Linear stability analysis, attractors, limit cycles, Poincare-Bendixson
theorem, relaxation oscillations, elements of bifurcation theory:
saddle-node, transcritical, pitchfork, Hopf bifurcations,
integrability, Hamiltonian systems, Lotka-Volterra equations,
Lyapunov function & direct method for stability, dissipative systems,
Lorenz system, chaos & its measures, Lyapunov exponents, strange
attractors, simple maps, period-doubling bifurcations, Feigenbaum
Both flows (continuous time systems) & discrete time systems (simple
maps) will be discussed.
Assignments will include numerical simulations.
Prerequisites, if any: familiarity with linear algebra - matrices, and
ordinary differential equations
Desirable: ability to write codes for solving simple problems.
Suggested books and references:
S. Strogatz, Nonlinear Dynamics and Chaos: with Applications to physics, Biology, Chemistry, and Engineering, Westview, 1994.
S. Wiggins, Introduction to applied nonlinear dynamics & chaos, Springer-Verlag, 2003.
K. Alligood, T. Sauer, & James A.Yorke, Chaos: An Introduction to Dynamical Systems, Springer-Verlag, 1996.
M.Tabor, Chaos and Integrability in Non-linear Dynamics, 1989.
L. Ya. Adrianova, Introduction to Linear Systems of Differential Equations, AMS 1995.
Morris W. Hirsch, Robert L. Devaney, Stephen Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press 2012.