Results 181 to 190 of about 27,933 (207)

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 ...
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Melnikov functions for singularly perturbed ordinary differential equations

Nonlinear Analysis: Theory, Methods & Applications, 1992
The author studies the homoclinic and periodic orbits for a system of singularly perturbed equations of the form \(\varepsilon y'=g(x,y)+\varepsilon f_ 1(x,y,t)\), \(x'=f_ 0(x,y)+\varepsilon f_ 3(x,y,t)\), \(y\in R^ m\), \(x\in R^ n\), where \(f_ 1\), \(f_ 3\) are \(T\)-periodic in \(t\), and \(g(x,0)=0\).
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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
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Evans' Function, Melnikov's Integral, and Solitary Wave Instabilities

1993
Publisher Summary This chapter describes the Evans' function, Melnikov's integral and solitary wave instabilities. The chapter mentions recent results on (1) the method for detecting the eigenvalues of systems of ordinary differential equations with asymptotically constant coefficients, (2) applications of this method to the detection of ...
Robert L. Pego, Michael I. Weinstein
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A FAST-MANIFOLD APPROACH TO MELNIKOV FUNCTIONS FOR SLOWLY VARYING OSCILLATORS

International Journal of Bifurcation and Chaos, 1996
A new approach to obtaining the Melnikov function for homoclinic orbits in slowly varying oscillators is proposed. The present method applies the usual two-dimensional Melnikov analysis to the “fast” dynamics of the system which lie on an invariant manifold. It is shown that the resultant Melnikov function is the same as that obtained in the usual way
Chen, Shyh-Leh, Shaw, Steven W.
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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 ...
Feng Liang, Maoan Han
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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.
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Reply to ‘‘Melnikov function and homoclinic chaos induced by weak perturbations’’

Physical Review E, 1993
For the case of weak perturbations, our theory is shown to reproduce exactly the results of Simiu and Frey [Phys. Rev. E 48, 3185 (1993)]. In the presence of weak noise, the two approaches yield different results. This can be traced to the neglect of diffusion effects in the Simiu-Frey theory; the inclusion of these effects, via an ensemble-averaged ...
, Bulsara, , Schieve, , Jacobs
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Sub-harmonic Melnikov function for a high-dimensional non-smooth coupled system

Chaos, Solitons and Fractals, 2022
RuiLan Tian
exaly  

Melnikov Functions and Homoclinic Orbits for Functional Differential Equations

2020
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