Dynamical System

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

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:

• 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:

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.

• 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

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🏷 Category=Dynamical System