# Math6340 Winter 2017 Differential Equations

Course Information:

Instructor Huaiping Zhu

 Office: Ross Building, N618 Phone: (416)736-2100, Ext: 66095 Email: huaiping@mathstat.yorku.ca
Class meetings: Monday 2:30-5:30pm CB 120
Grade policy: Assignment 50% + Final exam 50%
Office Hours: Wednesday 3:30-4:30 pm or by appointment
Final Exam: TBA
Text Book Differential Equations and Dynamical Systems. by Lawrence, Perko

Formatted lecture notes will be distributed. If you are interested in the notes, contact me.

Lectures: (Updated weekly)

• Chapter 1. Introduction
Typical examples from physics, biology and chemstry will be presented. Mass-spring oscillations, simple pendulum, Lienard systems, van der Pol oscillators, Duffing equation, predator-prey models with different type of response functions, compartmental models for infectious diseases. Equivalent forms between Lineard systems and predator-prey type of systems. Hilbert’s 16th problems.
• Chapter 2. Linear systems
• 2.1 Linear systems in plane. Equilibrium and classfication. Straight Line Solutions. T-D plane, phase plan and bifurcation of planar systems.
• 2.2. General theory of linear systems.
• 2.3. Linear systems with constant coefficients.
Exp(At). Jordan canonical form. Putzer’s Algorithm.

Syllabus

Math 6340 is a graduate course on ordinary differential equations and dynamical systems. Dynamical systems is a very active field of research that has a tremendous impact on our understanding of complicated, nonlinear phenomena in various systems of applied sciences. I will emphasize on both theoretical and applied aspects of differential equations and dynamical systems. We will cover most of the topics of the text book but the materials will be modified and reorganized.

Main topics

• General properties of Differential Equations
• Linear Systems and Stability
General theory of linear systems, Periodic coefficients and Floquet Theory, Stability of linear and nonlinear systems.
• Nonlinear Systems: Local Theory
Linearization, Invariant manifolds, Hartman-Grobman Theorem, Normal form theory.
• Nonlinear Systems: Global Theory
Limit sets aned attractors, limit periodic sets and limit cycles; Poincare map, Poincare-Bendixson Theory.
Lotka-Volterra system, Predator-prey system, Lineard systems.
Introduction of Hilbert’s 16th problem.
• Nonlinear Systems: Bifurcation theory
Saddle-node bifurcation, Transcritical bifurcation, Pitchfork bifurcation, Hopf bifurcations, homoclinic bifurcations, Bogdanov-Takens bifurcation of codimension 2.
• Numerical tools (software) for normal forms and bifurcation diagrams.

 References: H. Amann Ordinary Differential Equations: an introduction to nonlinear analysis, Berlin ; New York : W. de Gruyter, 1990. V. I. Arnold Ordinary Differential Equations, Springer Verlag, 1992 G. Birkhoff, and G. C Rota Ordinary Differential Equations, 4th ed. Wiley, 1988. Carmen Chicone Ordinary Differential Equations with Applications, 2nd Edition Texts in Applied Mathematics, New York: Springer-Verlag, 2006. Page from the author with Errata J. Guckenheimer, P. Holmes Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences 42, Springer-Verlag, New York-Berlin, 1983. J. K. Hale Ordinary Differential Equations, Krieger, Malabar, Florida, 1980. J. K. Hale and H. Kocak Dynamics and Bifurcations, Springer 1991. James Hetao Liu, A First Course in the Qualitative Theory of Differential Equations, Prentice Hall 2003. M. W. Hirsch and S. Smale Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, 1974. D. W. Jordan & P. Smith, Nonlinear ordinary differential equations, Oxford University Press 1977. Luis Barriera and Claudia Valls, Ordinary Differential Equations: Qualitative Theory, American Mathematical Society Graduate Studies in Mathematics 137, 2010. Y.A. Kuznetsov Elements of Applied Bifurcation Theory, Springer-Verlag, 2004. Ferdinand Verhulst, Nonlinear Differential Equations and Dynamical Systems, 2nd. ed., Springer Universitext, 2006. Vladimir Arnold, Ordinary Differential Equations, 3rd. ed., Springer Universitext, 1992. David Betounes, Differential Equations: Theory and Applications, 2nd. ed., Springer 2010. Stephen Salaff and Shing-Tung Yau, Ordinary Differential Equations, 2nd ed., International Press, 1998.