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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1994 • Question 4

Question

Math 6410 • ODE Fall 1994 • Question 4

Let the system \(x'=f(x)\) define a \(C^1\) flow \(\phi _t\) on the open set \(E\) contained in \(\mathbb {R}^n\). Prove that the \(\omega \) limit set of a trajectory \(\Gamma \) is closed and if it is also bounded, then it is nonvoid, compact and connected.

Raw LaTeX

Let the system \(x'=f(x)\) define a \(C^1\) flow \(\phi_t\) on the
open set \(E\) contained in \(\R^n\). Prove that the \(\omega\) limit set of a trajectory
\(\Gamma\) is closed and if it is also bounded, then it is nonvoid, compact and
connected.

Topic Tags

Invariant Sets, Attracting Sets, and Omega-Limit Sets