Nonautonomous bifurcation of bounded solutions I: A
Lyapunov-Schmidt approach

Pötzsche, Christian

doi:10.3934/dcdsb.2010.14.739

Cited by 33 publications

(49 citation statements)

References 46 publications

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“…8], [54,Sect. 5.12] or [43]). • Provided S 1 looses its invertibility but remains onto, then the surjective implicit function theorem (cf.…”

Section: Nonautonomous Hyperbolicitymentioning

confidence: 99%

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Fine Structure of the Dichotomy Spectrum

Pötzsche

2012

Integr. Equ. Oper. Theory

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The dichotomy spectrum is a crucial notion in the theory of dynamical systems, since it contains information on stability and robustness properties. However, recent applications in nonautonomous bifurcation theory showed that a detailed insight into the fine structure of this spectral notion is necessary. On this basis, we explore a helpful connection between the dichotomy spectrum and operator theory. It relates the asymptotic behavior of linear nonautonomous difference equations to the point, surjectivity and Fredholm spectra of weighted shifts. This link yields several dynamically meaningful subsets of the dichotomy spectrum, which not only allows to classify and detect bifurcations, but also simplifies proofs for results on the long term behavior of difference equations with explicitly time-dependent right-hand side.

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“…8], [54,Sect. 5.12] or [43]). • Provided S 1 looses its invertibility but remains onto, then the surjective implicit function theorem (cf.…”

Section: Nonautonomous Hyperbolicitymentioning

confidence: 99%

“…[43,44,46]). In doing so, we observe that the spectral theory for equations on the whole integer axis is richer than its previous counterpart dealing with semiaxes.…”

Section: Dichotomy Spectramentioning

confidence: 99%

Fine Structure of the Dichotomy Spectrum

Pötzsche

2012

Integr. Equ. Oper. Theory

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“…There are at present few general, theoretical results about bifurcations in nonautonomous dynamical systems, e.g., [7,8,10,11,14,15,16]. The above results allow us to make a preliminary investigation to show that what could be considered to be a bifurcation has occurred.…”

Section: Bifurcation In a Nonautonomous Systemmentioning

confidence: 99%

Negatively Invariant Sets and Entire Solutions

Kloeden

Marín-Rubio

2010

J Dyn Diff Equat

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Negatively invariant compact sets of autonomous and nonautonomous dynamical systems on a metric space, the latter formulated in terms of processes, are shown to contain a strictly invariant set and hence entire solutions. For completeness the positively invariant case is also considered. Both discrete and continuous time systems are considered. In the nonautonomous case, the various types of invariant sets are in fact families of subsets of the state space that are mapped onto each other by the process. A simple example shows the usefulness of the result for showing the occurrence of a bifurcation in a nonautonomous system.

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“…The required Fredholm theory is provided by means of dynamical properties for the variational equation along a nonhyperbolic reference solution. This enabled us to derive nonautonomous versions of the classical fold, transcritical and pitchfork bifurcation patterns in [33]. Furthermore, a crossing curve bifurcation (generalizing transcritical and pitchfork patterns) and a degenerate fold bifurcation have been obtained in [35] on the basis of abstract analytical results due to [27].…”

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confidence: 98%

“…We considered this as motivation and starting point to investigate the bifurcation behavior of bounded entire solutions in [33,35] using tools from analytical branching theory (cf. [17] or [43,Chapter 8]), like Lyapunov-Schmidt reduction.…”

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confidence: 99%

Persistence and imperfection of nonautonomous bifurcation patterns

Pötzsche

2011

Journal of Differential Equations

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For nonautonomous dynamical systems a bifurcation can be understood as topological change in the set of bounded entire solutions to a given time-dependent evolutionary equation. Following this idea, a Fredholm theory via exponential dichotomies on semiaxes enables us to employ tools from analytical branching theory yielding nonautonomous versions of fold, transcritical and pitchfork patterns. This approach imposes the serious hypothesis that precise quantitative information on the dichotomies is requiredan assumption hard to satisfy in applications. Thus, imperfect bifurcations become important. In this paper, we discuss persistence and changes in the previously mentioned bifurcation scenarios by including an additional perturbation parameter. While the unperturbed case captures the above bifurcation patterns, we obtain their unfolding and therefore the local branching picture in a whole neighborhood of the system. Using an operator formulation of parabolic differential, Carathéodory differential and difference equations, this will be achieved on the basis of recent abstract analytical techniques due to Shi (1999) and Liu, Shi and Wang (2007).

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Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach

Cited by 33 publications

References 46 publications

Fine Structure of the Dichotomy Spectrum

Fine Structure of the Dichotomy Spectrum

Negatively Invariant Sets and Entire Solutions

Persistence and imperfection of nonautonomous bifurcation patterns

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