Homoclinic Theorem Research Articles

Chaotic motions in the restricted four body problem via Devaney's saddle-focus homoclinic tangle theorem

We prove the existence of chaotic motions in an equilateral planar circular restricted four body problem (CRFBP), establishing that the system is not integrable. The proof works by verifying the hypotheses of a topological forcing theorem for Hamiltonian vector fields on R4 which hypothesizes the existence of a transverse homoclinic orbit in the energy manifold of a saddle focus equilibrium. We develop mathematically rigorous computer assisted arguments for verifying these hypotheses, and provide an implementation for CRFBP. Due to the Hamiltonian structure, this also establishes the existence of a “blue sky catastrophe”, and hence an analytic family of periodic orbits of arbitrarily long period at nearby energy levels.Our method works far from any perturbative regime and requires no mass symmetry. Additionally, the method is constructive and yields additional byproducts such as the locations of transverse connecting orbits, quantitative information about the invariant manifolds, and bounds on transport times.

Melnikov processes and chaos in randomly perturbed dynamical systems

We consider a wide class of randomly perturbed systems subjected to stationary Gaussian processes and show that chaotic orbits exist almost surely under some nondegenerate condition, no matter how small the random forcing terms are. This result is very contrasting to the deterministic forcing case, in which chaotic orbits exist only if the influence of the forcing terms overcomes that of the other terms in the perturbations. To obtain the result, we extend Melnikov’s method and prove that the corresponding Melnikov functions, which we call the Melnikov processes, have infinitely many zeros, so that infinitely many transverse homoclinic orbits exist. In addition, a theorem on the existence and smoothness of stable and unstable manifolds is given and the Smale–Birkhoff homoclinic theorem is extended in an appropriate form for randomly perturbed systems. We illustrate our theory for the Duffing oscillator subjected to the Ornstein–Uhlenbeck process parametrically.

Smale Horseshoes and Symbolic Dynamics in the Buck–Boost DC–DC Converter

DC–DC converter, which is used to change and control voltage magnitude, is a really important part of the power electronic conditioning systems. Study of nonlinear dynamics and chaos in power electronics are becoming increasingly significant in the power quality and stability of distributed power generation systems. In this paper, the mechanisms that give rise to chaotic dynamical in the buck–boost power converter is studied. The Poincare map of a buck–boost converter system possesses the invariant manifolds having a delightfully complicated structure. Based on the Smale–Birkhoff homoclinic theorem, the structure of invariant manifolds of a fixed point indicates the presence of Smale horseshoe. The horseshoe map of the system is topologically conjugate with the shift map of the symbolic dynamics. The system can be identified with the Smale horseshoe by means of its symbolic descriptions. Furthermore, since the Smale horseshoe map implies sensitive dependence on the initial condition, homoclinic intersection of stable and unstable manifolds may be applied to illustrate the existence of chaos.

Comments on “Shilnikov chaos and Hopf bifurcation in three-dimensional differential system”

In the commented paper, the authors consider a three-dimensional system and analyze the presence of Shilnikov chaos as well as a Hopf bifurcation. On the one hand, they state that the existence of a chaotic attractor is verified via the homoclinic Shilnikov theorem. The homoclinic orbit of this system is determined by using the undetermined coefficient method, introduced by Zhou et al. in [Chen's attractor exists, Int. J. Bifurcation Chaos 14 (2004) 3167–3178], a paper that presents very serious shortcomings. However, it has been cited dozens of times and its erroneous method has been copied in lots of papers, including the commented paper where an even expression for the first component of the homoclinic connection is used. It is evident that this even expression cannot represent the first component of a Shilnikov homoclinic connection, an orbit which is necessarily non-symmetric. Consequently, the results stated in Section 3, the core of the paper, are worthless. On the other hand, the study of the Hopf bifurcation presented in Section 4 is also wrong because the first Lyapunov coefficient provided by the authors is incorrect.

Study on Chaotic Behavior of the Restricted Four-Body Problem with an Equilateral Triangle Configuration

In this paper, the chaotic behavior of a planar restricted four-body problem with an equilateral triangle configuration is analytically studied. Firstly, according to the perturbation method of Melnikov, the planar restricted four-body problem is regarded as a perturbation of the two-body model. Then, we show that the Melnikov integral function has a simple zero, arriving at the existence of transversal homoclinic orbits. Afterwards, since the standard Smale–Birkhoff homoclinic theorem cannot be directly applied to the case of a degenerate saddle, we alternatively construct an invertible map [Formula: see text] and check that [Formula: see text] is a Smale horseshoe map, showing that our restricted four-body problem possesses chaotic behavior of the Smale horseshoe type.

