General introduction
The notion of a dynamical system and examples of simple phase portraits. Relationship between
continuous and discrete systems. Reduction to autonomous systems. Initial value problems, uniqueness,
finite-time blowup, examples. Flows, orbits, invariant sets, limit sets and topological equivalence.[3]
Fixed points of flows
Linearization. Classification of fixed points in R2
, Hamiltonian case. Effects of nonlinearity;hyperbolic and non-hyperbolic cases; Stable-manifold theorem (statement only), stable and unstable manifolds in R2. Phase-plane sketching. [3]
Stability
Lyapunov, quasi-asymptotic and asymptotic stability of invariant sets. Lyapunov and bounding functions. Lyapunov’s 1st theorem; La Salle’s invariance principle. Local and global stability. [2]
Periodic orbits in R2
The Poincare index; Dulac’s criterion; the Poincar´e–Bendixson theorem (*and proof*). Nearly Hamiltonian flows.Stability of periodic orbits; Floquet multipliers. Examples; van der Pol oscillator. [5]
Bifurcations in flows and maps
Non-hyperbolicity and structural stability. Local bifurcations of fixed points: saddle-node, transcritical,
pitchfork and Andronov-Hopf bifurcations. Construction of centre manifold and normal forms. Examples. Effects of symmetry and symmetry breaking. *Bifurcations of periodic orbits.* Fixed points and
periodic points for maps. Bifurcations in 1-dimensional maps: saddle-node, period-doubling, transcritical and pitchfork bifurcations. The logistic map. [5]
Chaos
Sensitive dependence on initial conditions, topological transitivity. Maps of the interval, the sawtooth map, horseshoes, symbolic dynamics. Period three implies chaos, the occurrence of N-cycles,
Sharkovsky’s theorem (statement only). The tent map. Unimodal maps and Feigenbaum’s constant.
[6]
Appropriate books
D.K. Arrowsmith and C.M. Place Introduction to Dynamical Systems. CUP 1990
P.G. Drazin Nonlinear Systems. CUP1992
† P.A. Glendinning Stability, Instability and Chaos. CUP1994
D.W. Jordan and P. Smith Nonlinear Ordinary Differential Equations. OUP 1999
J. Guckenheimer and P. Holmes Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector
Fields. Springer, second edition 1986
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