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Lecture

Stability Analysis: Linear Systems

In course

PHYS-460: Nonlinear dynamics, chaos and complex systems

The course provides students with the tools to approach the study of nonlinear systems and chaotic dynamics. Emphasis is given to concrete examples and numerical applications are carried out during the exercise sessions. ...

Description

This lecture covers the stability analysis of linear systems, focusing on the influence of eigenvalues and eigenvectors on the system's behavior. It discusses the concept of stable manifolds and how they determine the system's stability. The lecture also delves into phase portraits, bifurcations, and the nature of stationary points in the system.

Instructor

Paolo Ricci

Paolo Ricci earned his masters degree in nuclear engineering at the Politecnico di Torino, Turin (Italy) in 2000. His doctoral studies were conducted at the Los Alamos National Laboratory, with focus on kinetic simulation of magnetic reconnection in the Earth's magnetotail. He spent two-and-a-half years as a postdoctoral researcher at Dartmouth College's Department of Physics and Astronomy, where he worked on gyrokinetic simulations of the Z pinch. He joined the EPFL's Swiss Plasma Center (SPC), as a EURATOM fellow in 2006, was named Tenure Track Assistant Professor in June 2010, and Associate Professor in August 2016. He is at the head of the SPC theory group. Paolo Ricci is the recipient of the 2016 Section de Physique Teaching Prize and of the 2021 Craie d'Or award from the EPFL physics bachelor students.

Official source

Ontological neighbourhood

Systems science and engineering

Systems theory: Chaos theory

Related concepts (32)

Eigenvalues and eigenvectors

In linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor. Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear.

Eigendecomposition of a matrix

In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem. Eigenvalue, eigenvector and eigenspace A (nonzero) vector v of dimension N is an eigenvector of a square N × N matrix A if it satisfies a linear equation of the form for some scalar λ.

Eigenvalue algorithm

In numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors. Eigenvalues and eigenvectors and Generalized eigenvector Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.

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.

Linear system

In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often be modeled by linear systems.

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