Symbolic coding for noninvertible systems: uniform approximation and numerical computation

It is well known that the homoclinic theorem, which conjugates a map near a transversal homoclinic orbit to a Bernoulli subshift, extends from invertible to specific noninvertible dynamical systems. In this paper, we provide a unifying approach that combines such a result with a fully discrete analog of the conjugacy for finite but sufficiently long orbit segments. The underlying idea is to solve appropriate discrete boundary value problems in both cases, and to use the theory of exponential dichotomies to control the errors. This leads to a numerical approach that allows us to compute the conjugacy to any prescribed accuracy. The method is demonstrated for several examples where invertibility of the map fails in different ways.

Shilnikov chaos and Hopf bifurcation in three-dimensional differential system

In this paper, a three-dimensional differential system with only one equilibrium point is proposed. The existence of chaotic attractor in the system is verified via the homoclinic Silnikov theorem. By using the undetermined coefficient method, the homoclinic orbit of the system is determined. The Hopf bifurcation of the system is also investigated by analyzing the characteristic equation at equilibrium point and computing the first Lyapunov coefficient. Numerical simulations support the correctness of the theoretical results.

The existence of a Smale horseshoe in a planar circular restricted four-body problem

In this paper we study the existence of a Smale horseshoe in a planar circular restricted four-body problem. For this planar four-body system there exists a transversal homoclinic orbit, but the fixed point is a degenerate saddle, so that the standard Smale–Birkhoff homoclinic theorem cannot be directly applied. We therefore apply the Conley–Moser conditions to prove the existence of a Smale horseshoe. Specifically, we first use the transversal structure of stable and unstable manifolds to make a linear transformation and then introduce a nonlinear Poincare map $$P$$ by considering the truncated flow near the degenerate saddle; based on this Poincare map $$P$$ , we define an invertible map $$f$$ , which is a composite function; by carefully checking the satisfiability of the Conley–Moser conditions for $$f$$ we finally prove that $$f$$ is a Smale horseshoe map, which implies that our restricted four-body problem has the chaotic dynamics of the Smale horseshoe type.

The Related Extension and Application of the Ši'lnikov Theorem

The traditional Ši'lnikov theorems provide analytic criteria for proving the existence of chaos in high-dimensional autonomous systems. We have established one extended version of the Ši'lnikov homoclinic theorem and have given a set of sufficient conditions under which the system generates chaos in the sense of Smale horseshoes. In this paper, the extension questions of the Ši'lnikov homoclinic theorem and its applications are still discussed. We establish another extended version of the Ši'lnikov homoclinic theorem. In addition, we construct a new three-dimensional chaotic system which meets all the conditions in this extended Ši'lnikov homoclinic theorem. Finally, we list all well-known three-dimensional autonomous quadratic chaotic systems and classify them in the light of the Ši'lnikov theorems.

Inflated deterministic chaos and Smale's horseshoe

We consider a smooth diffeomorphism possessing a hyperbolic fixed point with a transversal homoclinic orbit. The aim of this paper is to investigate multivalued perturbations of the form where G is a compact convex-valued Lipschitz map from into itself and parameter is small. Our main result is a multivalued, parametrized version of the well-known Birkhoff–Smale homoclinic theorem asserting the existence of deterministic chaos with the corresponding Smale horseshoes. A similar result is given for multivalued, time-periodic perturbations of time-periodic ordinary differential equations. For autonomous ordinary differential equations, the resulting multivalued, parametrized homoclinic bifurcation problem is also discussed at some length.

ŠIL'NIKOV HOMOCLINIC ORBITS IN TWO CLASSES OF 3D AUTONOMOUS NONLINEAR SYSTEMS

The Šil'nikov homoclinic theorem provides one analytic criterion for proving the existence of chaos in three-dimensional autonomous nonlinear systems. In applications of the theorem, however, the existence of a homoclinic orbit that usually determines the geometric structure of the chaotic attractor is not easily verified mainly because there are no available efficient methods. In this paper, based on the undetermined coefficient approach we present a framework of how to find homoclinic orbits in two classes of three-dimensional autonomous nonlinear systems of normal forms, including how to set a reasonable form of expanding series of the homoclinic orbit, how to determine all coefficients in the expansion, and how to find a numerical homoclinic orbit. Numerical examples show that the proposed framework in combination with computer simulation is very efficient.

Distribution of Maps with Transversal Homoclinic Orbits in a Continuous Map Space

This paper is concerned with distribution of maps with transversal homoclinic orbits in a continuous map space, which consists of continuous maps defined in a closed and bounded set of a Banach space. By the transversal homoclinic theorem, it is shown that the map space contains a dense set of maps that have transversal homoclinic orbits and are chaotic in the sense of both Li‐Yorke and Devaney with positive topological entropy.

