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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2024 • Question 6

Question

Math 6410 • ODE Fall 2024 • Question 6

[25] Consider the system

\[ \dot {x}=x^2-xy \]
\[ \dot {y}=-y+x^2 \]

Determine the stability at the origin by constructing an approximate center manifold of the form \(y=ax^2+bx^3+\cdots \) and analyzing the flow on the approximate center manifold. Sketch the phase portrait showing the center and stable manifolds.

Raw LaTeX

[25] Consider the system

\[
\dot{x}=x^2-xy
\]

\[
\dot{y}=-y+x^2
\]


Determine the stability at the origin by constructing an approximate center manifold of
the form \(y=ax^2+bx^3+\cdots\) and analyzing the flow on the approximate center manifold.
Sketch the phase portrait showing the center and stable manifolds.

Topic Tags

Asymptotic and Exponential Stability Center Manifold Theory and Invariant Manifolds Phase Plane Analysis Stability of Linear Systems