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).

**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.

Note:

- In ODEs, F(x) is known; the following is not an ODE: $$\frac{\text{d} y}{\text{d} x} = y(y'(x)+1)$$
- 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:

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

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.

Solution methods:

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

Special functions:

Reaction-diffusion system:

Notes on Reaction-diffusion system

Discontinuity: Hyperbolic Conservation Laws:

Notes on Hyperbolic Conservation Laws