Dynamical systems and chaos

The long-term behaviour of systems that evolve: equilibria, stability, attractors, bifurcations and chaos — and the limits they impose on prediction.

7 topics

Topics

Dynamical systems

A state and a rule that moves it forward: continuous ( $\dot x = f(x)$ ) or discrete ( $x_{k+1} = g(x_k)$ ). The questions change from "find the formula" to "what happens in the long run?" — and every iterative algorithm, from gradient descent to a recurrent network, is a discrete dynamical system.

Advanced

Phase space

The space of all possible states (for a pendulum: angle and angular velocity). Each initial state traces a trajectory; drawing them all — the phase portrait — shows the whole behaviour at a glance.

Advanced

Equilibria and stability

An equilibrium $f(x^\ast) = 0$ is stable if nearby states stay nearby (and asymptotically stable if they return). Linearize: if every eigenvalue of the Jacobian $J_f(x^\ast)$ has negative real part, it is asymptotically stable.

Advanced

Attractors

Sets that nearby trajectories approach and stay in: a point, a closed orbit (limit cycle) or a fractal "strange attractor" like Lorenz's butterfly. The set of initial states that end up there is its basin.

Advanced

Bifurcations

Qualitative changes in behaviour as a parameter crosses a critical value: an equilibrium loses stability, a cycle is born, periods double on the road to chaos. Tipping points in climate and ecology are bifurcations.

Specialization

Chaos and sensitivity to initial conditions

Deterministic systems whose nearby trajectories separate exponentially, $|\delta(t)| \approx |\delta_0|e^{\lambda t}$ with $\lambda > 0$ . Prediction is possible only up to a horizon of about $\frac1\lambda\ln\frac{\text{tolerance}}{|\delta_0|}$ : better measurements buy time only logarithmically.

Specialization

Fractals

Sets with detail at every scale and non-integer dimension. The Mandelbrot set — the $c$ for which $z \mapsto z^2 + c$ stays bounded from $z_0 = 0$ — is the most famous; strange attractors are fractal too.

Specialization

Where this area leads in computing

Topics