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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2006 • Question 5

Question

Math 6410 • ODE Fall 2006 • Question 5

State the center manifold theorem and briefly explain its importance in bifurcation theory. Consider the system \[ \dot {x}=-2x+y-x^2,\quad \dot {y}=x(y-x). \] Construct an approximation to the center manifold at the origin and hence determine the local behavior of solutions. [Hint: first perform a change of variables in order to diagonalize the linear part of the system].

Raw LaTeX

State the center manifold theorem and briefly explain its importance in
bifurcation theory. Consider the system
\[
  \dot{x}=-2x+y-x^2,\quad \dot{y}=x(y-x).
\]
Construct an approximation to the center manifold at the origin and hence determine the
local behavior of solutions. [Hint: first perform a change of variables in order to
diagonalize the linear part of the system].

Topic Tags

Bifurcation Theory Center Manifold Theory and Invariant Manifolds Linear Systems and Matrix Exponentials Phase Plane Analysis