Existence Of Homoclinic Orbits Research Articles

Homoclinic Chaos in a Four-Dimensional Manifold Piecewise Linear System

The existence of homoclinic orbits is discussed analytically for a class of four-dimensional manifold piecewise linear systems with one switching manifold. An interesting phenomenon is found, that is, under the same parameter setting, homoclinic orbits and chaos appear simultaneously in the system. In addition, homoclinic chaos can be suppressed to a periodic orbit by adding a nonlinear control switch with memory. These theoretical results are illustrated with numerical simulations.

Homoclinic Orbits in Several Classes of Three-Dimensional Piecewise Affine Systems with Two Switching Planes

The existence of homoclinic orbits or heteroclinic cycle plays a crucial role in chaos research. This paper investigates the existence of the homoclinic orbits to a saddle-focus equilibrium point in several classes of three-dimensional piecewise affine systems with two switching planes regardless of the symmetry. An analytic proof is provided using the concrete expression forms of the analytic solution, stable manifold, and unstable manifold. Meanwhile, a sufficient condition for the existence of two homoclinic orbits is also obtained. Furthermore, two concrete piecewise affine asymmetric systems with two homoclinic orbits have been constructed successfully, demonstrating the method’s effectiveness.

Homoclinic orbits of sub-linear Hamiltonian systems with perturbed terms

By using variational methods, we obtain the existence of homoclinic orbits for perturbed Hamiltonian systems with sub-linear terms. To the best of our knowledge, there is no published result focusing on the perturbed and sub-linear Hamiltonian systems.

Canard explosion, homoclinic and heteroclinic orbits in singularly perturbed generalist predator–prey systems

This paper studies the dynamics of the generalist predator–prey systems modeled in [E. Alexandra, F. Lutscher and G. Seo, Bistability and limit cycles in generalist predator–prey dynamics, Ecol. Complex. 14 (2013) 48–55]. When prey reproduces much faster than predator, by combining the normal form theory of slow-fast systems, the geometric singular perturbation theory and the results near non-hyperbolic points developed by Krupa and Szmolyan [Relaxation oscillation and canard explosion, J. Differential Equations 174(2) (2001) 312–368; Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions, SIAM J. Math. Anal. 33(2) (2001) 286–314], we provide a detailed mathematical analysis to show the existence of homoclinic orbits, heteroclinic orbits and canard limit cycles and relaxation oscillations bifurcating from the singular homoclinic cycles. Moreover, on global stability of the unique positive equilibrium, we provide some new results. Numerical simulations are also carried out to support the theoretical results.

Homoclinic orbits and Jacobi stability on the orbits of Maxwell–Bloch system

In this paper, we analytically and geometrically investigate the complexity of Maxwell–Bloch system by giving new insight. In the first place, the existence of homoclinic orbits is rigorously proved by means of the generalized Melnikov method. More precisely, for 6a−2b>c and d>0, it is certified analytically that Maxwell–Bloch system has two nontransverse homoclinic orbits. Secondly, Jacobi stability on the orbits of Maxwell–Bloch system is examined in view point of Kosambi–Cartan–Chern theory (KCC-theory). In other words, in the light of the deviation curvature tensor of the five corresponding invariant associated to the reformulated Maxwell–Bloch system, we further proved that Jacobi stability of all equilibria under appropriate parameters. Moreover, the deviation vector, as well as the curvature of the deviation vector near equilibrium points, is focused to interpret the chaotic behavior of Maxwell–Bloch system in Finsler geometry.

Hopf Bifurcation Analysis and Existence of Heteroclinic Orbit and Homoclinic Orbit in an Extended Lorenz System

In this paper, we have considered a Lorenz-like model with slight changes in the nonlinear terms. Here we have studied the system dynamics for different range of values of parameters $$\sigma , r$$ . The Hopf bifurcation analysis of the system has been done using center manifold theorem for $$\sigma = -1, r > 1$$ . Phase portraits of solutions of the system are plotted for various system parameters to substantiate the change in dynamics. The bifurcation diagram and the Lyapunov exponent evaluation plots also help to explain the behaviour of the system. Using Fishing principle, we have shown the existence of homoclinic orbit and consequently, observed the existence of homoclinic as well as heteroclinic orbits in the numerical simulation for $$\sigma> 0, r > 1$$ .

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.

Existence of Homoclinic Solutions for a Class of Second-Order Hamiltonian Systems with Locally Subquadratic Potentials

In this paper, we mainly consider the existence of homoclinic orbits for the following second-order Hamiltonian systems $$\begin{aligned} \ddot{u}(t)-L(t)u(t)+\nabla W\bigl (t,u(t)\bigr )=f(t), \end{aligned}$$where L(t) is a positive definite and symmetric matrix for all $$t\in \mathbb {R}$$ and the potential function W(t, u) is locally subquadratic. Here, the coefficient of the upper bound for W is a positive constant, whereas in the previous literature the corresponding coefficient need to be some integrable functions a(t) on $$\mathbb {R}$$.

Existence of homoclinic orbits and conditions for the onset of chaotic behavior in a perturbed double-well oscillator

Existence of homoclinic orbits and conditions for the onset of chaotic behavior in a perturbed double-well oscillator

Two homoclinic orbits for some second-order Hamiltonian systems

This paper is concerned with the existence of homoclinic orbits for a class of second order Hamiltonian systems considering a non-periodic potential and a weaker Ambrosetti-Rabinowitz condition. By considering an auxiliary problem, we show the existence of two different approximative sequences of periodic solutions, the first one of mountain pass type and the second one of local minima. We obtain two different homoclinic orbits by passing to the limit in such sequences. As a relevant application, we obtain another homoclinic solution for the Hamiltonian system studied in \cite{IJ}.

