S. H. Strogatz, Nonlinear Dynamics & Chaos,
by S. H. Strogatz, Addison-Wesley, 1994. (Check out Prof. Strogatz's homepage
(*) For graduate students only
I.1 One-dimensional flows: fixed points, phase diagram,
stability analysis, bifurcations, examples.
I.2 Flow on the circle: examples, phase diagram,
II.1: Two-dimensional flows: linear systems, classification
of fixed points.
II.2: Two-dimensional flows: orbits, phase portraits,
fixed points and linearization, conservative systems, reversible systems,
index theory, examples.
II.3: Two-dimensional flows: limit cycles, Poincaré-Bendixson
II.4: Bifurcations of two-dimensional flows: saddle-node,
transcritical, pitchfork bifurcations. Hopf bifurcations. Global bifurcations
of limit cycles. Quasiperiodicity. Examples.
II.5: Advanced Topic*: flows, local flows, invariant
sets, non-wandering sets, limit sets. Topological conjugacy, topological
equivalence of flows, Poincaré maps. Hyperbolic linear flows, local
(global) stable (unstable) manifolds. Hyperbolic fixed point of non-linear
flows, global stable and unstable manifolds. Hyperbolic closed orbit. Non-hyperbolic
fixed points. Center manifold theory.
III.1 Chaos: Lorenz equations, Lorenz map, bifurcation
III.2 One-dimensional discrete maps: fixed points,
periodic orbits, stability. Logistic map and its bifurcation diagram. Renormalization,
III.3 Fractals and strange attractors. Chaos in Hamiltonian
III.4 Routes to Chaos:quasiperiodicity, subharmonic