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

For a point $x$ in the phase space, $\Phi_x(t): T \to M$ is the trajectory through $x$, while $\gamma_x = \{ \Phi_x(t) \mid t \in T \}$ is the orbit through $x$.

Table: Classes of Dynamical Systems

Direction \ Index Discrete-time Continuous-time
Forward semi-cascade semi-flow
Bi-directional cascade/map flow

Governing Equations of Dynamic System is a closed set of differential equations.

Dynamical systems normally refer to ordinary differential equations, with only time derivatives (no spatial derivatives).

## General

Invariant manifold is a topological manifold that is invariant under the action of the dynamical system, such as slow manifold, (un)stable manifold, (sub)center manifold, and inertial manifold.

Stable manifold theorem: The (un)stable set of hyperbolic fixed point of a smooth map is a (un)stable manifold.

(Un)stable manifold is a smooth manifold whose tangent space has the same dimension as the (un)stable space of the linearized the map at the point.

Center manifold of a fixed point of a dynamical system consists of orbits whose behavior around the fixed point is not controlled by either the attraction of the stable manifold or the repulsion of the unstable manifold.

## Ordinary Differential Equations

Note:

1. In ODEs, F(x) is known; the following is not an ODE: $$\frac{\text{d} y}{\text{d} x} = y(y'(x)+1)$$
2. General solutions are not necessarily all the solutions. For example, ODE $y^2 + y'^2 = 1$ has general solution $y = \sin(x+c)$ and extra solutions (singular solutions) $y=\pm 1$.

Solutions:

General Theory:

Qualitative Theory: 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.

## Partial Differential Equations

Solution methods:

Special functions:

Reaction-diffusion system:

Notes on Reaction-diffusion system

Discontinuity: Hyperbolic Conservation Laws:

Notes on Hyperbolic Conservation Laws