This course provides an introduction to applied dynamical systems and the qualitative study of differential equations. Dynamical systems are abundant in the physical world and include for example, the motion of the planets, the weather, electrical circuits and the behaviour of living organisms. Some systems like planetary motion are regular and predictable, while others such as the weather are believed to be chaotic in the sense that even small discrepancies in the initial state will inevitably result in huge discrepancies over time.
This module builds upon the stage 3 ordinary differential equations module. Topics covered include Lyapunov stability, invariant manifolds, periodic orbits, bifurcations and chaos in dynamical systems. The motivation behind new concepts and their application to problems in science and engineering is emphasized.
Linear dynamical systems
Introduction and preparatory material. Linear versus nonlinear systems. Equilibria, diagonalization, multiple eigenvalues, stability, stable, unstable, and center subspaces. Nonhomogeneous systems. Classification of planar systems.
Solutions of nonlinear dynamical systems
Solutions of initial value problems, existence and uniqueness of solutions, continuous dependence on initial conditions and parameters.
Linearization methods for nonlinear dynamical systems
Linearization, invariant manifolds. Stable, unstable and center manifolds. Hartman-Grobman theorem.
Lyapunov stability theory for nonlinear dynamical systems
Lyapunov functions. Stability and instability theorems. Exponential stability.
Global theory of nonlinear dynamical systems
Periodic orbits. Limit cycles, attractors. Poincaré-Bendixon theorem. Poincaré maps.
Bifurcation theory for nonlinear dynamical systems
Bifurcations of vector fields. Saddle-node, transcritical, pitchfork and Hopf bifurcations. Stability under perturbations. Structural stability.
Sensitivity to initial conditions
Density of periodic orbits. Chaos and strange attractors. Linking of periodic orbits.
Lectures supported by problem-solving sessions.
|Module Content & Assessment