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# Department of Mathematical Sciences

# MATH3091 Dynamical Systems III

Dynamical systems is the mathematical study of systems which evolve in time. One classical example is Newtonian dynamics, but the applicability is in fact much wider than this.

This course mainly deals with systems described by (coupled) ordinary differential equations. We start by studying the local behaviour or solutions, both in time and in the neighbourhoods of fixed points. Although one might think that such systems could be solved in closed form with sufficient effort, Poincaré realised a century ago that there are fundamental obstacles to global exact solvability and that most dynamical systems are in fact unsolvable in closed form for all time. This leads us to develop methods to study the global qualitative behaviour of such systems, finishing with a brief introduction to chaotic dynamical systems.

## Outline of Course

Aim: To provide an introduction to modern analytical methods for nonlinear ordinary differential equations in real variables.

### Term 1

• Introduction: Smooth direction fields in phase space. Existence and uniqueness theorem and initial-value dependence of trajectories. Orbits. Phase portraits. Equilibrium and periodic solutions. Orbital derivative, first integrals.
• Linear autonomous systems: Classification of linear systems in two and higher dimensions. The exponential map.
• Nonlinear systems near equilibrium: Hyperbolic fixed points, stable and unstable manifolds. Stable-manifold theorem, Hartman-Grobman theorem.
• Stability of fixed points: Definitions of stability, orbital derivative, first integrals, Lyapounov functions and stability theorems.

### Term 2

• Local bifurcations: Classification for 1d systems, some 2d examples, robustness of bifurcations, example of a global bifurcation
• Orbits and limit sets: Omega-limit sets, Poincare-Bendixson theorem, index theorem, absorbing sets and limit cycles
• Lorenz system and introduction to chaos

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.