Results 171 to 180 of about 5,152,512 (204)

The Evans Function and Generalized Melnikov Integrals

SIAM Journal on Mathematical Analysis, 1999
Summary: The Evans function, \(E(\lambda)\), is an analytic function whose zeros coincide with the eigenvalues of the operator \(L\), obtained by linearizing about a travelling wave. The algebraic multiplicity of the eigenvalue \(\lambda_0\) is equal to the order of the zero of \(E(\lambda)\).
openaire   +2 more sources

Melnikov functions and Bautin ideal

Qualitative Theory of Dynamical Systems, 2001
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
openaire   +2 more sources

Applications of the signs of Melnikov's function

Applied Mathematics and Mechanics, 1992
The Melnikov function technique is applied to study the existence and stability of periodic solutions of a planar autonomous system under a small perturbation.
openaire   +2 more sources

A note on higher order Melnikov functions

Qualitative Theory of Dynamical Systems, 2005
The paper comments some facts related to the weakened 16th Hilbert's problem about limit cycles. The authors deal with small polynomial perturbations of Hamiltonian systems in the plane \(dH-\varepsilon \omega=0\) such that the first displacement map near a periodic orbit of the unperturbed system is of the form \(\Delta H=\varepsilon^kM_k(h)+O ...
Jebrane, Ahmed, Żołądek, Henryk
openaire   +1 more source

Perturbed normalizers and Melnikov functions

Journal of Mathematical Analysis and Applications, 2018
Consider a \(C^{\infty}\) smooth family of real planar vector fields \(\left(X_\varepsilon\right)_\varepsilon\) of the form \[ X_\varepsilon (x) \, = \, X_0(x) + \varepsilon X_1(x) + \varepsilon^2 X_2(x) + \, \ldots \, + \varepsilon^n X_n(x) + \mathcal{O}\left(\varepsilon^{n+1}\right), \] for some integer \(n \geq 1\), \(x \in \mathbb{R}^2\) and ...
openaire   +2 more sources

Nonsmooth homoclinic orbits, Melnikov functions and chaos in discontinuous systems

Physica D: Nonlinear Phenomena, 2012
We study the problem of chaotic behaviour in time-perturbed discontinuous systems whose unperturbed part has a piecewise image homoclinic solution transversally crossing the discontinuity manifolds.
Flaviano Battelli, Michal Fečkan
exaly   +2 more sources

Melnikov function and homoclinic chaos induced by weak perturbations

Physical Review E, 1993
The effect of noise on the possible occurrence of chaos in systems with a homoclinic orbit (e.g., the Duding equation) was recently considered by Bulsara, Schieve, and Jacobs [Phys. Rev. A 41, 668 (1990)], and Schieve and Bulsara [Phys. Rev. A 41, 1172 (1990)], who adopted an approach based on a redefinition of the Melnikov function.
, Simiu, , Frey
openaire   +2 more sources

On the Equivalence of the Melnikov Functions Method and the Averaging Method

Qualitative Theory of Dynamical Systems, 2016
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
openaire   +1 more source

Fundamental Theory of the Melnikov Function Method

2012
Chapter 6 introduces the fundamental theory of Melnikov function method. Basic definitions and fundamental lemmas are presented. A main theory on the number of limit cycles is given.
Maoan Han, Pei Yu
openaire   +1 more source

ASYMPTOTIC EXPANSIONS OF MELNIKOV FUNCTIONS AND LIMIT CYCLE BIFURCATIONS

International Journal of Bifurcation and Chaos, 2012
In the study of the perturbation of Hamiltonian systems, the first order Melnikov functions play an important role. By finding its zeros, we can find limit cycles. By analyzing its analytical property, we can find its zeros. The main purpose of this article is to summarize some methods to find its zeros near a Hamiltonian value corresponding to an ...
openaire   +2 more sources