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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2000 • Question 4

Question

Math 6410 • ODE Fall 2000 • Question 4

Consider \(\dot {x}=\mathbf {f}(x)\) where \(x\in \mathbb {R}^n\) and \(\mathbf {f}\in C^1(\mathbb {R}^n,\mathbb {R}^n)\). Suppose that the solution through any point \(x_0\in \mathbb {R}^n\) exists for all \(t\in \mathbb {R}\).

  1. Define the \(\alpha \)- and \(\omega \)-limit sets of a point \(x_0\in \mathbb {R}^n\).
  2. Consider the nonlinear spring equation \[ x''+\mu x'+g(x)=0 \] where \(g(x)\) is an odd function of \(x\), \(g'(x)>0\) and \(\mu \ge 0\). Translate this equation to the \(\dot {x}=\mathbf {f}(x)\) form and linearize about the trivial equilibrium. What conclusions can be made about the stability of this equilibrium?
  3. Define what is meant by a Lyapunov function for the above equation; derive one and verify that it is a Lyapunov function. Describe the \(\alpha \)- and \(\omega \)-limit sets of a point \(x_0\in \mathbb {R}^n\) for the case \(\mu =0\) and for the case \(\mu >0\). Make a conclusion about the stability of the trivial equilibrium for each of these two cases.

Raw LaTeX

Consider \(\dot{x}=\mathbf{f}(x)\) where \(x\in\R^n\) and
\(\mathbf{f}\in C^1(\R^n,\R^n)\). Suppose that the solution through any point
\(x_0\in\R^n\) exists for all \(t\in\R\).
\begin{enumerate}
\item Define the \(\alpha\)- and \(\omega\)-limit sets of a point \(x_0\in\R^n\).
\item Consider the nonlinear spring equation
\[
  x''+\mu x'+g(x)=0
\]
where \(g(x)\) is an odd function of \(x\), \(g'(x)>0\) and \(\mu\ge0\). Translate this equation
to the \(\dot{x}=\mathbf{f}(x)\) form and linearize about the trivial equilibrium. What
conclusions can be made about the stability of this equilibrium?
\item Define what is meant by a Lyapunov function for the above equation; derive one and
verify that it is a Lyapunov function. Describe the \(\alpha\)- and \(\omega\)-limit sets of
a point \(x_0\in\R^n\) for the case \(\mu=0\) and for the case \(\mu>0\). Make a conclusion
about the stability of the trivial equilibrium for each of these two cases.
\end{enumerate}

Topic Tags

Contraction Mapping and Nonlinear Equations Invariant Sets, Attracting Sets, and Omega-Limit Sets Lyapunov Stability and Lyapunov Functions Stability of Linear Systems