Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1998 • Question 3

Question

Math 6410 • ODE Fall 1998 • Question 3

Consider \[ x'=f(x) \tag {3.1} \] where \(x\in \mathbb {R}^n\) and \(f\in C^1(\mathbb {R}^n,\mathbb {R}^n)\). Suppose that the solution through any point \(x_0\in \mathbb {R}^n\) exists for all \(t\in \mathbb {R}\).

Define the positive and negative semiorbits of (3.1) through \(x_0\in \mathbb {R}^n\).
Define the \(\alpha \)- and \(\omega \)-limit sets of a point \(x_0\in \mathbb {R}^n\).
Consider the nonlinear spring equation \[ x''+g(x)=0 \tag {3.2} \] where \(g(x)\) is an odd function of \(x\) and \(g'(x)>0\). Describe the \(\alpha \)- and \(\omega \)-limit set for any point on the positive \(x\)-axis.
Consider \[ x''+x'+g(x)=0 \tag {3.3} \] where \(g\) is as in (c). Define what is meant by a Liapunov function for this equation and derive one (verify that it is a Liapunov function). Describe the \(\omega \)-limit set for any point in the plane.

Consider
\[
  x'=f(x)
  \tag{3.1}
\]
where \(x\in\R^n\) and \(f\in C^1(\R^n,\R^n)\). Suppose that the solution through any point
\(x_0\in\R^n\) exists for all \(t\in\R\).
\begin{enumerate}
\item Define the positive and negative semiorbits of (3.1) through \(x_0\in\R^n\).
\item Define the \(\alpha\)- and \(\omega\)-limit sets of a point \(x_0\in\R^n\).
\item Consider the nonlinear spring equation
\[
  x''+g(x)=0
  \tag{3.2}
\]
where \(g(x)\) is an odd function of \(x\) and \(g'(x)>0\). Describe the \(\alpha\)- and
\(\omega\)-limit set for any point on the positive \(x\)-axis.
\item Consider
\[
  x''+x'+g(x)=0
  \tag{3.3}
\]
where \(g\) is as in (c). Define what is meant by a Liapunov function for this equation
and derive one (verify that it is a Liapunov function). Describe the \(\omega\)-limit set
for any point in the plane.
\end{enumerate}

Topic Tags

Contraction Mapping and Nonlinear Equations Distributions and Weak Solutions Invariant Sets, Attracting Sets, and Omega-Limit Sets Poincare Maps and Periodic Solutions