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Nonlinear Dynamical Systems

650

This course provides an introduction to nonlinear ordinary differential equations and dynamical systems theory with an emphasis on applications. Topics covered include the existence and uniqueness of solutions to initial value problems, as well as the qualitative behavior of solutions, including existence of equilibria, periodic and connecting orbits and their stability. Additional topics include an introduction to bifurcation theory and chaos.

Text: 

Nonlinear Dynamics & Chaos, Strogatz, 2nd edition, Westview Press

Prerequisite: 

Math 122 or MATH 127 or MATH 142 or MATH 147, and MATH 220 or MATH 221 or MATH 320, and MATH 290 or MATH 291.

Credit Hours: 
3

This is a second course in the study of differential equations. While these equations originally arose in the study of celestial mechanics, today they play a fundamental and promenade role in every branch of science and engineering. Ocean currents, blood flow, financial markets, electrical circuits, and fiber optical communication are all examples of dynamical processes which may be modeled by collections of differential equations. Unfortunately, these equations are often too difficult to solve explicitly and so a considerable portion of modern applied mathematics is devoted to their study both analytically and computationally.

Topics to be covered:

  1. Flows on the line.
  2. Bifurcations in one-dimensional systems.
  3. Flows on the circle.
  4. Two dimensional linear systems.
  5. Nonlinear systems in the plane.
  6. Limit cycles.
  7. Bifurcations in two-dimensional systems.
  8. Introduction to chaos: Lorenz equations, fractals and strange attractors.

(Johnson 2016 )


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