On existence of homoclinic orbits for some types of autonomous quadratic systems of differential equations

The new existence conditions of homoclinic orbits for the system of ordinary quadratic differential equations are founded. Further, the realization of these conditions together with the Shilnikov Homoclinic Theorem guarantees the existence of a chaotic attractor at 3D autonomous quadratic system. Examples of the chaotic attractors are given.

AN EXTENDED ŠIL'NIKOV HOMOCLINIC THEOREM AND ITS APPLICATIONS

The classical Šil'nikov homoclinic theorem provides an analytic criterion for proving the existence of chaos in three-dimensional autonomous systems, but it can only be applied to systems with fixed points of the saddle-focus type. This paper extends this powerful theorem to a degenerate case where one of the eigenvalues of the Jacobian evaluated at an equilibrium point is zero and the other two are a pair of conjugate complex numbers, and consequently establishes a set of criteria for proving the existence of chaos in the sense of having Smale horseshoes. Based on this new extended Šil'nikov homoclinic theorem, a new chaotic system is constructed, whose corresponding bounded chaotic attractor is first verified numerically through phase trajectories, Lyapunov exponents, bifurcation routes and Poincaré mappings, followed by theoretical analysis on the existence of one homoclinic orbit, the key component of the extended Šil'nikov homoclinic theorem.

Chaotification for a Class of Nonlinear Systems with Backlash Functions

The problem of creating chaos for a class of nonlinear systems with backlash function is studied in this paper. By utilizing the characteristic of backlash function, some criteria for guaranteing the nonlinear system to be horseshoe type of chaos are derived based on Ši'lnikov heterclinic theorem and Ši'lnikov homoclinic theorem. Examples and simulations are given to illustrate the effectiveness of the theoretical results.

Homoclinic and heteroclinic behavior in an infinite-degree-of-freedom Hamiltonian system: Chaotic free vibrations of an undamped, buckled beam

We study an infinite-degree-of-freedom Hamiltonian system representing a mathematical model for an undamped, buckled beam. Using recent results by Melnikov-type techniques for two or more degrees of freedom Hamiltonian systems, we show that very complicated behavior occurs in this system: there exist orbits transversely homoclinic to periodic orbits, and orbits transversely homoclinic and heteroclinic to invariant tori consisting of quasiperiodic orbits. This leads to the nonintegrability of the system in appropriate meanings. The existence of orbits transversely homoclinic to periodic orbits also immediately yields chaotic dynamics by the Smale–Birkhoff homoclinic theorem. Moreover, the orbits transversely heteroclinic to invariant tori construct transition chains and pairs of heteroclinic cycles with different itineraries, so that transitions between small neighborhoods of invariant tori, which is similar to Arnold diffusion, and chaotic dynamics generated by a mechanism different from the Smale–Birkhoff homoclinic theorem can occur.

Homoclinic crossing in open systems: chaos in periodically perturbed monopole plus quadrupolelike potentials.

The Melnikov method is applied to periodically perturbed open systems modeled by an inverse-square-law attraction center plus a quadrupolelike term. A compactification approach that regularizes periodic orbits at infinity is introduced. The (modified) Smale-Birkhoff homoclinic theorem is used to study transversal homoclinic intersections. A larger class of open systems with degenerated (nonhyperbolic) unstable periodic orbits after regularization is also briefly considered.

An analogue to the Smale—Birkhoff homoclinic theorem for iterated entire mappings

An analogue to the Smale—Birkhoff homoclinic theorem for iterated entire mappings

An analogue to the Smale—Birkhoff homoclinic theorem for iterated entire mappings

A snapback repeller of an analytic mapping is defined as a full orbit which tends to an unstable fixed point backwards in time and snaps back to the same fixed point. This note gives a rather elementary proof that unstable periodic orbits accumulate near snapback repellers. The proof is entirely selfcontained and uses only standard elementary tools. We exploit that the global semiconjugacy of the entire analytic map to a linear map is itself an entire analytic function and apply the Theorem of Rouche to its zeros. We also generalize Marotto's result about the chaotic motion near a snapback repeller to include the degenerate case.

Heteroclinic and homoclinic bifurcations in bistable reaction diffusion systems

We apply a general heteroclinic and homoclinic bifurcation theory to the study of bifurcations of travelling waves of bistable reaction diffusion systems. Using the notion of separation, we first prove the existence of a cusp point of the set of travelling front solutions in the parameter space. This as well as the symmetry of the system yields a coexisting pair of front and back solutions which undergoes the homoclinic bifurcation producing a pulse solution. All the hypotheses imposed on the general heteroclinic and homoclinic bifurcation theorem are rigorously verified for a system of bistable reaction diffusion equations containing a small parameter ε by using singular perturbation techniques, especially the SLEP method. A relation between the stability of front (or back) solutions and the intersecting manner of the stable and unstable manifolds is also given by means of the separation.