Timing of transients: quantifying reaching times and transient behavior in complex systems (original) (raw)

Extracting dynamics from threshold-crossing interspike intervals: Possibilities and limitations

Physical Review E, 2000

In this paper we estimate dynamical characteristics of chaotic attractors from sequences of thresholdcrossing interspike intervals, and study how the choice of the threshold level ͑which sets the equation of a secant plane͒ influences the results of the numerical computations. Under quite general conditions we show that the largest Lyapunov exponent can be estimated from a series of return times to the secant plane, even in the case when some of the loops of the phase space trajectory fail to cross this plane.

Metric for attractor overlap

Journal of Fluid Mechanics, 2019

We present the first general metric for attractor overlap (MAO) facilitating an unsupervised comparison of flow data sets. The starting point is two or more attractors, i.e. ensembles of states representing different operating conditions. The proposed metric generalizes the standard Hilbert-space distance between two snapshot-to-snapshot ensembles of two attractors. A reduced-order analysis for big data and many attractors is enabled by coarse graining the snapshots into representative clusters with corresponding centroids and population probabilities. For a large number of attractors, MAO is augmented by proximity maps for the snapshots, the centroids and the attractors, giving scientifically interpretable visual access to the closeness of the states. The coherent structures belonging to the overlap and disjoint states between these attractors are distilled by a few representative centroids. We employ MAO for two quite different actuated flow configurations: a two-dimensional wake ...

Temporal Dynamics

Assembly Time, 2023

This article delves into the fascinating realm of temporal dynamics within complex systems, examining key concepts such as absement, assembly rate, temporal coherence, dependencies, feedback, resilience, and emergence time. Drawing upon the principles of assembly time theory, we explore how these temporal aspects shape the behavior, evolution, and organization of complex systems. Through the lens of assembly time theory methods, we investigate the intricate interplay between time and system dynamics, shedding light on their underlying mechanisms and providing novel insights into their temporal complexities.

Singularities of transition processes in dynamical systems: Qualitative theory of critical delays

Electron. J. Diff. Eqns. Monograph 5, 2004, 2004

This monograph presents a systematic analysis of the singularities in the transition processes for dynamical systems. We study general dynamical systems, with dependence on a parameter, and construct relaxation times that depend on three variables: Initial conditions x, parameters k of the system, and accuracy e of the relaxation. We study the singularities of relaxation times as functions of (x,k) under fixed e, and then classify the bifurcations (explosions) of limit sets. We study the relationship between singularities of relaxation times and bifurcations of limit sets. An analogue of the Smale order for general dynamical systems under perturbations is constructed. It is shown that the perturbations simplify the situation: the interrelations between the singularities of relaxation times and other peculiarities of dynamics for general dynamical system under small perturbations are the same as for the Morse-Smale systems.

A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks

Entropy, 2021

The combination of network sciences, nonlinear dynamics and time series analysis provides novel insights and analogies between the different approaches to complex systems. By combining the considerations behind the Lyapunov exponent of dynamical systems and the average entropy of transition probabilities for Markov chains, we introduce a network measure for characterizing the dynamics on state-transition networks with special focus on differentiating between chaotic and cyclic modes. One important property of this Lyapunov measure consists of its non-monotonous dependence on the cylicity of the dynamics. Motivated by providing proper use cases for studying the new measure, we also lay out a method for mapping time series to state transition networks by phase space coarse graining. Using both discrete time and continuous time dynamical systems the Lyapunov measure extracted from the corresponding state-transition networks exhibits similar behavior to that of the Lyapunov exponent. In...

Can Lyapunov exponent predict critical transitions in biological systems?

Nonlinear Dynamics, 2017

Transitions from one dynamical regime to another one are observed in many complex systems, especially biological ones. It is possible that even a slight perturbation can cause such a transition. It is clear that this can happen to an object when it is close to a tipping point. There is a lot of interest in finding ways to recognize that a tipping point (in which a bifurcation occurs) is near. There is a possibility that in complex systems, a phenomenon known as "critical slowing down" may be used to detect the vicinity of a tipping point. In this paper, we propose Lyapunov exponents as an indicator of "critical slowing down."