Methods of Applied Mathematics I Topics
Applied and Computational Mathematics at Florida State University
References:
- Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems, D. W. Jordan and P. Smith, Oxford, 1999, Third Edition
- Nonlinear Dynamics and Chaos: with applications to physics, biology, chemistry, and engineering, Steven H. Strogatz, Addison-Wesley, 1994 (hardcover) or 2001 (paperback)
- Introduction to Perturbation Techniques, A. H. Nayfeh, Wiley-VCH, 2004
Topics:
- Phase portraits
- 1-d and 2-d
- trajectories
- periodic and unbounded orbits
- homoclinic and heteroclinic orbits
- separatrices
- existence and uniqueness
- fixed points
- basin of attraction
- Stability
- linear stability analysis
- Lyapunov stability and Lyapunov method
- Bifurcations
- saddle-node
- transcritical
- pitchfork
- Hopf and imperfect bifurcation
- hysteresis
- catastrophe (fold and cusp)
- Nonlinear oscillations
- limit cycles
- Poincare-Bendixson Theorem
- Lienard systems
- relaxation oscillations
- weakly nonlinear oscillators
- Van der Pol and Duffing oscillators
- energy analysis
- Conservative systems
- gradient systems.
- Applications
- mechanical systems
- pendulum
- mass on a spring
- bead on a rotating wire
- soft and hard springs
- population models
- predator-prey systems
- chemical reactors
- Perturbation methods
- Poincare-Lindstedt method
- two-timing
- Perturbation methods applied to algebraic systems
- Perturbation methods applied to ordinary differential equations
