Dynamical System

A dynamical system is a manifold \(M\) called the phase space (or state space) endowed with a family of smooth evolution functions \(\Phi(t): M \rightarrow M \) that for any evolution parameter \(t \in T\).

\(\Phi_x(t)\), trajectory through \(x\); \(\gamma_x\), orbit through \(x\).

Table: Classification of Dynamical Systems by the Evolution Variable

Time	Bi-directional	Non-negative
Real line	flow	semi-flow
Integers	map/cascade	semi-cascade

The study of dynamical systems is largely qualitative, i.e. on properties that do not change under smooth coordinate transformations.

Nonlinear dynamical systems are typically chaotic.

Ordinary Differential Equations

• ODE Concepts

Solutions

• 1-st order ODEs
• Singular solutions
• n-th order & system of ODEs
• Solution of some common ODEs
• Hyperbolic functions

General Theory

• Existence & Uniqueness
• Continuity & Differentiability

Qulitative Theory

• Qualitative Theory of Differential Equations
• Qualitative theory: ODE
• Boundary value problem

Partial Differential Equations

• Introduction
• Notes on PDE
• Classification of 2nd Order PDE

Solution methods

• Separation of variables
• Integral transforms - notes missing.
• Fundamental solution

Special functions

• Bessel functions & Legendre polynomials

Reaction-diffusion system

Notes on Reaction-diffusion system

Discontinuity: Hyperbolic Conservation Laws

Notes on Hyperbolic Conservation Laws