Concept# Dynamical system

Summary

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it.
At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related publications (100)

Loading

Loading

Loading

Related people (67)

Related concepts (168)

Chaos theory

Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initi

Attractor

In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get

Phase space

In dynamical systems theory and control theory, a phase space or state space is a space in which all possible "states" of a dynamical system or a control system are represented, with each possib

Related courses (128)

ME-273: Introduction to control of dynamical systems

Cours introductif à la commande des systèmes dynamiques. On part de quatre exemples concrets et on introduit au fur et à mesure un haut niveau d'abstraction permettant de résoudre de manière unifiée les problèmes d'asservissement et de régulation, en particulier les questions de stabilité.

ME-221: Dynamical systems

Provides the students with basic notions and tools for the analysis of dynamic systems. Shows them how to develop mathematical models of dynamic systems and perform analysis in time and frequency domains.

ME-326: Control systems and discrete-time control

Ce cours inclut la modélisation et l'analyse de systèmes dynamiques, l'introduction des principes de base et l'analyse de systèmes en rétroaction, la synthèse de régulateurs dans le domain fréquentiel et dans l'espace d'état, et la commande de systèmes discrets avec une approche polynomiale.

Related units (42)

Related lectures (306)

Coupled dynamical systems are omnipresent in everyday life. In general, interactions between
individual elements composing the system are captured by complex networks. The latter
greatly impact the way coupled systems are functioning and evolving in time. An important
task in such a context, is to identify the most fragile components of a system in a fast and
efficient manner. It is also highly desirable to have bounds on the amplitude and duration
of perturbations that could potentially drive the system through a transition from one equi-
librium to another. A paradigmatic model of coupled dynamical system is that of oscillatory
networks. In these systems, a phenomenon known as synchronization where the individual
elements start to behave coherently may occur if couplings are strong enough. We propose
frameworks to assess vulnerabilities of such synchronous states to external perturbations. We
consider transient excursions for both small-signal response and larger perturbations that can
potentially drive the system out of its initial basin of attraction.
In the first part of this thesis, we investigate the robustness of complex network-coupled
oscillators. We consider transient excursions following external perturbations. For ensemble
averaged perturbations, quite remarkably we find that robustness of a network is given by
a family of network descriptors that we called generalized Kirchhoff indices and which are
defined from extensions of the resistance distance to arbitrary powers of the Laplacian matrix
of the system. These indices allow an efficient and accurate assessment of the overall vulnera-
bility of an oscillatory network and can be used to compare robustness of different networks.
Moreover, a network can be made more robust by minimizing its Kirchhoff indices. Then for
specific local perturbations, we show that local vulnerabilities are captured by generalized
resistance centralities also defined from extensions of the resistance distance. Most fragile
nodes are therefore identified as the least central according to resistance centralities. Based on
the latter, rankings of the nodes from most to least vulnerable can be established. In summary,
we find that both local vulnerabilities and global robustness are accurately evaluated with
resistance centralities and Kirchhoff indices. Moreover, the framework that we define is rather
general and may be useful to analyze other coupled dynamical systems.
In the second part, we focus on the effect of larger perturbations that eventually lead the sys-
tem to an escape from its initial basin of attraction. We consider coupled oscillators subjected
to noise with various amplitudes and correlation in time. To predict desynchronization and
transitions between synchronous states, we propose a simple heuristic criterion based on the
distance between the initial stable fixed point and the closest saddle point. Surprisingly, we
find numerically that our criterion leads to rather accurate estimates for the survival probability and first escape time. Our criterion is general and may be applied to other dynamical
systems.