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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Spring 2011 • Question 4

Question

Math 6410 • ODE Spring 2011 • Question 4

Suppose that \(H:\mathbb {R}^2\to \mathbb {R}\) is a smooth function. Consider the Hamiltonian System \[ \begin {aligned} \dot {x} &= H_y(x,y),\\ \dot {y} &= -H_x(x,y). \end {aligned} \] Suppose that \(\Gamma _0\) is a nonconstant periodic orbit for this system. Show that \(\Gamma _0\) cannot be Asymptotically Orbitally Stable. (This is sometimes referred to as asymptotic stability of the periodic orbit \(\Gamma _0\) or asymptotic Poincarè stability.)

Raw LaTeX

Suppose that \(H:\R^2\to\R\) is a smooth function. Consider the Hamiltonian System
\[
\begin{aligned}
  \dot{x} &= H_y(x,y),\\
  \dot{y} &= -H_x(x,y).
\end{aligned}
\]
Suppose that \(\Gamma_0\) is a nonconstant periodic orbit for this system. Show that
\(\Gamma_0\) cannot be Asymptotically Orbitally Stable. (This is sometimes referred to
as asymptotic stability of the periodic orbit \(\Gamma_0\) or asymptotic Poincar\`e
stability.)

Topic Tags

Asymptotic and Exponential Stability Periodic Orbits, Limit Cycles, and Orbital Stability Poincare Maps and Periodic Solutions Stability of Linear Systems