Lecture

Dynamical Systems: Equilibrium Points and Stability

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Description

This lecture introduces dynamical systems, focusing on equilibrium points, stability, and phase plots. It covers the representation of dynamical systems, types of equilibrium points, and stability analysis using Lyapunov functions. The lecture also discusses the trajectory of a dynamical system, solution to ordinary differential equations, and the concept of phase plots. Various examples, including the pendulum system, are used to illustrate these concepts.

Instructor

Aude Billard

Expertise RoboticsMachine LearningArtificial IntelligenceHuman-Robot InteractionMechatronicsDynamical Systems ControlComputational Neuroscience Aude Billard is full professor and head of the LASA laboratory at the School of Engineering at the Swiss Institute of Technology Lausanne (EPFL), and the Director of the Swiss Innovation Booster on Robotics. Dr. Billard acts as the President of the IEEE Robotics and Automation Society (RAS), after serving in several roles in the administrative and executive committe

Official source

Ontological neighbourhood

Systems science and engineering

Systems theory: Chaos theory

Related concepts (32)

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).

Lyapunov function

In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state space Markov chains, usually under the name Foster–Lyapunov functions.

Phase portrait

In mathematics, a phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each set of initial conditions is represented by a different point or curve. Phase portraits are an invaluable tool in studying dynamical systems. They consist of a plot of typical trajectories in the phase space. This reveals information such as whether an attractor, a repellor or limit cycle is present for the chosen parameter value.

Dynamical systems theory

Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle.

Differential equation

In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

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