Results 111 to 120 of about 774 (145)

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.
Adriana Buică
exaly   +2 more sources

Exponential dichotomies and Melnikov functions for singularly perturbed systems

Nonlinear Analysis: Theory, Methods & Applications, 1999
Consider the singularly perturbed differential system \[ dx/dt = f(x,y)+ \varepsilon h_1 (t,x,y,\varepsilon),\quad \varepsilon dy/dt = g (x,y) + \varepsilon h_2 (t,x,y,\varepsilon),\tag \(*\) \] with \(x \in \mathbb{R}^n\), \(y\in \mathbb{R}^m\) and where \(\varepsilon \geq 0\) is a small parameter, \(f,g, h_1\) and \(h_2\) are \(C^2\)-smooth, and ...
Weiyao, Zeng, Jiaowan, Luo
exaly   +3 more sources

Melnikov Functions in Quadratic Perturbations of Generalized Lotka–Volterra Systems

Journal of Dynamical and Control Systems, 2015
The author presents a detailed analysis of Melnikov functions which arise in quadratic perturbations of generalized Lotka-Volterra vector fields with the first integral \(x^{\alpha}y^{\beta}(1-x-y)\) and, in particular, proves that the maximal number of limit cycles in the generic case is equal to 2 and in the Hamiltonian triangle case is equal to 3.
Henryk Zołądek
exaly   +2 more sources

Expansion coefficients and their relation for Melnikov functions near polycycles

Journal of Differential Equations
Assuming a particular condition, the authors present novel results concerning expansion coefficients and their interrelation within the first-order Melnikov functions. These results, derived for m-polycycles (where m is a positive integer) with hyperbolic saddles, lead to a comprehensive bifurcation theory for predicting limit cycles near these ...
Maoan Han
exaly   +2 more sources

Melnikov function and Poincaré map

Applied Mathematics and Mechanics, 1988
The following ODE is investigated: \(x''+g(x)=\epsilon \mu f(x,x')+\epsilon \delta h(x,x',\omega t)\) where \(h(x,x',\omega t)\) is periodic in t. A relationship between the Melnikov function and the Poincaré mapping is established and a new proof for the Melnikov method is given. Some illustrative examples are also presented.
Xu, Zhenyuan, Li, Li
openaire   +1 more source

Equivalence of the Melnikov Function Method and the Averaging Method

Qualitative Theory of Dynamical Systems, 2015
In this paper, the authors study the problem of equivalence between the Melnikov method and the averaging method for studying the number of limit cycles which can bifurcate from the period annulus of planar analytic differential systems.
Maoan Han, Valery G. Romanovski, Xiang Zhang   +2 more
openaire   +1 more source

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

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