On the Hopf-zero bifurcation of the Michelson system

Llibre, Jaume; Zhang, Xiang

doi:10.1016/j.nonrwa.2010.10.019

Cited by 25 publications

(18 citation statements)

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“…The averaging theory for computing periodic solutions of a differential system is one of the best analytical tools for the study of the periodic solutions, see for instance the papers [23,13,19,12,14,17,18,20,10,11,8].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

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Improving the averaging theory for computing periodic solutions of the differential equations

Llibre

Novaes

2014

Z. Angew. Math. Phys.

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Abstract. For m = 1, 2, 3, we consider differential systems of the formwherefunctions, and T -periodic in the first variable, being D an open subset of R n , and ε a small parameter. For such system we assume that the unperturbed system x = F 0 (t, x) has a k-dimensional manifold of periodic solutions with k ≤ n. We weaken the sufficient assumptions for studying the periodic solutions of the perturbed system when |ε| > 0 is sufficiently small.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…The same observation made about the value of m for the expression (16) is applied for the expressions (19) and (20). In order to apply Lemma 3 to function (18) we compute…”

Section: Proof Of Lemma 3 First Of All We Considermentioning

confidence: 99%

Improving the averaging theory for computing periodic solutions of the differential equations

Llibre

Novaes

2014

Z. Angew. Math. Phys.

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“…Then, for j"j > 0 sufficiently small, there exists a T periodic solution '.t, "/ of system (16) such that '.0, "/ ! p when " !…”

Section: Remark 10mentioning

confidence: 99%

Zero‐Hopf bifurcation in the FitzHugh–Nagumo system

Euzébio

Llibre

Vidal

2014

Math Methods in App Sciences

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Communicated by A. MiranvilleWe characterize the values of the parameters for which a zero-Hopf equilibrium point takes place at the singular points, namely, O (the origin), P C , and P in the FitzHugh-Nagumo system. We find two two-parameter families of the FitzHugh-Nagumo system for which the equilibrium point at the origin is a zero-Hopf equilibrium. For these two families, we prove the existence of a periodic orbit bifurcating from the zero-Hopf equilibrium point O. We prove that there exist three two-parameter families of the FitzHugh-Nagumo system for which the equilibrium point at P C and at P is a zero-Hopf equilibrium point. For one of these families, we prove the existence of one, two, or three periodic orbits starting at P C and P .

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“…Note that when c = 0 the Michelson system has a unique singular point at the origin with eigenvalues 0, ±i. In [18] it is proved that for c > 0 sufficiently small the Michelson system (7) has a Hopf-zero bifurcation at the origin for c = 0. Here we shall reproduce the short proof of [18] because it is necessary for proving our result:…”

Section: Proof Of Theorems 5 6 7 Andmentioning

confidence: 99%