Computing Covariant Lyapunov Vectors in Hilbert spaces

<p style='text-indent:20px;'>Covariant Lyapunov Vectors (CLVs) are intrinsic modes that describe long-term linear perturbations of solutions of dynamical systems. With recent advances in the context of semi-invertible multiplicative ergodic theorems, existence of CLVs has been proved for various infinite-dimensional scenarios. Possible applications include the derivation of coherent structures via transfer operators or the stability analysis of linear perturbations in models of increasingly higher resolutions.</p><p style='text-indent:20px;'>We generalize the concept of Ginelli's algorithm to compute CLVs in Hilbert spaces. Our main result is a convergence theorem in the setting of [<xref ref-type="bibr" rid="b19">19</xref>]. The theorem relates the speed of convergence to the spectral gap between Lyapunov exponents. While the theorem is restricted to the above setting, our proof requires only basic properties that are given in many other versions of the multiplicative ergodic theorem.</p>

Information Gain in Event Space Reflects Chance and Necessity Components of an Event

Information flow for occurrences in phase space can be assessed through the application of the Lyapunov characteristic exponent (multiplicative ergodic theorem), which is positive for non-linear systems that act as information sources and is negative for events that constitute information sinks. Attempts to unify the reversible descriptions of dynamics with the irreversible descriptions of thermodynamics have replaced phase space models with event space models. The introduction of operators for time and entropy in lieu of traditional trajectories has consequently limited—to eigenvectors and eigenvalues—the extent of knowable details about systems governed by such depictions. In this setting, a modified Lyapunov characteristic exponent for vector spaces can be used as a descriptor for the evolution of information, which is reflective of the associated extent of undetermined features. This novel application of the multiplicative ergodic theorem leads directly to the formulation of a dimension that is a measure for the information gain attributable to the occurrence. Thus, it provides a readout for the magnitudes of chance and necessity that contribute to an event. Related algorithms express a unification of information content, degree of randomness, and complexity (fractal dimension) in event space.

Notes on the multiplicative ergodic theorem

The Oseledets multiplicative ergodic theorem is a basic result with numerous applications throughout dynamical systems. These notes provide an introduction to this theorem, as well as subsequent generalizations. They are based on lectures at summer schools in Brazil, France, and Russia.

Conditions for equality between Lyapunov and Morse decompositions

Let$Q\rightarrow X$be a continuous principal bundle whose group$G$is reductive. A flow${\it\phi}$of automorphisms of$Q$endowed with an ergodic probability measure on the compact base space$X$induces two decompositions of the flag bundles associated to$Q$: a continuous one given by the finest Morse decomposition and a measurable one furnished by the multiplicative ergodic theorem. The second is contained in the first. In this paper we find necessary and sufficient conditions so that they coincide. The equality between the two decompositions implies continuity of the Lyapunov spectra under perturbations leaving unchanged the flow on the base space.