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Dynamical systems and chaos
Long-term behaviour of flows and iterated maps: fixed points, linearisation and stability, phase portraits, limit cycles and the Poincaré–Bendixson theorem, bifurcations, the logistic map, period doubling, sensitive dependence and strange attractors.
math.analysis.dynamical-systems
Se basa en
Requisitos previos obligatorios, hasta el fondo: 38
Aprenderás a
- Classify fixed points of planar linear systems and use linearisation to study nonlinear ones.
- Apply the Poincaré–Bendixson theorem to establish limit cycles, and explain why chaos cannot occur for planar flows.
- Analyse the logistic map: fixed points, period-doubling bifurcations and numerical evidence of chaos, distinguishing evidence from proof.
Definición
- A fixed point x* of a map f satisfies f(x*) = x*; it is attracting if nearby orbits converge to it.
- A limit cycle is an isolated periodic orbit of a flow.
- A map f has sensitive dependence on initial conditions if there is δ > 0 such that for every point x and every ε > 0 there is a point y within ε of x and an n with |fⁿ(x) − fⁿ(y)| ≥ δ.
- A bifurcation is a qualitative change in the behaviour of a family of systems as a parameter varies.
Teorema
- A fixed point x* of a C¹ map of ℝ with |f′(x*)| < 1 is attracting, and repelling if |f′(x*)| > 1.
- Hartman–Grobman: near a hyperbolic fixed point a flow is topologically conjugate to its linearisation.
- Poincaré–Bendixson: a bounded forward orbit of a planar C¹ flow whose limit set contains no fixed point approaches a periodic orbit.
- For the logistic map x ↦ r x (1 − x), the fixed point 1 − 1/r is attracting for 1 < r < 3 and loses stability at r = 3 in a period-doubling bifurcation.
Errores frecuentes
- Thinking chaos means randomness; chaotic systems are deterministic, yet unpredictable in practice.
- Believing a computer simulation proves a system is chaotic; rounding errors and finite time make simulation evidence, not proof.
Conduce a
Descubrimientos opcionales
- Ergodic theory Ergodic theory studies the statistical behaviour of dynamical systems: time averages versus space averages, recurrence and mixing.
- Differential topology The Poincaré–Hopf theorem says the indices of the zeros of a vector field on a closed surface add up to its Euler characteristic, constraining every flow on that surface.
- ← Sequences and series
- ← Differential topology
Itinerarios de aprendizaje
Lecturas complementarias
- Steven H. Strogatz, Nonlinear Dynamics and Chaos (1994)
- Robert L. Devaney, An Introduction to Chaotic Dynamical Systems (1986)