Homoclinic Trajectories Research Articles

Hidden extreme multistability generated from a fractional-order chaotic system

A new 4D fractional-order chaotic system with no equilibrium is proposed in this paper. One salient feature of this system is that it has no equilibrium point, which leads to the lack of homo-clinic or hetero-clinic trajectory. Thus, Shil’nikov theorem is not suitable to prove the existence of chaos in this system. However, the system can exhibit complex dynamic behaviors when its system parameter and fractional derivative order are varied. More interesting, the dynamics depending on initial values are also investigated and a more complex phenomenon of hidden extreme multistability is revealed. Furthermore, by selecting the appropriate system parameters, the state transition behaviors can be also found in this system. Finally, in order to verify the feasibility of the proposed fractional-order chaotic system, the corresponding physical circuit is designed by using the modular design method. The hardware experimental results are in good agreement with numerical analysis.

THE CONSTRUCTION OF SOLUTIONS OF PIECEWISE-LINEAR PHASE SYSTEMS

The creation of methods for the study of nonlinear phase systems has a long history, since the 60s of the last century (V.I. Tikhonov, V. Lindsay, M.V. Kapranov, B.I. Shakhtarin, etc.). By now, rigorous and approximate analysis methods of such systems have been developed. However, most methods are limited to the analysis of low order systems. Only in recent years attempts have been made to create methods, which allow to carry out the analysis of high-order phase systems. The material of this article deals with these methods. The article considers the construction of solutions of phase systems on the example of phase-locked frequency of arbitrary dimension with piecewise linear approximation of the nonlinear function. This approximation allowed to use an explicit form of solutions in the linearity and to obtain analytical conditions for the existence of various types of system behavior. The analytical conditions for the existence of solutions leading to the emergence of complex limit sets of the trajectories of phase systems and their bifurcations are obtained. These are homoclinic trajectories in the case of the saddle-focus equilibrium state, which play a decisive role in the occurrence of chaos. It is also shown that it is possible to obtain analytical conditions for the bifurcation of the birth and the existence of multi-pass rotational cycles in a piecewise linear phase system, on the basis of which a criterion for the transition to chaos through bifurcations cascade of doubling the stable cycle period can be obtained; in accordance with the Sharkovsky theorem it ends with the bifurcation of the cycle birth of the period three and the occurrence of developed chaos. It should be noted that the research methods of piecewise linear systems described in the paper were applied by the authors not only to phase systems, but, for example, to the Chua system, allowing various chaotic behavior.

Semiclassical asymptotic distribution of resonances created by homoclinic trajectories

Semiclassical asymptotic distribution of resonances created by homoclinic trajectories

Heteroclinic and Homoclinic Structures in the System of Four Identical Globally Coupled Phase Oscillators with Nonpairwise Interactions

Systems of $N$ identical globally coupled phase oscillators can demonstrate a multitude of complex behaviours. Such systems can have chaotic dynamics for $N>4$ when a coupling function is biharmonic. The case $N = 4$ does not possess chaotic attractors when the coupling is biharmonic, but has them when the coupling includes nonpairwise interactions of phases. Previous studies showed that some of chaotic attractors in this system are organized by heteroclinic networks. In present paper we discuss which heteroclinic cycles are forbidden and which are supported by this particular system. We also discuss some of the cases regarding homoclinic trajectories to saddle-foci equilibria.

Bifurcation of heteroclinic orbits via an index theory

Heteroclinic orbits for one-parameter families of nonautonomous vectorfields appear in a very natural way in many physical applications. Inspired by a recent bifurcation result for homoclinic trajectories of nonautonomous vectorfield proved by author in [13], we define a new $$\mathbf {Z}_2$$ -index and we construct a index theory for heteroclinic orbits of nonautonomous vectorfield. We prove an index theorem, by showing that, under some standard transversality assumptions, the $$\mathbf {Z}_2$$ -index is equal to the parity, a homotopy invariant for paths of Fredholm operators of index 0. As a direct consequence of the index theory developed in this paper, we get a new bifurcation result for heteroclinic orbits.

