Li–Yorke chaos in nonautonomous Hopf bifurcation patterns—I

Abstract

We analyze the characteristics of the global attractor of a type of dissipative nonautonomous dynamical systems in terms of the Sacker and Sell spectrum of its linear part. The model gives rise to a pattern of nonautonomous Hopf bifurcation which can be understood as a generalization of the classical autonomous one. We pay special attention to the dynamics at the bifurcation point, showing the possibility of occurrence of Li-Yorke chaos in the corresponding attractor and hence of a high degree of unpredictability.

Non-autonomous scalar linear-dissipative and purely dissipative parabolic PDEs over a compact base flow

“…Besides, if P does not contain any minimal and periodic subsets, then following the methods mentioned in Núñez and Obaya [23] it is possible to find a map a ∈ C(P ) with P a dν = 0 such that the family of problems (3.1) over P with linear coefficient h(p•t, x) + a(p•t) has a global attractor which is fiber-chaotic in measure with respect to ν in the sense of Li-Yorke. If ν 1 and ν 2 are two ergodic measures on P with null Lypunov exponent and supp(ν 1 )∩supp(ν 2 ) = ∅, the previous construction is congruent in the sense that it is possible to choose the same map a ∈ C(P ) for both measures.…”

Generalized Pitchfork Bifurcations in D-Concave Nonautonomous Scalar Ordinary Differential Equations

“…The interest that the description of nonautonomous bifurcation patterns arouses in the scientific community has increased significantly in recent years, as evidenced by the works [1], [2], [3], [6], [11], [12], [16], [17], [20], [21], [24,25], [27,28], [30,31], [32], and references therein. This paper constitutes an extension of the work initiated in [11], were we describe several possibilities for the global bifurcation diagrams of certain types of one-parametric variations of a coercive equation.…”