Heteroclinic Chaotic Threshold in a Nonsmooth System with Jump Discontinuities

In smooth systems, the form of the heteroclinic Melnikov chaotic threshold is similar to that of the homoclinic Melnikov chaotic threshold. However, this conclusion may not be valid in nonsmooth systems with jump discontinuities. In this paper, based on a newly constructed nonsmooth pendulum, a kind of impulsive differential system is introduced, whose unperturbed part possesses a nonsmooth heteroclinic solution with multiple jump discontinuities. Using the recursive method and the perturbation principle, the effects of the nonsmooth factors on the behaviors of the nonsmooth dynamical system are converted to the integral items which can be easily calculated. Furthermore, the extended Melnikov function is employed to obtain the nonsmooth heteroclinic Melnikov chaotic threshold, which implies that the existence of the nonsmooth heteroclinic orbits may be due to the breaking of the nonsmooth heteroclinic loops under the perturbation of damping, external forcing and nonsmooth factors. It is worth pointing out that the form of the nonsmooth heteroclinic Melnikov function is different from the one of the nonsmooth homoclinic Melnikov function, which is quite different from the classical Melnikov theory.

A tale of two approaches to heteroclinic solutions for Φ-Laplacian systems

In this article, the existence of heteroclinic solution of a class of generalized Hamiltonian system with potential $V : {\open R}^{n} \longmapsto {\open R}$ having a finite or infinite number of global minima is studied. Examples include systems involving the p-Laplacian operator, the curvature operator and the relativistic operator. Generalized conservation of energy is established, which leads to the property of equipartition of energy enjoyed by heteroclinic solutions. The existence problem of heteroclinic solution is studied using both variational method and the metric method. The variational approach is classical, while the metric method represents a more geometrical point of view where the existence problem of heteroclinic solution is reduced to that of geodesic in a proper length metric space. Regularities of the heteroclinic solutions are discussed. The results here not only provide alternative solution methods for Φ-Laplacian systems, but also improve existing results for the classical Hamiltonian system. In particular, the conditions imposed upon the potential are very mild and new proof for the compactness is given. Finally in ℝ2, heteroclinic solutions are explicitly written down in closed form by using complex function theory.

Existence of heteroclinic solutions for a class of problems involving the fractional Laplacian

In this paper, we study the existence of heteroclinic solution for a class of nonlocal problems of the type [Formula: see text] where [Formula: see text], [Formula: see text] are continuous functions verifying some technical conditions. For example [Formula: see text] can be asymptotically periodic and potential [Formula: see text] can be the Ginzburg–Landau potential, that is, [Formula: see text].

Existence of a Heteroclinic Solution for a~Double Well Potential Equation in an Infinite Cylinder of ℝ N

This paper is concerned with the existence of a heteroclinic solution for the following class of elliptic equations: -\Delta{u}+A(\epsilon x,y)V^{\prime}(u)=0\quad\mbox{in }\Omega, where {\epsilon>0} , {\Omega=\mathbb{R}\times\mathcal{D}} is an infinite cylinder of {\mathbb{R}^{N}} with {N\geq 2} . Here, we consider a large class of potentials V that includes the Ginzburg–Landau potential {V(t)=(t^{2}-1)^{2}} and two geometric conditions on the function A. In the first condition we assume that A is asymptotic at infinity to a periodic function, while in the second one A satisfies 0<A_{0}=A(0,y)=\inf_{(x,y)\in\Omega}A(x,y)<\liminf_{|(x,y)|\to+\infty}A(x,y)=A% _{\infty}<\infty\quad\text{for all }y\in\mathcal{D}.

Heteroclinic Bifurcation Analysis of Duffing-Van Der Pol System by the Hyperbolic Lindstedt-Poincaré Method

The heteroclinic bifurcation of the Duffing-Van der Pol oscillatory System is studied by the hyperbolic Lindstedt-Poincaré method. The heteroclinic solution can be solved analytically by the method. And the critical value of the bifurcation parameter under which heteroclinic orbit forms can be determined by the perturbation procedure. Typical applications are studied in detail and compared with numerical results to illustrate the accuracy of the present method.

Heteroclinic Solutions to Asymptotically Autonomous Equations via Continuation Methods

By means of a continuation argument, we prove the existence of at least one increasing heteroclinic solution to a scalar equation of the kind ẍ = a(t)V′(x), where V is a non-negative double well potential, and a(t) is a positive, measurable coefficient, which is definitively monotone with respect to |t|, converges to l as |t| diverges and fulfils one of the two following assumptions: a(t) ≥ l everywhere, or a(t) − l converges to 0, as |t| → +∞, more slowly than a suitable exponential term.