MATH-4400, ORDINARY DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS

# ORDINARY DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS

Time: Monday, Thursday, 10:00 to 11:50 AM
Room: Carnegie 101
Instructor: Gregor Kovacic
Office: 419 Amos Eaton
Phone: 276-6908
E-mail: kovacg at rpi dot edu

• This document contains the list of topics to be presented in this course, a list of textbooks covering some of the topics, and some comments and recommendations about these textbooks.
• Click on each topic title to download the notes for that topic.

## Topics

Click on each topic title to download the notes for that topic.

First order equations: Separable equations, linear equations, initial-value problems, explicit and implicit solutions, exact equations and integrating factors, autonomous equations and equilibrium points, stability.

Higher order equations and systems: Higher-order equations and nXn systems, linear equations, Wronskians and Liouville's theorem, higher-order linear equations and systems with constant coefficients, the exponential of a matrix, phase portraits, stable, unstable and center subspaces, multiple roots, Jordan normal form, inhomogeneous systems, method of undetermined coefficients, variation of constants, mechanical oscillations.

Calculus of variations: Functionals and their extrema, Euler's equations, extrema with side conditions, geodesics, Newtonian mechanics, Lagrange's equations, Lagrangian and Hamiltonian functions, Kepler's problem.

Phase space analysis of higher-order systems: Autonomous systems, phase space, equilibrium points, limit cycles, first integrals, conservative systems with one-degree-of-freedom, separatrices, small oscillations, equations on the torus, linearization, Lyapounov stability, asymptotic stability.

Theory of planar systems: Predator-prey models, linearization about and survival of hyperbolic equilibria, index theory, Poincare-Bendixson theorem, Bendixson criterion, small perturbations of conservative systems.

Periodic solutions and Poincare maps: Surface of section, Poincare return map, stability of periodic orbits, linearized Poincare map, Floquet multipliers.

Bifurcations and chaotic dynamics: Saddle-node, transcritical, pitchfork, period doubling, Hopf, unimodal maps and the Feigenbaum cascade, symbolic dynamics, chaos in unimodal maps.

Existence and uniqueness: Contraction mappings, Picard approximations, Lipschitz condition, local existence and uniqueness.

The class notes from Fall 2013 are deposited here.
Here is the set of notes from Fall 2012.

## Textbooks

The following textbooks contain material similar to that presented in this course:

K. T. Aligood, T. D. Sauer, and J. A. Yorke, Chaos, an introduction to dynamical systems, Springer-Verlag.
A. A. Andronov, Qualitative Theory of Second-Order Dynamic Systems, Wiley.
A. A. Andronov, S. E. Khaikin, and A. A. Vitt, Theory of Oscillations, Dover.
V. I. Arnold, Ordinary Differential Equations, MIT press or Springer Verlag.
G. Birkhoff and G. Rota, Ordinary Differential Equations, Wiley.
W. E Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary-Value Problems, Wiley.
F. Brauer and J. A. Nohel, Ordinary Differential Equations, Benjamin.
F. Brauer and J. A. Nohel, Qualitative Theory of Ordinary Differential Equations, Benjamin.
G. F. Carrier and C. E. Pearson, Ordinary Differential Equations, Blaisdell.
E. A. Coddington, An Introduction to Ordinary Differential Equations, Prentice Hall.
R. Courant and D. Hilbert, Methods of Mathematical Physics, Part I, Wiley-Interscience.
R. L Devaney, An Introduction to Chaotic Dynamical Systems, Benjamin-Cummings.
C. Fox, An introduction to the calculus of variations, Dover.
M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press.
E. L. Ince, Ordinary Differential Equations, Dover.
I. Percival and D. Richards, Introduction to Dynamics, Cambridge University Press.
L. S. Pontryagin, Ordinary Differential Equations, Dover.
D. A. Sanchez, Ordinary Differential Equations and Stability Theory: an Introduction, Dover.
G. F. Simmons, Differential Equations with Applications and Historical Notes, McGraw-Hill.
S. H. Strogatz, Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering, Addison-Wesley.
R. A. Struble, Nonlinear Differential Equations, McGraw-Hill.
W. Walter, Ordinary Differential Equations, Springer-Verlag.
R. Weinstock, Calculus of Variations, With Applications to Physics and Engineering, Dover.
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer Verlag.