MA 278: Introduction to Dynamical Systems Theory
Credits: 3:0
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 constants, fractals. 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.