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The exhaustive list of topics in Dynamical Systems in which we provide Help with Homework Assignment and Help with Project is as follows:

  • 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.
  • 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: Lorentz and Rossler attractors. Crises, crisis-induced intermittency, strange non-chaotic attractors.
  • Degrees of stochasticity: ergodicity, mixing, K, C. and Bernoulli systems. Elements of symbolic dynamics.
  • Measures of chaos. Liapunov exponents. Fractal sets and dimensions. Multifractals, generalized dimensions, K-S entropy.
  • Hamiltonian systems. Canonical transformations. Liouville-Arnold integrability. Action-angle variables. Dynamical symmetry. Adiabatic invariants, Poincare sections, area-preserving mappings. Elementary ideas on perturbation theory and the