Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov–Schmidt reduction

“…Proof. Expression (4) was obtained in [6] by using the Faá di Bruno's formula for the L-th derivative of a composite function. This claim follows by applying the version of Faá di Bruno's formula in terms of Bell polynomials (see [6,22]).…”

Section: Claim 1 the Bifurcation Functionmentioning

confidence: 99%

Bifurcations from families of periodic solutions in piecewise differential systems

Llibre

Novaes

Rodrigues

2020

Physica D: Nonlinear Phenomena

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“…This result has been used for studying bifurcation of periodic orbits into the Lorenz and FitzHugh-Nagumo system [14]. In [15] this formulation is presented in a more general way. Assume m < n, and let π : R m × R n−m → R m and π ⊥ : R m × R n−m → R n−m denote the projections onto the first m coordinates and onto the last n − m coordinates, respectively.…”

Section: The Averaging Theorymentioning

confidence: 99%

Zero-Hopf bifurcations in 3-dimensional differential systems with no equilibria

Cândido

Llibre

2018

Mathematics and Computers in Simulation

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Recently sixteen 3-dimensional differential systems exhibiting chaotic motion and having no equilibria have been studied, and it has been graphically observed that these systems have a period-doubling cascade of periodic orbits providing a route to chaos. Here using new results on the averaging theory we prove that these systems exhibit, for some values of their parameters different to the ones having chaotic motion, either a zero-Hopf or a Hopf bifurcation, and graphically we observed that the periodic orbit starting in those bifurcations is at the beginning of the mentioned period-doubling cascade.

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“…It is well known that the existence problem of periodic solutions is one of center topics in the qualitative theory of deterministic differential equations for its significance in the physical science [15]. There has been a large amount of work (see for example [4,8,27] and the references therein). However, for SDEs, the existence of periodic solutions is thought to be a challenging problem.…”

Section: Introductionmentioning

confidence: 99%

Periodic solutions for SDEs through upper and lower solutions

Ji¹,

Yang²,

Yong³

2020

Discrete &Amp; Continuous Dynamical Systems - B

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We study a kind of better recurrence than Kolmogorov's one: periodicity recurrence, which corresponds periodic solutions in distribution for stochastic differential equations. On the basis of technique of upper and lower solutions and comparison principle, we obtain the existence of periodic solutions in distribution for stochastic differential equations (SDEs). Hence this provides an effective method how to study the periodicity of stochastic systems by analyzing deterministic ones. We also illustrate our results.In the present paper, we will touch the problem. We find that the answer will be affirmative if the ODE has upper and lower solutions. This conclusion is somewhat unexpected, because no additional condition is added to the diffusion term besides the usual one. We believe that it is best possible to pose periodic solutions in distribution. As one pointed out ( see, for example, [21]), it is impossible to obtain periodic solutions in probability or moment for SDEs due to the effects of diffusion.Let us recall that there are many topological and analytic methods, such as degree theory, fixed point theorems in studying the existence of periodic solutions of deterministic differential equations. But for SDEs, these nonlinear methods do not work in general, due to lack of compactness. Khasminskii [14] defined periodic solutions in the sense of periodic Markov process. Recently, Ji et al. [11] studied periodic probability solutions to be periodic analogs of stationary measures for stationary Fokker-Planck equations. Chen et al. [5] gave a criterion analogous to Halanay's criterion to prove the existence of periodic solutions in distribution. Liu and Sun [19] established the existence of solutions which are almost automorphic in distribution for some semilinear SDEs with Lévy noise. Liu and Wang [20] obtained almost periodicity in distribution by Favard separation method. Tudor [28] proved the almost periodicity of the one-dimensional distributions of solutions under some hypotheses. Prato and Tudor [24] showed the existence of periodic and almost periodic solutions in distribution of semilinear stochastic equations on a separable Hilbert space. Zhao and Zheng [29] and Feng and Zhao [6, 7] made some interesting investigation on periodic solutions for SDEs in a kind of local periodicity.Additionally, there are a lot literature about monotone methods and comparison arguments in deterministic dynamical systems (see, for example, [26]). Especially, the upper and lower solutions method is an effective tool in dealing with periodic solutions [13,16,17,18,25]. In this paper, on the basis of upper and lower solutions, using stochastic comparison technique we try to prove the existence of periodic solutions in distribution. Of course comparison principle [1,10,23] is also powerful to investigate dynamics of nonlinear systems. However, to our knowledge, it seems the first time to give periodic solutions in distribution for SDEs by combining these technique. Therefore this paper provides some way to tackle the existence o...

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Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov–Schmidt reduction

Cited by 34 publications

References 20 publications

Bifurcations from families of periodic solutions in piecewise differential systems

Bifurcations from families of periodic solutions in piecewise differential systems

Zero-Hopf bifurcations in 3-dimensional differential systems with no equilibria

Periodic solutions for SDEs through upper and lower solutions

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