Invariant Sets in Quasiperiodically Forced Dynamical Systems

This paper addresses structures of state space in quasiperiodically forced dynamical systems. We develop a theory of ergodic partition of state space in a class of measure-preserving and dissipative flows, which is a natural extension of the existing theory for measure-preserving maps. The ergodic partition result is based on eigenspace at eigenvalue 0 of the associated Koopman operator, which is realized via time-averages of observables, and provides a constructive way to visualize a low-dimensional slice through a high-dimensional invariant set. We apply the result to the systems with a finite number of attractors and show that the time-average of a continuous observable is well-defined and reveals the invariant sets, namely, a finite number of basins of attraction. We provide a characterization of invariant sets in the quasiperiodically forced systems. A theoretical result on uniform boundedness of the invariant sets is presented. The series of theoretical results enables numerical analysis of invariant sets in the quasiperiodically forced systems based on the ergodic partition and time-averages. Using this, we analyze a nonlinear model of complex power grids that represents the short-term swing instability, named the coherent swing instability. We show that our theoretical results can be used to understand stability regions in such complex systems.Comment: 23 pages, 4 figure

Nonautonomous saddle-node bifurcations: random and deterministic forcing

We study the effect of external forcing on the saddle-node bifurcation pattern of interval maps. By replacing fixed points of unperturbed maps by invariant graphs, we obtain direct analogues to the classical result both for random forcing by measure-preserving dynamical systems and for deterministic forcing by homeomorphisms of compact metric spaces. Additional assumptions like ergodicity or minimality of the forcing process then yield further information about the dynamics. The main difference to the unforced situation is that at the critical bifurcation parameter, two alternatives exist. In addition to the possibility of a unique neutral invariant graph, corresponding to a neutral fixed point, a pair of so-called pinched invariant graphs may occur. In quasiperiodically forced systems, these are often referred to as 'strange non-chaotic attractors'. The results on deterministic forcing can be considered as an extension of the work of Novo, Nunez, Obaya and Sanz on nonautonomous convex scalar differential equations. As a by-product, we also give a generalisation of a result by Sturman and Stark on the structure of minimal sets in forced systems.Comment: 17 pages, 5 figure

Multistability and nonsmooth bifurcations in the quasiperiodically forced circle map

It is well-known that the dynamics of the Arnold circle map is phase-locked in regions of the parameter space called Arnold tongues. If the map is invertible, the only possible dynamics is either quasiperiodic motion, or phase-locked behavior with a unique attracting periodic orbit. Under the influence of quasiperiodic forcing the dynamics of the map changes dramatically. Inside the Arnold tongues open regions of multistability exist, and the parameter dependency of the dynamics becomes rather complex. This paper discusses the bifurcation structure inside the Arnold tongue with zero rotation number and includes a study of nonsmooth bifurcations that happen for large nonlinearity in the region with strange nonchaotic attractors.Comment: 25 pages, 22 colored figures in reduced quality, submitted to Int. J. of Bifurcation and Chaos, a supplementary website (http://www.mpipks-dresden.mpg.de/eprint/jwiersig/0004003/) is provide

Almost periodic structures and the semiconjugacy problem

The description of almost periodic or quasiperiodic structures has a long tradition in mathematical physics, in particular since the discovery of quasicrystals in the early 80's. Frequently, the modelling of such structures leads to different types of dynamical systems which include, depending on the concept of quasiperiodicity being considered, skew products over quasiperiodic or almost-periodic base flows, mathematical quasicrystals or maps of the real line with almost-periodic displacement. An important problem in this context is to know whether the considered system is semiconjugate to a rigid translation. We solve this question in a general setting that includes all the above-mentioned examples and also allows to treat scalar differential equations that are almost-periodic both in space and time. To that end, we study a certain class of flows that preserve a one-dimensional foliation and show that a semiconjugacy to a minimal translation flow exists if and only if a boundedness condition, concerning the distance of orbits of the flow to those of the translation, holds

Chaotic transport in the homoclinic and heteroclinic tangle regions of quasiperiodically forced two-dimensional dynamical systems

The authors generalize notions of transport in phase space associated with the classical Poincare map reduction of a periodically forced two-dimensional system to apply to a sequence of nonautonomous maps derived from a quasiperiodically forced two-dimensional system. They obtain a global picture of the dynamics in homoclinic and heteroclinic tangles using a sequence of time-dependent two-dimensional lobe structures derived from the invariant global stable and unstable manifolds of one or more normally hyperbolic invariant sets in a Poincare section of an associated autonomous system phase space. The invariant manifold geometry is studied via a generalized Melnikov function. Transport in phase space is specified in terms of two-dimensional lobes mapping from one to another within the sequence of lobe structures, which provides the framework for studying several features of the dynamics associated with chaotic tangles