow M$ having an isolated non-saddle set. We see that the complexity of the region of influence of an isolated non-saddle set $K$ depends on the way in which $K$ sits on the phase space at the cohomological level. We construct flows in surfaces having isolated non-saddle sets with a prescribed structure for its region of influence. We also study parametrized families of smooth flows and continuations of non-saddle sets.

Bifurcations of Chaotic Attractors in a Piecewise Smooth Lorenz-Type System

We study the dynamics of a piecewise-smooth system of differential equations for which the existence of a strange Lorenz-type attractor had been rigorously proved previously and bifurcation mechanisms of its birth had been obtained. In this work we discuss the destruction of this attractor due to the appearance of sliding motions in its structure. Using qualitative and numerical methods, we study a complex sequence of attractor bifurcations that leaves in the system a globally stable limit cycle. We show that this sequence is based on C-bifurcations and bifurcations of multi-loop homoclinic trajectories.

Bifurcation analysis of nonlinear and supernonlinear dust–acoustic waves in a dusty plasma using the generalized (r, q) distribution function for ions and electrons

AbstractBifurcation analysis of dust acoustic (DA) periodic waves in three components, unmagnetized dusty plasma system is investigated using the generalized (r, q) distribution function for ions and electrons. Depending on the different parameters of the system considered, all possible phase portraits, including periodic, homoclinic, superperiodic, and superhomoclinic trajectories, are presented. The existence of rarefactive and compressive solitary waves is proved. Also, the plasma system under consideration supports both nonlinear and supernonlinear DA periodic waves. It has been found that the double spectral indices r and q play a decisive effect on the bifurcation of the waves.

On the Existence of Homoclinic Orbits in Nonautonomous Second-Order Differential Equations

or the second-order differential equation ẍ + f(t) ẋ + g(t)x = 0, the method of Lyapunov functions is used to obtain sufficient conditions for the existence of homoclinic trajectories, i.e., solutions x(t), ẋ (t) satisfying the conditions limt→±∞x(t) = 0 and limt→±∞ẋ (t) = 0. The specific case in which all the solutions of this differential equation are homoclinic is considered.

Bifurcations of small-amplitude supernonlinear waves of the mKdV and modified Gardner equations in a three-component electron-ion plasma

Small-amplitude supernonlinear ion-acoustic waves (SIAWs) are examined in a multicomponent electron-ion plasma that is composed of fluid cold ions and two temperature q-nonextensive hot and cold electrons. Implementing the reductive perturbation method, four nonlinear evolution equations are derived: the Korteweg-de-Vries (KdV) equation, the modified KdV (mKdV) equation, the further modified KdV equation, and the modified Gardner (mG) equation. Employing the traveling wave transformation, the nonlinear evolution equations are deduced to their corresponding planar dynamical systems. Applying phase plane theory of dynamical systems, phase portrait profiles including nonlinear homoclinic trajectories, nonlinear periodic trajectories from the KdV equation, and additional supernonlinear periodic trajectories are presented for ion-acoustic waves (IAWs) from the modified KdV equation. Furthermore, supersolitons corresponding to the supernonlinear homoclinic trajectory of IAWs under the modified Gardner equation are shown in a phase plane and confirmed by the potential plot with a specified set of physical parameters q, σc, σh, f, and U. Nonlinear and SIAWs are displayed using computation for distinct parametric values.

ЗАКОН СПІВІСНУВАННЯ ГОМОКЛІНІЧНИХ ТРАЄКТОРІЙ ДЛЯ ВІДОБРАЖЕНЬ НАД ПРОСТОРОМ НЕПЕРЕРВНИХ ФУНКЦІЙ

The paper deals with problem of ordering coexistence of homoclinic trajectories for maps of segment into itself, which are generated by a certain continuous function of the segment.

Suppressing homoclinic chaos for a weak periodically excited non-smooth oscillator

In this work, some new effective methods for suppressing homoclinic chaos in a weak periodically excited non-smooth oscillator are studied, and the main idea is to modify slightly the Melnikov function such that the zeros are eliminated. Firstly, a general form of planar piecewise-smooth oscillators is given to approximatively model many nonlinear restoring force of smooth oscillators subjected to all kinds of damping and periodic excitations. In the absence of controls, the Melnikov method for non-smooth homoclinic trajectories within the framework of a piecewise-smooth oscillator is briefly introduced without detailed derivation. This analytical tool is useful to detect the threshold of parameters for the existence of homoclinic chaos in the non-smooth oscillator. After some methods of state feedback control, self-adaptive control and parametric excitations control are, respectively, considered, sufficient criteria for suppressing homoclinic chaos are derived by employing the Melnikov function of non-smooth systems. Finally, the effectiveness of strategies for suppressing homoclinic chaos is analytically and numerically demonstrated through a specific example.

The order of coexistence of homoclinic trajectories for the maps of an interval

A nonperiodic trajectory of a discrete dynamical system is called n-homoclinic if its α- and ω-limit sets coincide and form the same cycle of period n. We prove the statement that the ordering 1 ⊳ 3 ⊳ 5 ⊳ 7 ⊳ ... ⊳ 2 ⋅ 1 ⊳ 2 ⋅ 3 ⊳ 2 ⋅ 5 ⊳ ... ⊳ 22 ⋅ 1 ⊳ 22 ⋅ 3 ⊳ 22 ⋅ 5 ⊳ ... determines the coexistence of homoclinic trajectories of one-dimensional systems in a sense that if a onedimensional dynamical system possesses an n-homoclinic trajectory, then it also has an m-homoclinic trajectory for each m such that n ⊳ m. It is also proved that every one-dimensional dynamical system with a cycle of period n ≠ 2i also possesses an n-homoclinic trajectory.