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

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