MATH 558 - Ordinary Differential Equation
Fall 2013, Section 001
Instruction Mode: Section 001 is (see other Sections below)
Subject: Mathematics (MATH)
Department: LSA Mathematics
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Requirements & Distribution:
Advisory Prerequisites:
MATH 451.
This course counts toward the 60 credits of math/science required for a Bachelor of Science degree.
May not be repeated for credit.
Primary Instructor:


Differential equations model systems throughout science and engineering and display rich dynamical behavior. This course empahsizes the qualitative and geometric ideas which characterize the modern (post Poincaré) theory. The course surveys a broad range of topics with emphasis on techniques, and results that are useful in applications.


  • Phase line. Dynamics in dimension 1 and 1.5. Bifurcations. Poincar´e map.
  • Existence, uniqueness and perturbations.
  • Theory of linear systems.
  • Linearization at equilibria. Theory of constant coefficient systems. Spectral theorems.
    (The case of multiple eigenvalues and the ergodic properties of coupled oscillators complement introductory treatments).
  • The geometry of phase plane of linear systems..
  • Stable and unstable manifolds.
  • Conjugation of sinks/sources.
  • Lyapunov’s method. LaSalle’s invariance principal.
  • Gradient flows and hamiltonian systems.
  • Periodic solutions, stability, Poincar´e map, !-limit set, Poincaré-Bendixson.
  • Bifurcation theory of equilibria. Pitchfork and Hopf.
  • Introduction to chaotic dynamics. Definitions and first examples.

Text. M. Hirsh, S. Smale, and R. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, 2nd ed., Elsevier. N.B. The UM library has an electronic version of this text which can be freely consulted as many times as you like.


MATH 558 - Ordinary Differential Equation
Schedule Listing
001 (LEC)
 In Person
TuTh 10:00AM - 11:30AM

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