Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Harmonic Responses to Non-Harmonic Signals

In course

ME-523: Nonlinear Control Systems

Les systèmes non linéaires sont analysés en vue d'établir des lois de commande. On présente la stabilité au sens de Lyapunov, ainsi que des méthodes de commande géométrique (linéarisation exacte). Divers exemples illustrent la théorie (exercices papier cra ...

Description

This lecture covers the harmonic responses of systems to non-harmonic signals, focusing on distinct balance points, attractors, and trajectories in chaotic systems. It also discusses eigenvalues, eigenvectors, and stability analysis.

Instructor

Philippe Müllhaupt

Official source

Related concepts (32)

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 close enough to the attractor values remain close even if slightly disturbed. In finite-dimensional systems, the evolving variable may be represented algebraically as an n-dimensional vector. The attractor is a region in n-dimensional space.

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 initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization.

Complex system

A complex system is a system composed of many components which may interact with each other. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication systems, complex software and electronic systems, social and economic organizations (like cities), an ecosystem, a living cell, and ultimately the entire universe.

Rössler attractor

The Rössler attractor ˈrɒslər is the attractor for the Rössler system, a system of three non-linear ordinary differential equations originally studied by Otto Rössler in the 1970s. These differential equations define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor. Rössler interpreted it as a formalization of a taffy-pulling machine. Some properties of the Rössler system can be deduced via linear methods such as eigenvectors, but the main features of the system require non-linear methods such as Poincaré maps and bifurcation diagrams.

Lyapunov stability

Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point stay near forever, then is Lyapunov stable. More strongly, if is Lyapunov stable and all solutions that start out near converge to , then is said to be asymptotically stable (see asymptotic analysis).

Related lectures (39)

Chaos and Lyapunov Exponents: Analyzing Predictability

Covers Lyapunov exponents, chaos measurement, and perturbation analysis in dynamical systems.

Nonlinear Dynamics: Stability and Chaos

Explores fixed point stability, Lyapunov functions, Lotka-Volterra models, and nonlinear dynamics in complex systems.

Deterministic Chaos and Statistics

Explores the Lorenz System, sensitivity to initial conditions, chaotic systems, and topological mixing.

Chaos and Lyapunov Exponents: Properties and Dynamics

Covers chaos properties, focusing on Lyapunov exponents and their role in chaotic dynamics and predictability.

Attractors and Stability