General introduction to linear and nonlinear evolution equations: Flows and maps, types of dynamical behaviour, conservative versus dissipative systems, sensitivity to initial conditions, deterministic chaos.
Dynamical variables, phase space, phase trajectories and their properties. Linear autonomous systems. Phase plane analysis of 2D systems. Classification of singular points. Stability and asymptotic stability. Stable, unstable and centre manifolds. Hartman-Grobman theorem. Limitations of linear stability analysis. Centre manifold theorem. Limit cycles. Poincare-Bendixson theorem.
Dissipative systems. 1D maps: Bernoulli, tent and logistic maps. Period-doubling route to chaos. Intermittency. Frobenius-Perron equation, invariant density. 2D maps: Baker's transformation, cat map, standard map. Quasiperiodicity and mode-locking.
Notion of normal forms. Elementary classification of bifurcations for maps and flows.
Routes to chaos in dissipative systems. Elementary ideas on turbulence. Strange attractors
- Teacher: Neelima M Gupte