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Perturbed normalizers and Melnikov functions
Journal of Mathematical Analysis and Applications, 2018Consider a \(C^{\infty}\) smooth family of real planar vector fields \(\left(X_\varepsilon\right)_\varepsilon\) of the form \[ X_\varepsilon (x) \, = \, X_0(x) + \varepsilon X_1(x) + \varepsilon^2 X_2(x) + \, \ldots \, + \varepsilon^n X_n(x) + \mathcal{O}\left(\varepsilon^{n+1}\right), \] for some integer \(n \geq 1\), \(x \in \mathbb{R}^2\) and ...
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Some properties of Melnikov functions near a cuspidal loop
Science China Mathematics, 2023Limit cycle bifurcations of a near-Hamiltonian system near a cuspidal loop is studied. A method using first-order Melnikov functions is employed. A general method to obtain the number of limit cycles near the cuspidal loop is presented. The results are exemplified on a Liénard-like system.
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MELNIKOV FUNCTIONS AND PERTURBATION OF A PLANAR HAMILTONIAN SYSTEM
Chinese Annals of Mathematics, 1999zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Equivalence of the Melnikov Function Method and the Averaging Method
Qualitative Theory of Dynamical Systems, 2015In 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.
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