Global Instability in the Restricted Planar Elliptic Three Body Problem

The restricted planar elliptic three body problem (RPETBP) describes the motion of a massless particle (a comet or an asteroid) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter) revolving around their center of mass on elliptic orbits with some positive eccentricity. The aim of this paper is to show the existence of orbits whose angular momentum performs arbitrary excursions in a large region. In particular, there exist diffusive orbits, that is, with a large variation of angular momentum. The leading idea of the proof consists in analyzing parabolic motions of the comet. By a well-known result of McGehee, the union of future (resp. past) parabolic orbits is an analytic manifold $${\mathcal{P}^+}$$ (resp. $${\mathcal{P}^-}$$ ). In a properly chosen coordinate system these manifolds are stable (resp. unstable) manifolds of a manifold at infinity $${\mathcal{P}_\infty}$$ , which we call the manifold at parabolic infinity. On $${\mathcal{P}_\infty}$$ it is possible to define two scattering maps, which contain the map structure of the homoclinic trajectories to it, i.e. orbits parabolic both in the future and the past. Since the inner dynamics inside $${\mathcal{P}_\infty}$$ is trivial, two different scattering maps are used. The combination of these two scattering maps permits the design of the desired diffusive pseudo-orbits. Using shadowing techniques and these pseudo orbits we show the existence of true trajectories of the RPETBP whose angular momentum varies in any predetermined fashion.

New global bifurcation diagrams for piecewise smooth systems: Transversality of homoclinic points does not imply chaos

In this paper we consider some piecewise smooth 2-dimensional systems having a possibly non-smooth homoclinic γ→(t). We assume that the critical point 0→ lies on the discontinuity surface Ω0. We consider 4 scenarios which differ for the presence or not of sliding close to 0→ and for the possible presence of a transversal crossing between γ→(t) and Ω0. We assume that the systems are subject to a small non-autonomous perturbation, and we obtain 4 new bifurcation diagrams. In particular we show that, in one of these scenarios, the existence of a transversal homoclinic point guarantees the persistence of the homoclinic trajectory but chaos cannot occur. Further we illustrate the presence of new phenomena involving an uncountable number of sliding homoclinics.

Heteroclinic Trajectory and Hopf Bifurcation in an Extended Lorenz System

This note revisits an extended Lorenz system, which was presented in the paper entitled “Hopf bifurcations in an extended Lorenz system” by Zhou et al. [2017]. On the one hand, one points out and corrects some wrong results in that paper on the Hopf bifurcation at the symmetric equilibria [Formula: see text] and [Formula: see text]. On the other hand, combining Lyapunov function and the concepts of [Formula: see text]- and [Formula: see text]-limit sets, it is rigorously proved that there exists two and only two heteroclinic trajectories but no homoclinic trajectories under some certain conditions of its parameters and initial values. In addition, numerical simulations illustrate the consistence with the theoretical conclusions. The results together not only improve and complement the known ones, but also provide support in some future applications.

A Novel Four-Dimensional No-Equilibrium Hyper-Chaotic System With Grid Multiwing Hyper-Chaotic Hidden Attractors

By using a simple state feedback control technique and introducing two new nonlinear functions into a modified Sprott B system, a novel four-dimensional (4D) no-equilibrium hyper-chaotic system with grid multiwing hyper-chaotic hidden attractors is proposed in this paper. One remarkable feature of the new presented system is that it has no equilibrium points and therefore, Shil'nikov theorem is not suitable to demonstrate the existence of chaos for lacking of hetero-clinic or homo-clinic trajectory. But grid multiwing hyper-chaotic hidden attractors can be obtained from this new system. The complex hidden dynamic behaviors of this system are analyzed by phase portraits, the time domain waveform, Lyapunov exponent spectra, and the Kaplan–York dimension. In particular, the Lyapunov exponent spectra are investigated in detail. Interestingly, when changing the newly introduced nonlinear functions of the new hyper-chaotic system, the number of wings increases. And with the number of wings increasing, the region of the hyper-chaos is getting larger, which proves that this novel proposed hyper-chaotic system has very rich and complicated hidden dynamic properties. Furthermore, a corresponding improved module-based electronic circuit is designed and simulated via multisim software. Finally, the obtained experimental results are presented, which are in agreement with the numerical simulations of the same system on the matlab platform.

