Related papers: Melnikov's method in String Theory
We have applied the Melnikov criterion to examine a global homoclinic bifurcation and transition to chaos in a case of a double well dynamical system with a nonlinear fractional damping term and external excitation. The usual double well…
The chaotic dynamics and its control under power law noise in Micro-electromechanical Systems (MEMS) resonators with electrostatic excitation are probed. On the basis of the stochastic Melnikov method in the mean-square sense and the mean…
We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds (NHIMs). The method is based on a new geometric proof of the normally…
We use a third-order perturbation theory and Melnikov's method to prove the existence of chaos in spinning circular disks subject to a lateral point load. We show that the emergence of transverse homoclinic and heteroclinic points…
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.…
We analyze, by means of Melnikov method, the possibility of modifying the threshold of homoclinic chaos in general 1-dimensional problems, by introducing small periodic resonant modulations. We indicate in particular a prescription in order…
In this work we consider a two-dimensional piecewise smooth system, defined in two domains separated by the switching manifold $x=0$. We assume that there exists a piecewise-defined continuous Hamiltonian that is a first integral of the…
A class of n-dimensional Poisson systems reducible to an unperturbed harmonic oscillator shall be considered. In such case, perturbations leaving invariant a given symplectic leaf shall be investigated. Our purpose will be to analyze the…
Dynamic behavior of a weightless rod with a point mass sliding along the rod axis according to periodic law is studied. This is the simplest model of child's swing. Melnikov's analysis is carried out to find bifurcations of homoclinic,…
The Melnikov-Arnold integrals (MA-integrals) is a well-known instrument used to measure the splitting of separatrices in Hamiltonian systems. In this article, we explore how calculation of MA-integrals can be used as well to estimate sizes…
In this paper, by means of the Melnikov functions we consider bifurcations of harmonic or subharmonic solutions from a periodic solution of a planar Hamiltonian system under impulsive perturbation. We give some sufficient conditions under…
We consider a $2$-dimensional autonomous system subject to a $1$-periodic perturbation, i.e. $$ \dot{\vec{x}}=\vec{f}(\vec{x})+\epsilon\vec{g}(t,\vec{x},\epsilon),\quad \vec{x}\in\Omega .$$ We assume that for $\epsilon=0$ there is a…
We consider the one degree-of-freedom Hamiltonian system defined by the Morse potential energy function (the "Morse oscillator"). We use the geometry of the level sets to construct explicit expressions for the trajectories as a function of…
Using a completely analytic procedure - based on a suitable extension of a classical method - we discuss an approach to the Poincar\'e-Mel'nikov theory, which can be conveniently applied also to the case of non-hyperbolic critical points,…
In this paper we consider a piecewise smooth $2$-dimensional system \[ \dot{\vec{x}}=\vec{g} (\vec{x})+\varepsilon\vec{g}(t,\vec{x},\varepsilon) \] where $\varepsilon>0$ is a small parameter and $\vec{f}$ is discontinuous along a curve…
In this work we study the splitting distance of a rapidly perturbed pendulum $H(x,y,t)=\frac{1}{2}y^2+(\cos(x)-1)+\mu(\cos(x)-1)g\left(\frac{t}{\varepsilon}\right)$ with $g(\tau)=\sum_{|k|>1}g^{[k]}e^{ik\tau}$ a $2\pi$-periodic function and…
The anisotropic Manev problem, which lies at the intersection of classical, quantum, and relativity physics, describes the motion of two point masses in an anisotropic space under the influence of a Newtonian force-law with a relativistic…
By a classical result of Kathleen Alligood and James Yorke we know that as we isotopically deform a map $f:ABCD\to\mathbb{R}^2$ to a Smale horseshoe map we should often expect the dynamical complexity to increase via a period--doubling…
We illustrate a completely analytic approach to Mel'nikov theory, which is based on a suitable extension of a classical method, and which is parallel and -- at least in part -- complementary to the standard procedure. This approach can be…
We have studied a chaotic transport in a two-dimensional periodic vortical flow under a time-dependent perturbation with period T where the global diffusion occurs along the stochastic web. By using the Melnikov method we construct the…