A **dynamical system** is a manifold \(M\) called the **phase space** (or state space)
endowed with a family of smooth **evolution functions** \(\Phi(t): M \rightarrow M \)
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, center manifold, (un)stable manifold, subcenter 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