A biographical sketch of Robert J. Sacker

Sell, George R.

doi:10.1080/10236190903058950

Cited by 1 publication

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“…It is well known that there exists at least one entire trajectory through each point a ∈ A that is contained in A, i.e., there exists a mapping χ : R → X such that χ(t + s) = ϕ(s, χ(t)) for all t ∈ R and s ∈ R + , with χ(0) = a and χ(t) ∈ A for all t ∈ R. Positively invariant sets are often encountered from absorbing sets, which is a first step to prove the existence of an attractor. Negatively invariant sets are not discussed directly so often in the literature, e.g., [13], but are present in many unstable situations such as following the loss of stability in a bifurcation or on an unstable manifold about an equilibrium point as well as in discretization and persistency problems, e.g., [2,3,4,6,12,17,18,19].…”

Section: Introductionmentioning

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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