The main course objective is to learn the basics of the qualitative analysis of ordinary differential equations (ODE). Most ODE cannot be solved analytically. In this course the students will see how one can understand the behavior of solutions to ODE without actually being able to solve it. The mathematical theory will be illustrated by biological and physical examples.

__Classes:__MWF 2:00pm-2:50pm, NDSU South Engineering, Rm 118__Office hours:__MWF 12:00pm-12:50pm (Minard 408E22)- Syllabus
__Textbook:__No textbook is required. Lecture notes will be provided.

- Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering by Steven H. Strogatz
- Chaos: An Introduction to Dynamical Systems, by Kathleen T. Alligood, Tim D. Sauer, James A. Yorke
- Differential Equations, Dynamical Systems, and an Introduction to Chaos, by Morris W. Hirsch, Stephen Smale, Robert L. Devaney

- Autonomous equations. Elementary bifurcations.
- Linear equations. Phase plane.
- Stability by linearization and Lyapunov functions.
- Classical mechanics with one degree of freedom and elements of the Calculus of Variations.
- Limit cycles and Poincare-Bendixson theorem.
- Poincare-Andronov-Hopf bifurcation.
- Strange attractors and chaos.
- The evolutionary game theory and the replicator equation.

- 01. Review of ODE theory
- 02. First order autonomous ODE
- 03. Elementary bifurcations
- 04. Insect outbreak model
- 05. Lotka-Volterra system
- 06. Planar autonomous differential equations
- 07. Linear equations on the plane
- 08. Linear equations in
*d*dimensions - 09. Elements of classical mechanics (updated 10.18.2017)
- 10. Linearization and null-clines
- 11. Predator-prey model with intraspecific competition
- 12. Competition models
- 13. Lyapunov functions
- 14. Index theory (for an accesible introduction see Section 6.8 of Strogatz book)
- 15. Limit sets and Poincare-Bendixson theorem
- 16. Poincare-Andronov-Hopf bifurcation

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