A Modified Poincaré Method for the Persistence of Periodic Orbits and Applications

Hale, Jack K.; Raugel, Geneviève

doi:10.1007/s10884-009-9155-4

Cited by 15 publications

(9 citation statements)

References 25 publications

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“…When the system is not smooth in finite time, regularity (in space or time) of the elements J can be very difficult to prove or could be even false. Note that regularity results are primordial in the theory of perturbations of invariant sets and in particular of periodic orbits, as shown in [24] (see also [25]). Numerous authors have shown regularity properties for J in the case of dynamical systems which are not smoothness in finite time.…”

Section: ) Then There Exists a Setmentioning

confidence: 92%

Global Attractor of Nonlocal Nonlinear Schrödinger Equation on R

Chao-sheng¹

2016

AAN

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This paper is devoted to the large time behavior and especially to the regularity of the global attractor of the dissipative 1D nonlinear Schrödinger equation with nonlocal integral term on R. We first prove that the existence of the global attractor Aγ in the strong topology of H 1 (R) and the existence of the exponential attractor M which contains the global attractor Aγ, are still finite dimensional, and attract the trajectories exponentially fast. We also show that the global attractor Aγ is regular, i.e., Aγ is included, bounded and compact in H 2 (R) assuming that the forcing term f (x) is of class H 2 (R). Furthermore we estimate the number of the determining modes for this equation. Moreover, we show that the solution trajectories and the global attractor of the nonlocal Schrödinger equation converge to those of the usual Schrödinger equation, as the coefficient of the nonlocal integral term goes to zero.

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Section: ) Then There Exists a Setmentioning

confidence: 92%

Global Attractor of Nonlocal Nonlinear Schrödinger Equation on R

Chao-sheng¹

2016

AAN

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“…Let p 0 ptq be a simple periodic solution of period ω 0 ą 0 of (1.1) for f " f 0 . Assume that sup tPr0,ω 0 q }p 0 ptq} L 8 ď m and sup tPr0,ω 0 q }∇p 0 ptq} L 8 ď m. The statement of the persistence of a simple periodic solution p f ptq near p 0 ptq with period ω f close to ω 0 and also of the uniqueness (up to a time translation) of this periodic solution, if f belongs to a small enough neighborhood of f 0 in C r pΩˆr´2m, 2msˆr´2m, 2ms d q, is a direct consequence of [31,Theorem 8.3.2]; it is proved by using the method of Poincaré sections and the implicit function theorem or the fixed point theorem of strict contraction (for further results in the case where the perturbations are less regular, see also [23] and [24]). One concludes like in the proof of the statement 1) by using the restriction mapping R of Section 2.1.…”

Section: Proofmentioning

confidence: 99%

Generic Transversality of Heteroclinic and Homoclinic Orbits for Scalar Parabolic Equations

2019

Self Cite

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In this paper, we consider the scalar reaction-diffusion equationson a bounded domain Ω Ă R d of class C 2,γ . We show that the heteroclinic and homoclinic orbits connecting hyperbolic equilibria and hyperbolic periodic orbits are transverse, generically with respect to f . One of the main ingredients of the proof is an accurate study of the singular nodal set of solutions of linear parabolic equations. Our main result is a first step for proving the genericity of Kupka-Smale property, the generic hyperbolicity of periodic orbits remaining unproved.

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“…First, the question, whether a nondegenerate time-periodic solution to a dissipative nonlinear wave equation is locally unique (up to time shifts in the autonomous case) and whether it depends smoothly on the system parameters, is much more delicate than for ODEs or parabolic PDEs (cf., e.g., [14,15]). One reason for that is the so-called loss of derivatives for hyperbolic PDEs.…”

Section: The Problemmentioning

confidence: 99%

Hopf Bifurcation for General 1D Semilinear Wave Equations with Delay

Kmit,

Recke

2021

J Dyn Diff Equat

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We consider boundary value problems for 1D autonomous damped and delayed semilinear wave equations of the type $$\begin{aligned} \partial ^2_tu(t,x)- a(x,\lambda )^2\partial _x^2u(t,x)= b(x,\lambda ,u(t,x),u(t-\tau ,x),\partial _tu(t,x),\partial _xu(t,x)), \; x \in (0,1) \end{aligned}$$ ∂ t 2 u ( t , x ) - a ( x , λ ) 2 ∂ x 2 u ( t , x ) = b ( x , λ , u ( t , x ) , u ( t - τ , x ) , ∂ t u ( t , x ) , ∂ x u ( t , x ) ) , x ∈ ( 0 , 1 ) with smooth coefficient functions a and b such that $$a(x,\lambda )>0$$ a ( x , λ ) > 0 and $$b(x,\lambda ,0,0,0,0) = 0$$ b ( x , λ , 0 , 0 , 0 , 0 ) = 0 for all x and $$\lambda $$ λ . We state conditions ensuring Hopf bifurcation, i.e., existence, local uniqueness (up to time shifts), regularity (with respect to t and x) and smooth dependence (on $$\tau $$ τ and $$\lambda $$ λ ) of small non-stationary time-periodic solutions, which bifurcate from the stationary solution $$u=0$$ u = 0 , and we derive a formula which determines the bifurcation direction with respect to the bifurcation parameter $$\tau $$ τ . To this end, we transform the wave equation into a system of partial integral equations by means of integration along characteristics and then apply a Lyapunov-Schmidt procedure and a generalized implicit function theorem. The main technical difficulties, which have to be managed, are typical for hyperbolic PDEs (with or without delay): small divisors and the “loss of derivatives” property. We do not use any properties of the corresponding initial-boundary value problem. In particular, our results are true also for negative delays $$\tau $$ τ .

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A Modified Poincaré Method for the Persistence of Periodic Orbits and Applications

Cited by 15 publications

References 25 publications

Global Attractor of Nonlocal Nonlinear Schrödinger Equation on R

Global Attractor of Nonlocal Nonlinear Schrödinger Equation on R

Generic Transversality of Heteroclinic and Homoclinic Orbits for Scalar Parabolic Equations

Hopf Bifurcation for General 1D Semilinear Wave Equations with Delay

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