The number of periodic solutions of 2-dimensional periodic systems

Nakajima, Fumio; Seifert, George

doi:10.1016/0022-0396(83)90005-0

Cited by 20 publications

(14 citation statements)

References 2 publications

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“…The number of periodic solutions of every integer period is divisible by twice the period if all periodic solutions are hyperbolic (Levinson [3], Massera [4]), and is finite iff is real analytic in x and the trace of the Jacobian matrix off is negative (Nakajima and Seifert [5]). …”

Section: Introductionmentioning

confidence: 99%

The number and linking of periodic solutions of periodic systems

Matsuoka

1982

Invent Math

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Section: Introductionmentioning

confidence: 99%

The number and linking of periodic solutions of periodic systems

Matsuoka

1982

Invent Math

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“…The finitness of the number of 2T -periodic solutions follows from the Nakajima-Seifert theorem [22] upon the following observation. The result [22,Theorem,p. 431] formally assumes that the system under consideration is dissipative, that is not granted in our case.…”

Section: The Gause Model With Negative Divergencementioning

confidence: 99%

“…431] formally assumes that the system under consideration is dissipative, that is not granted in our case. However, the only fact that is used in the proof in [22] out of dissipativity is that the set of 2T -periodic solutions is bounded 2 . Moreover this set should not necessary be the set of all 2T -periodic solutions, but some bounded set of 2T -periodic solutions of interest isolated from other 2T -periodic solutions, which we do have in (0, ∞) × (0, ∞) according to lemma 2.3.…”

Section: The Gause Model With Negative Divergencementioning

confidence: 99%

Topological degree in the generalized Gause prey–predator model

Makarenkov

2014

Journal of Mathematical Analysis and Applications

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We consider a generalized Gause prey-predator model with T -periodic continuous coefficients. In the case where the Poincaré map P over time T is well defined, the result of the paper can be explained as follows: we locate a subset U of R 2 such that the topological degree d(I − P, U) equals to +1. The novelty of the paper is that the later is done under only continuity and (some) monotonicity assumptions for the coefficients of the model. A suitable integral operator is used in place of the Poincaré map to cope with possible non-uniqueness of solutions. The paper, therefore, provides a new framework for studying the generalized Gause model with functional differential perturbations and multi-valued ingredients.

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“…Conditions to guarantee that the fixed points considered in Theorem 6.2 be simple were proposed in [33] and [41]; they made essential use of the differentiability of the right-hand sides of the system (1.2), which was assumed to be in (A P ). A development of Theorem 6.2 may be obtained based on [34].…”

Section: Periodic Solutions Of Perturbed Autonomous Systems 29mentioning

confidence: 99%

Poincaré index and periodic solutions of perturbed autonomous systems

Makarenkov

2009

Trans. Moscow Math. Soc.

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Abstract. Classical conditions for the bifurcation of periodic solutions in perturbed auto-oscillating and conservative systems go back to Malkin and Mel'nikov, respectively. These authors' papers were based on the Lyapunov-Schmidt reduction and the implicit function theorem, which lead to the requirement that both the cycles and the zeros of the bifurcation functions be simple. In this paper a geometric approach is put forward which does not assume these requirements, but imposes a certain condition on the Poincaré index of a generating cycle with respect to some auxiliary vector field. The approach is based on calculating the topological degree of the Poincaré operator of the perturbed system with respect to interior and exterior neighbourhoods of a generating cycle, as a consequence of which the conclusion of the main theorem guarantees bifurcation of a certain number of periodic solutions towards the interior of the cycle, and of a certain number of periodic solutions towards the exterior of the cycle. Concrete examples are given, where this approach either establishes bifurcation of a greater number of periodic solutions compared with the known classical results, or provides additional information on the location of these solutions.

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The number of periodic solutions of 2-dimensional periodic systems

Cited by 20 publications

References 2 publications

The number and linking of periodic solutions of periodic systems

The number and linking of periodic solutions of periodic systems

Topological degree in the generalized Gause prey–predator model

Poincaré index and periodic solutions of perturbed autonomous systems

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