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Homoclinic chaos and the Poincaré-Melnikov method

In this thesis is to describe the use of the Poincaré-Melnikov method in the detection of homoclinic phenomena, and hence chaotic dynamics. After a short review of the theory of dynamical system, it is introduced the Poincaré-Melnikov method and its extension to the case of heteroclinic orbits.
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DOCOMOMO : First international conference, September 12-15, 1990, Eindhoven [PDF]

, 1991

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High order Melnikov method: Pendulums

Journal of Differential Equations, 2022
This paper is concerned with the computation of the second order Melnikov function for periodically perturbed pendulum equations given by \[ \frac{dx}{dt}=y,\qquad \frac{dy}{dt}=-\sin x +\varepsilon\cos^2(x/2)\cdot P(t), \] where \(P(t)\) is a periodic function in \(t\) and \(\varepsilon\) is a small parameter.
Oksasoglu, Ali, Wang, Qiudong
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Higher-order Melnikov method

Applied Mathematics and Mechanics, 1991
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Guo You-zhong, Liu Zeng-rong, Jiang Xia-mei, Han Zhi-bin   +3 more
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Melnikov method for parabolic orbits

NoDEA : Nonlinear Differential Equations and Applications, 2003
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Casasayas, Josefina, Faisca, Patrícia, Nunes, Ana   +2 more
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High order Melnikov method: Theory and application

Journal of Differential Equations, 2019
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Chen, Fengjuan, Q. D. Wang, Qiudong
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A numerical implementation of Melnikov's method

Physics Letters A, 1987
Abstract A numerical implementation of Melnikov's method is proposed. The procedure is based on the convergence of the integral and the uniqueness of the boundary of the horseshoe region in the parameter space under certain conditions. Several examples are calculated.
F.H. Ling, G.W. Bao
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A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

International Journal of Bifurcation and Chaos, 1992
This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare ...
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EXPONENTIAL DICHOTOMIES, HOMOCLINIC ORBITS AND METHODS OF MELNIKOV

Acta Mathematica Scientia, 1996
The main goal of this paper is to investigate the existence of transversal homoclinic orbits of the perturbed equation \[ dx/dt= g(x)+ \varepsilon h(t,x,\varepsilon).\tag{\(*\)} \] The authors construct a Melnikov-type function which yields transversal homoclinic orbits also when the perturbation does not depend on \(t\).
Zeng, Weiyao, Qian, Xiangzheng, Wang, Zhicheng   +2 more
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Melnikov's method and averaging

Celestial Mechanics, 1982
Consider the differential equation \(\dot x=f^ 0(x)+\epsilon f^ 1(\omega t,x;\epsilon)\), \(x\in D\subset R^ n\) where \(f^ 0\) and \(f^ 1\) are sufficiently smooth, \(f^ 1\) is \(2\pi\)-periodic in \(\omega\) t and \(\epsilon\) is a small positive parameter. Let the unperturbed system \(\dot x=f^ 0(x)\) have a hyperbolic point.
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