MA 360 - Intermediate Differential Equations
This course offers more in-depth coverage of differential equations. Topics include ordinary differential equations as finite-dimensional dynamical systems; vector fields and flows in phase space; existence/uniqueness theorems; invariant manifolds; stability of equilibrium points; bifurcation theory; Poincaré-Bendixson Theorem and chaos in both continuous and discrete dynamical systems; and applications to physics, biology, economics, and engineering.
Prerequisites: MA 221
This course deals with differential equations from both analytic and geometric point of view. It introduces basic concepts and techniques of dynamical system theory, as well as reviews and strengthens the analytical methods covered in Ma 221. The aim of the course is for students to combine analytic and qualitative techniques to understand dynamical systems and their solutions with applications to mathematical models arising in sciences and engineering problems.
By the end of the semester, students should be able to:
- use basic techniques to analyze one-dimensional dynamical systems, such as plotting slope fields and phase lines, and identifying stability of equilibrium points;
- recognize bifurcations and find bifurcation points in dynamical systems;
- use geometric approach to two-dimensional dynamical systems such as plotting vector fields, trajectories, and phase planes;
- understand properties of two-dimensional linear systems, including superposition principles, stable or unstable equilibrium points based on the types of eigenvalues of the dynamical systems, as well as special features of phase portrait;
- stability analysis of equilibrium points of nonlinear dynamical system, including linear stability analysis and phase portrait;
- apply geometric techniques to investigating Hamiltonian systems and dissipative systems;
- use both analytic and geometric techniques to analyze mathematical models such as population dynamics, harmonic oscillations with forcing or dissipation;
- interpret Existence and Uniqueness Theorem as a geometric property of dynamical systems. That means there is a unique trajectory going through every non-equilibrium point in the phase portrait.
- use software packages, such as Matlab, and numerical methods to plot phase plane, trajectories and visualize dynamical systems.
Paul Blanchard, Glen Hall and Robert Devaney, Differential Equations. Brooks/Cole, 2005.
Steven Strogatz, Nonlinear Dynamics and Chaos, with Applications to Physics, Biology, Chemistry and Engineering.