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.
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Equilibria and stability
Stability describes whether nearby trajectories remain close to or approach an equilibrium.
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Discrete dynamical systems
Iteration of a map evolves a state through discrete time steps.
Linearization
The Jacobian approximates a smooth system near an equilibrium; zero-real-part eigenvalues need further analysis.
Logistic map
The quadratic iteration rx(1-x) exhibits fixed points, periodic cycles and chaotic regimes as r varies.
Lyapunov exponents
Asymptotic growth rates quantify separation of nearby trajectories.
Lyapunov functions
A suitably decreasing scalar function can establish stability without solving trajectories explicitly.
Knowledge connections
- Phase portraits · Related · Map context
- Linearization · Related · Map context
- Lyapunov functions · Related · Map context
- Logistic map · Related · Map context
- Discrete dynamical systems · Related · Map context
- Lyapunov exponents · Related · Map context
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Ordinary Differential Equations
Explore first-order equations; linear equations and forcing; systems and phase space. Concepts and connections are available independently of video coverage.