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Dynamical Systems and Chaos
Explore iteration and flows; stability and bifurcations; chaos and long-term behavior. Concepts and connections are available independently of video coverage.
Prerequisites and connections
Functions
A function assigns exactly one output to each input in its domain; its domain matters when comparing expressions.
5 reviewed resources
Iteration and flows
Discrete dynamical systems
Iteration of a map evolves a state through discrete time steps.
Video explanations to be addedContinuous flows
An autonomous differential equation generates time-indexed state transformations where solutions exist uniquely.
Video explanations to be addedFixed and periodic points
A fixed point returns after one iterate; a periodic point returns after finitely many.
Video explanations to be addedOrbits
An orbit is the collection of states visited from a chosen initial state.
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Stability and bifurcations
Equilibria and stability
Stability describes whether nearby trajectories remain close to or approach an equilibrium.
Video explanations to be addedLyapunov functions
A suitably decreasing scalar function can establish stability without solving trajectories explicitly.
Video explanations to be addedBifurcations
Changing a parameter can qualitatively change equilibria or periodic behavior.
Video explanations to be addedLogistic map
The quadratic iteration rx(1-x) exhibits fixed points, periodic cycles and chaotic regimes as r varies.
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Chaos and long-term behavior
Lyapunov exponents
Asymptotic growth rates quantify separation of nearby trajectories.
Video explanations to be addedChaotic dynamics
Chaos combines forms of sensitive dependence and complex recurrence under specified definitions.
Video explanations to be addedAttractors
An attractor is an invariant set approached by a specified collection of trajectories.
Video explanations to be addedErgodicity
In measure-preserving systems, ergodicity means invariant measurable sets have trivial measure.
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