本科高年级Analysis课程已上线

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

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前置知识

Ordinary differential equations Real analysis

38 项必需的前置知识（一直追溯到底）

你将学会

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.

定义

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.

定理

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.

常见误解

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.

通向

Ergodic theory

可选发现

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

学习路径

Analysis, PDE and dynamics