Dynamics & Chaos

A dynamical system mathematically models how states evolve over time. It's expressed through differential or difference equations.

Chaos is unpredictable behavior in deterministic systems. It shows extreme sensitivity to initial conditions (butterfly effect), making long-term prediction impossible.

Fractals are geometric shapes where parts resemble the whole (self-similarity). They can have non-integer dimensions.

A fixed point is a state that doesn't change over time in a dynamical system. Fixed points can be stable (attractors), unstable, or saddle points.

Bifurcation is when qualitative behavior of a dynamical system suddenly changes as a parameter varies. New fixed points may appear or stability may change.

An ordinary differential equation (ODE) relates an unknown function of one variable to its derivatives. It's the fundamental language of physics and engineering.

A partial differential equation (PDE) involves partial derivatives of an unknown function with respect to multiple independent variables. Describes waves, heat transfer, quantum mechanics.

Phase space represents all possible states of a dynamical system as coordinates. System evolution can be visualized as trajectories.

An attractor is a state or set of states that trajectories converge to over time in a dynamical system. Types include point, periodic orbit, and strange attractors.