Lyapunov Functions in the Global Analysis of Chaotic Systems

We present an overview of the development of the direct Lyapunov method in the global analysis of chaotic systems and describe three directions of application of the Lyapunov functions: in the methods of localization of global attractors, where the estimates of dissipativity in Levinson’s sense are obtained, in the problems of existence of homoclinic trajectories, and in the estimation of the dimensions of attractors. The efficiency of construction of Lyapunov-type functions is demonstrated. In particular, the Lyapunov dimension formula is proved for the attractors of the Lorentz system.

Global Melnikov Theory in Hamiltonian Systems with General Time-Dependent Perturbations

We consider a mechanical system consisting of $n$ penduli and a $d$-dimensional generalized rotator subject to a time-dependent perturbation. The perturbation is not assumed to be either Hamiltonian, or periodic or quasi-periodic. The strength of the perturbation is given by a parameter $\epsilon\in\mathbb{R}$. For all $|\epsilon|$ sufficiently small, the augmented flow has a $(2d + 1)$-dimensional normally hyperbolic locally invariant manifold $\tilde\Lambda_\epsilon$. We define a Melnikov vector, which gives the first order expansion of the displacement of the stable and unstable manifolds of $\tilde\Lambda_0$ under the perturbation. We provide an explicit formula for the Melnikov vector in terms of convergent improper integrals of the perturbation along homoclinic orbits of the unperturbed system. We show that if the perturbation satisfies some explicit non-degeneracy conditions, then the stable and unstable manifolds of $\tilde\Lambda_\epsilon$, $W^s(\tilde\Lambda_\epsilon)$ and $W^u(\tilde\Lambda_\epsilon)$, respectively, intersect along a transverse homoclinic manifold, and, moreover, the splitting of $W^s(\tilde\Lambda_\epsilon)$ and $W^u(\tilde\Lambda_\epsilon)$ can be explicitly computed, up to the first order, in terms of the Melnikov vector. This implies that the excursions along some homoclinic trajectories yield a non-trivial increase of order $O(\epsilon)$ in the action variables of the rotator, for all sufficiently small perturbations. The formulas that we obtain are independent of the unperturbed motions and give, at the same time, the effects on periodic, quasi-periodic, or general orbits. When the perturbation is Hamiltonian, we express the effects of the perturbation, up to the first order, in terms of a Melnikov potential. In addition, if the perturbation is periodic, we obtain that the non-degeneracy conditions on the Melnikov potential are generic.

Nonlinear phenomena in vibro-impact dynamics: central collisions and energy jumps between two rolling bodies

Central collisions and energy jumps between two rolling bodies as singular nonlinear phenomena are investigated in vibro-impact system dynamics. For that purpose, an extension of theory of collision of two rigid bodies, in rolling dynamics, is presented. Under the authors’ use of the Petroviċ’s elements of mathematical phenomenology, especially mathematical analogy between kinetic parameters of central collision of two bodies in translator motions and central collision of two rolling different axial symmetrically bodies, new original expressions of two post-collision outgoing angular velocities for each of rolling bodies after collision are defined. Presented results are the generalization of author’s previous published results. Using this new and original result for collisions, the vibro-impact dynamics of two rolling heavy thin disks with different radiuses on the rotating circle trace in vertical plane in the period of series of collisions is investigated. Using the series of the elliptic integrals, new nonlinear equations for obtaining the angles, which defined the disks positions of successive collisions, are defined. Phase trajectories of the disks in vibro-impact dynamics are analytically and graphically presented. Two cases of vibro-impact dynamics when phase portraits contain trigger of coupled singularities and homoclinic phase trajectory in the form of number “eight,” as well as in the case without that trigger of coupled singularities, are discussed. Phase trajectory branches of dynamics of both of the rolling disks in the period from initial positions to first collision between them are presented. By using the pre-collisions impact (arrival) angular velocities and post-collision (outgoing) angular velocities of the rolling bodies, the rates of the total mechanical energy change for each disk in comparison under kinetic state that corresponds to pre- and post-collision are expressed, and also of both of the rolling bodies as system pre-collision and post-collision kinetic states. From the energy analysis, some conclusions are formulated. Full theory and new methodology for the investigation of the vibro-impact system dynamics with rolling bodies is found.

Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistability.

By using a simple state feedback controller in a three-dimensional chaotic system, a novel 4D chaotic system is derived in this paper. The system state equations are composed of nine terms including only one constant term. Depending on the different values of the constant term, this new proposed system has a line of equilibrium points or no equilibrium points. Compared with other similar chaotic systems, the newly presented system owns more abundant and complicated dynamic properties. What interests us is the observation that if the value of the constant term of the system is nonzero, it has no equilibria, and therefore, the Shil'nikov theorem is not suitable to verify the existence of chaos for the lack of heteroclinic or homoclinic trajectory. However, one-wing, two-wing, three-wing, and four-wing hidden attractors can be obtained from this new system. In addition, various coexisting hidden attractors are obtained and the complex transient transition behaviors are also observed. More interestingly, the unusual and striking dynamic behavior of the coexistence of infinitely many hidden attractors is revealed by selecting the different initial values of the system, which means that extreme multistability arises. The rich and complex hidden dynamic characteristics of this system are investigated by phase portraits, bifurcation diagrams, Lyapunov exponents, and so on. Finally, the new system is implemented by an electronic circuit. A very good agreement is observed between the experimental results and the numerical simulations of the same system on the Matlab platform.

The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos

We consider a special situation of the Hess-Appelrot case of the Euler-Poisson system which describes the dynamics of a rigid body about a fixed point. One has an equilibrium point of saddle type with coinciding stable and unstable invariant 2-dimensional separatrices. We show rigorously that, after a suitable perturbation of the Hess-Appelrot case, the separatrix connection is split such that only finite number of 1-dimensional homoclinic trajectories remain and that such situation leads to a chaotic dynamics with positive entropy and to the non-existence of any additional first integral.

Dynamics of Waves in the Cubically Nonlinear Model for Mutually Penetrating Continua

In this report we study the mathematical model for mutually penetrating continua. This model consists of the wave equation describing the carrying medium and the equation for oscillators forming the oscillating inclusion. Prescribing the constitutive equation of the carrying medium and kinetics of oscillator’s dynamics for model in question, the cubic nonlinearity is accounted. We are interested in the structure of wave solutions obeying the dynamical system of Hamiltonian type. This allows us to determine the peculiarities of the phase space of dynamical system, namely, the relation describing the homoclinic trajectories, the division of phase plane into the parts with equivalent orbits’ behavior, the conditions of bifurcations. To simulate the wave dynamics, we construct the three level finite-difference numerical scheme and study the evolution of solitary waves, their pair interactions and stability. Propagation of periodic waves is modeled as well.

Index theory for heteroclinic orbits of Hamiltonian systems

Index theory revealed its outstanding role in the study of periodic orbits of Hamiltonian systems and the dynamical consequences of this theory are enormous. Although the index theory in the periodic case is well-established, very few results are known in the case of homoclinic orbits of Hamiltonian systems. Moreover, to the authors’ knowledge, no results have been yet proved in the case of heteroclinic and halfclinic (i.e. parametrized by a half-line) orbits. Motivated by the importance played by these motions in understanding several challenging problems in Classical Mechanics, we develop a new index theory and we prove at once a general spectral flow formula for heteroclinic, homoclinic and halfclinic trajectories. Finally we show how this index theory can be used to recover all the (classical) existing results on orbits parametrized by bounded intervals.