Homoclinic solutions for singular Hamiltonian systems without the strong force condition

We consider the existence of homoclinic orbits at the origin of a Hamiltonian system q ¨ + V ′ ( q ) = 0 in R N ( N ≥ 3 ) where V has a strict global maximum at q = 0 and a singularity at a point e ≠ 0 , namely V ( q ) → − ∞ as q → e . We establish via variational methods the existence of a generalized homoclinic orbit q ˜ that may enter the singularity e without assuming the strong force condition of Gordon. Moreover when V ∼ − 1 / | q − e | α ( 0 < α < 2 ) near e , we give a bound for the number of collisions of q ˜ based on the Morse index of approximated solutions. As a consequence we obtain that q ˜ is classical (non-collision) orbit for α ∈ ] 1 , 2 [ and enters the singularity e at most one time in R if α ∈ ] 0 , 1 ] .

Homoclinic solutions for Hamiltonian system with impulsive effects

In this article, we investigate a class of impulsive Hamiltonian systems with a p-Laplacian operator. By establishing a series of new sufficient conditions, the existence of homoclinic solutions to such type of systems is revealed. We show the existence of homoclinic orbit induced by impulses by introducing some conditions. To illustrate the applications of the main results in this article, we create an example.

Periodic parametric perturbation control for a 3D autonomous chaotic system and its dynamics at infinity

Periodic parametric perturbation control and dynamics at infinity for a 3D autonomous quadratic chaotic system are studied in this paper. Using the Melnikov's method, the existence of homoclinic orbits, oscillating periodic orbits and rotating periodic orbits are discussed after transferring the 3D autonomous chaotic system to a slowly varying oscillator. Moreover, the parameter bifurcation conditions of these orbits are obtained. In order to study the global structure, the dynamics at infinity of this system are analyzed through Poincare compactification. The simulation results demonstrate feasibility of periodic parametric perturbation control technology and correctness of the theoretical results.

On the Homoclinic Orbits of the Lü System

In this paper, the existence of homoclinic orbits of the equilibrium point [Formula: see text] is demonstrated in the case of the Lü system for parameter values not reported by G. A. Leonov. In addition, some simulations are shown that agree with our theoretical analysis.

Homoclinic solutions for ordinary (q, p)-Laplacian systems with a coercive potential

Abstract A result for the existence of homoclinic orbits is obtained for (q, p)-Laplacian systems.

Homoclinic chaos in coupled SQUIDs

An rf superconducting quantum interference device (SQUID) consists of a superconducting ring interrupted by a Josephson junction (JJ). The induced supercurrents around the ring are determined by the JJ through the celebrated Josephson relations. We study the dynamics of a pair of parametrically-driven coupled SQUIDs lying on the same plane with their axes in parallel. The drive is through the alternating critical current of the JJs. This system exhibits rich nonlinear behavior, including chaotic effects. We take advantage of the weak damping that characterizes these systems to perform a multiple-scales analysis and obtain amplitude equations, describing the slow dynamics of the system. This picture allows us to expose the existence of homoclinic orbits in the dynamics of the integrable part of the slow equations of motion. Using high-dimensional Melnikov theory, we are able to obtain explicit parameter values for which these orbits persist in the full system, consisting of both Hamiltonian and non-Hamiltonian perturbations, to form so called Shilnikov orbits, indicating a loss of integrability and the existence of chaos.

Homoclinic orbits for a class of second order dynamic equations on time scales via variational methods

In this paper, we study the existence of nontrivial homoclinic orbits of a dynamic equation on time scales mathbb{T} of the form \t\t\t{(p(t)uΔ(t))Δ+qσ(t)uσ(t)=f(σ(t),uσ(t)),△-a.e. t∈T,u(±∞)=uΔ(±∞)=0.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\left \\{ \\textstyle\\begin{array}{l} ( p(t)u^{\\Delta}(t) ) ^{\\Delta}+q^{\\sigma}(t)u^{\\sigma}(t)= f(\\sigma(t),u^{\\sigma}(t)),\\quad \\triangle\\text{-a.e. } t\\in\\mathbb{T}, \\\\ u(\\pm\\infty)=u^{\\Delta}(\\pm\\infty)=0. \\end{array}\\displaystyle \\right . $$\\end{document} We construct a variational framework of the above-mentioned problem, and some new results on the existence of a homoclinic orbit or an unbounded sequence of homoclinic orbits are obtained by using the mountain pass lemma and the symmetric mountain pass lemma, respectively. The interesting thing is that the variational method and the critical point theory are used in this paper. It is notable that in our study any periodicity assumptions on p(t), q(t) and f(t,u) are not required.

Conditions for approaching the origin without intersecting the x-axis in the Liénard plane

We consider the Li?nard system in the plane and present general assumptions to obtain some new explicit conditions under which this system has or fails to have a positive orbit which starts at a point on the vertical isocline and approaches the origin without intersecting the x-axis. This arises naturally in the existence of homoclinic orbits and oscillatory solutions. Our investigation is based on the notion of orthogonal trajectories of orbits of the system.

Designing Chaotic Systems by Piecewise Affine Systems

Based on mathematical analysis, this paper provides a methodology to ensure the existence of homoclinic orbits in a class of three-dimensional piecewise affine systems. In addition, two chaotic generators are provided to illustrate the effectiveness of the method.

A new class of 3-dimensional piecewise affine systems with homoclinic orbits

Based on mathematical analysis, this paper proves the existence ofhomoclinic orbits in a new class of 3-dimensional piecewise affinesystems, and gives an example to illustrate the effectiveness of the method.