Results 141 to 150 of about 22,008 (172)

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 \(g(x,0)=0\).
Michal Fečkan
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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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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
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Second order Melnikov function and its application

Physics Letters A, 1990
Abstract Based on Melnikov's method, the second order Melnikov function for the study of subharmonic and ultrasubharmonic orbits in a class of planar Hamiltonian systems is derived. Using this function the existence criterion of subharmonic and ultrasubharmonic orbits is set. A nonlinear oscillator subject to perturbation as example is discussed.
Zengrong Liu, Guoqing Gu
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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 \(E(\lambda)\).
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Melnikov functions and Bautin ideal

Qualitative Theory of Dynamical Systems, 2001
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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
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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
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Melnikov vector function for high-dimensional maps

Physics Letters A, 1996
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Finding More Limit Cycles Using Melnikov Functions

2012
In Chap. 9, an idea for finding more limit cycles is introduced, which combines the bifurcation of limit cycles from centers, homoclinic and heteroclinic loops. A generalized theorem is presented. In particular, two polynomial systems are studied. By using the theorems and results obtained in Chaps.
Maoan Han, Pei Yu
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