On the existence of heteroclinic cycles in some class of 3-dimensional piecewise affine systems with two switching planes

Since complicated dynamical behavior can occur easily near homoclinic trajectory or heteroclinic cycle in dynamical systems with dimension not less than three, this paper investigates the existence of heteroclinic cycles in some class of 3-dimensional three-zone piecewise affine systems with two switching planes. Based on the exact determination of the stable manifold, unstable manifold and analytic solution, a rigorous analytic methodology of designing chaos generators is proposed, which may be of potential applications to chaos secure communication. Furthermore, we obtain three sufficient conditions for the existence of a single or two heteroclinic cycles in three different cases. Finally, some examples are given to illustrate our theoretical results.

Generalized Lorenz Equations for Acoustic-Gravity Waves in the Atmosphere. Attractors Dimension, Convergence and Homoclinic Trajectories

Attractors dimension of Lorenz-Stenflo system is estimated. Convergence criteria are proved. Fishing principle for existence of homoclinic trajectory is applied.

Arnold diffusion in the planar elliptic restricted three-body problem: mechanism and numerical verification

We present a diffusion mechanism for time-dependent perturbations of autonomous Hamiltonian systems introduced in Gidea (2014 arXiv:1405.0866). This mechanism is based on shadowing of pseudo-orbits generated by two dynamics: an ‘outer dynamics’, given by homoclinic trajectories to a normally hyperbolic invariant manifold, and an ‘inner dynamics’, given by the restriction to that manifold. On the inner dynamics the only assumption is that it preserves area. Unlike other approaches, Gidea (2014 arXiv:1405.0866) does not rely on the KAM theory and/or Aubry–Mather theory to establish the existence of diffusion. Moreover, it does not require to check twist conditions or non-degeneracy conditions near resonances. The conditions are explicit and can be checked by finite precision calculations in concrete systems (roughly, they amount to checking that Melnikov-type integrals do not vanish and that some manifolds are transversal).As an application, we study the planar elliptic restricted three-body problem. We present a rigorous theorem that shows that if some concrete calculations yield a non zero value, then for any sufficiently small, positive value of the eccentricity of the orbits of the main bodies, there are orbits of the infinitesimal body that exhibit a change of energy that is bigger than some fixed number, which is independent of the eccentricity.We verify numerically these calculations for values of the masses close to that of the Jupiter/Sun system. The numerical calculations are not completely rigorous, because we ignore issues of round-off error and do not estimate the truncations, but they are not delicate at all by the standard of numerical analysis. (Standard tests indicate that we get 7 or 8 figures of accuracy where 1 would be enough.) The code of these verifications is available. We hope that some full computer assisted proofs will be obtained in the near future since there are packages (CAPD) designed for problems of this type.

Invariant manifold connections via polyhedral representation

In recent years, manifold dynamics has assumed an increasing relevance for analysis and design of low-energy missions, both in the Earth-Moon system and in alternative multibody environments, and several space missions have already taken advantage of the results of the related studies. Recent efforts have been devoted to developing a suitable representation for the manifolds, which would be extremely useful for mission analysis and optimization. This work describes and uses a recently-introduced, intuitive polyhedral interpolative approach for each state component associated with manifold trajectories, both in two and in three dimensions. A grid of data, coming from the numerical propagation of a finite number of manifold trajectories, is used. This representation is employed for some invariant manifolds, associated with the two planar Lyapunov orbits at the collinear libration points located in the proximity of the Moon, and with a three-dimensional Halo orbit. Accuracy is evaluated, and is proven to be satisfactory, with the exclusion of limited regions of the manifolds. This paper describes the use of this polyhedral representation for the detection of homoclinic and heteroclinic connections. In particular, a variety of homoclinic trajectories connected with two Lyapunov orbits are detected. Then, the polyhedral interpolating technique is successfully employed for the determination of heteroclinic connections between the manifolds associated with the Lyapunov orbits. Lastly, near-homoclinic connections between the manifolds emanating from a Halo orbit are detected. The results achieved in this paper prove utility and effectiveness of the polyhedral interpolative technique for detecting manifold connections in the circular restricted three-body problem, and represent the premise for its application to space mission analysis involving invariant manifold dynamics.