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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Winter 2007 • Question 5

Question

Math 6410 • ODE Winter 2007 • Question 5

State the center manifold theorem and explain its importance in bifurcation theory. Consider the system \[ \dot {x}=-xy,\quad \dot {y}=-y-x^2. \] Construct an approximation to the center manifold at the origin and hence determine the local behavior of solutions.

Raw LaTeX

State the center manifold theorem and explain its importance in bifurcation
theory. Consider the system
\[
  \dot{x}=-xy,\quad \dot{y}=-y-x^2.
\]
Construct an approximation to the center manifold at the origin and hence determine the
local behavior of solutions.

Topic Tags

Bifurcation Theory Center Manifold Theory and Invariant Manifolds Phase Plane Analysis Poincare Maps and Periodic Solutions