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On the Equivalence of the Melnikov Functions Method and the Averaging Method
Qualitative Theory of Dynamical Systems, 2016zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Melnikov's method with applications
2009This thesis gives a detailed discussion of Melnikov's method, which is an analytical tool to study global bifurcations that occur in homoclinic or heteroclinic loops, or in one-parameter families of periodic orbits of a perturbed system. Basic results of the Melnikov theory relating the number, positions and multiplicities of the limit cycles by the ...
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A practical use of the Melnikov homoclinic method
Journal of Mathematical Physics, 2009Using cutoff functions and periodic extensions, we prove that the Melnikov homoclinic method gives a criterium to show that for a finite time interval [−T,T], with T arbitrarily large, the perturbed system is conjugated to a chaotic one for quite general classes of perturbation functions.
Castilho, César, Marchesin, Marcelo
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On some applications of Melnikov's method to chaos and subharmonics
Bulletin of the Australian Mathematical Society, 1996zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Melnikov’s method for a general nonlinear vibro-impact oscillator
Nonlinear Analysis: Theory, Methods & Applications, 2009The classical Melnikov's method is applied to a second order differential equation with impact effects. The equation is a perturbation of a Hamiltonian system with a homoclinic orbit to the origin. Then the first-order Melnikov function can be obtained analytically in the usual way. The method is applied to a double-well Duffing oscillator with impacts,
Xu, Wei, Feng, Jinqian, Rong, Haiwu
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Fundamental Theory of the Melnikov Function Method
2012Chapter 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
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A numerical study of the applicability of Melnikov's method
Physics Letters A, 1987Abstract Melnikov's method can be used to predict the appearance of the homoclinic and also the heteroclinic tangency. The applicability of the method is studied by using numerically calculated invariant manifolds of the Poincare map of a soft spring oscillator.
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Melnikov's method applied to the double pendulum
Zeitschrift f�r Physik B Condensed Matter, 1994Melnikov's method is applied to the planar double pendulum proving it to be a chaotic system. The parameter space of the double pendulum is discussed, and the integrable cases are identified. In the neighborhood of the integrable case of two uncoupled pendulums Melnikov's integral is evaluated using residue calculus.
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A Melnikov Method for Strongly Odd Nonlinear Oscillators
Japanese Journal of Applied Physics, 1998In this paper, explicit calculations that extend the applicability of the Melnikov method to include strongly odd nonlinear and large forcing amplitude oscillating systems, are presented. We consider the response of the strongly nonlinear oscillating system governed by an equation of motion containing a parameter ε which need not be small ...
Zheng-Ming Ge, Fu-Neng Ku
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Melnikov method for discontinuous planar systems
Nonlinear Analysis: Theory, Methods & Applications, 2007This paper deals with the existence of a homoclinic solution in planar systems with discontinuous right-hand side. In fact, these type of systems are more used in practical problems than the classical planar differential systems and this article serves to a generalization of the Melnikov function to these discontinuous right-hand side systems.
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