Mathematics Qualifying Exam Question Bank

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Question

Math 6410 • ODE

Fall 2007 • Question 4

Question

Math 6410 • ODE Fall 2007 • Question 4

Let \[ f:\mathbb {R}^N\to \mathbb {R}^N \] be a \(C^1\) mapping and consider the system \begin{equation} x'=f(x). \tag {0.6} \end{equation} Assume there exists a \(C^1\) functional \[ \phi :\mathbb {R}^N\to \mathbb {R} \] such that \[ \phi '(x):=\nabla \phi (x)\cdot f(x)\le 0,\quad x\in \mathbb {R}^N. \] Prove that the \(\omega \)-limit set of any bounded semiorbit of (0.6) is an invariant set for the flow associated with (0.6) and is contained in the set \[ E=\{x:\phi '(x)=0\}. \]

Apply this result to the system \[ \begin {array}{rcl} u'&=&-v\\ v'&=&-v+u^3\\ x&=&\begin {pmatrix}u\\ v\end {pmatrix}, \end {array} \] using \[ \phi (u,v)=\frac {v^2}{2}+\frac {u^4}{4}. \] What is your conclusion?

Let
\[
  f:\R^N\to\R^N
\]
be a \(C^1\) mapping and consider the system
\begin{equation}
  x'=f(x).
  \tag{0.6}
\end{equation}
Assume there exists a \(C^1\) functional
\[
  \phi:\R^N\to\R
\]
such that
\[
  \phi'(x):=\nabla\phi(x)\cdot f(x)\le0,\quad x\in\R^N.
\]
Prove that the \(\omega\)-limit set of any bounded semiorbit of (0.6) is an
invariant set for the flow associated with (0.6) and is contained in the set
\[
  E=\{x:\phi'(x)=0\}.
\]

Apply this result to the system
\[
\begin{array}{rcl}
  u'&=&-v\\
  v'&=&-v+u^3\\
  x&=&\begin{pmatrix}u\\ v\end{pmatrix},
\end{array}
\]
using
\[
  \phi(u,v)=\frac{v^2}{2}+\frac{u^4}{4}.
\]
What is your conclusion?

Topic Tags

Center Manifold Theory and Invariant Manifolds Contraction Mapping and Nonlinear Equations Dual Spaces and Bounded Linear Functionals Invariant Sets, Attracting Sets, and Omega-Limit Sets