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The scaling behavior of the maximal Lyapunov exponent in chaotic systems with time-delayed feedback is investigated. For large delay times it has been shown that the delay-dependence of the exponent allows a distinction between strong and…
We investigate the structure of the invariant measure of space-time chaos by adopting an "open-system" point of view. We consider large but finite windows of formally infinite one-dimensional lattices and quantify the effect of the…
Bifurcations mark qualitative changes of long-term behavior in dynamical systems and can often signal sudden ("hard") transitions or catastrophic events (divergences). Accurately locating them is critical not just for deeper understanding…
We propose a mechanism which produces periodic variations of the degree of predictability in dynamical systems. It is shown that even in the absence of noise when the control parameter changes periodically in time, below and above the…
In fluids and plasmas with zonal flow reversed shear, a peculiar kind of transport barrier appears in the shearless region, one that is associated with a proper route of transition to chaos. Using a symplectic nontwist maps, which model…
We consider the model describing the vertical motion of a ball falling with constant acceleration on a wall and elastically reflected. The wall is supposed to move in the vertical direction according to a given periodic function $f$. We…
From a kinematical point of view, the geometrical information of hamiltonian chaos is given by the (un)stable directions, while the dynamical information is given by the Lyapunov exponents. The finite time Lyapunov exponents are of…
Random neural networks are dynamical descriptions of randomly interconnected neural units. These show a phase transition to chaos as a disorder parameter is increased. The microscopic mechanisms underlying this phase transition are unknown,…
System identification of complex and nonlinear systems is a central problem for model predictive control and model-based reinforcement learning. Despite their complexity, such systems can often be approximated well by a set of linear…
The spiraling of adjacent trajectories in chaotic dynamical systems can be characterized by distribution of local angular velocities of rotation of the displacement vector, which is governed by linearized equations of motion. This…
We describe a method for analyzing the phase space structures of Hamiltonian systems. This method is based on a time-frequency decomposition of a trajectory using wavelets. The ridges of the time-frequency landscape of a trajectory, also…
Lyapunov stability of a mechanical system means that the dynamic response stays bounded in an arbitrarily small neighborhood of a static equilibrium configuration under small perturbations in positions and velocities. This type of stability…
We address the issue of how to identify the equations of a largely unknown chaotic system from knowledge about its state evolution. The technique can be applied to the estimation of parameters that drift slowly with time. To accomplish…
It has long been suspected that the Global Navigation Satellite Systems exist in a background of complex resonances and chaotic motion; yet, the precise dynamical character of these phenomena remains elusive. Recent studies have shown that…
We provide Lyapunov-like characterizations of boundedness and convergence of non-trivial solutions for a class of systems with unstable invariant sets. Examples of systems to which the results may apply include interconnections of stable…
Li-Yorke chaos is a popular and well-studied notion of chaos. Several simple and useful characterizations of this notion of chaos in the setting of linear dynamics were obtained recently. In this note we show that even simpler and more…
By tracking the divergence of two initially close trajectories in phase space in an Eulerian approach to forced turbulence, the relation between the maximal Lyapunov exponent $\lambda$, and the Reynolds number $Re$ is measured using direct…
We study the spatial patterns formed by interacting biological populations or reacting chemicals under the influence of chaotic flows. Multiple species and nonlinear interactions are explicitly considered, as well as cases of smooth and…
We review recent progress in applying information- and computation-theoretic measures to describe material structure that transcends previous methods based on exact geometric symmetries. We discuss the necessary theoretical background for…
Advances in machine learning have revolutionized capabilities in applications ranging from natural language processing to marketing to health care. Here, we demonstrate the efficacy of machine learning in predicting chaotic behavior in…