Non-Linear Dynamical Systems

Types of non-linear dynamical systems and connections between them.

Types of non-linear dynamical systems and connections between them; Poincare sections; conjugacy and flow equivalence; Review of portraits and the geometry of solutions to ordinary differential equations; Stability: Liapunov; quasi-asymptotic stability; Liapunov functions; Liapunov stability theorems and linear stability (for distinct eigenvalues); Stationary points in R2; Population models as examples; Periodic orbits in ordinary differential equations; Statement and explanation of Poincare-Bendixson theorem; Poincare index and Dulac criterion; Bifurcation theory (by Taylor's series) and Hopf bifurcation; Maps of the interval; Fixed points; periodic points and stability; Saddle-node and periodic doubling bifurcations; Chaos: Piecewise linear maps; the tent map; Transitivity and chaos (sensitive dependence on initial conditions); Brief description of the maps x - nx(1-x); particularly for n=4; and topological conjugacy; Period three implies existence of all periods; Statement of Sharkovskii theorem