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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2008 • Question 3

Question

Math 6410 • ODE Fall 2008 • Question 3

Consider the scalar equation \[ \ddot {x}+\dot {x}=-\epsilon (x^2-x),\qquad 0<\epsilon \ll 1 \] Using the method of multiple scales show that the \(O(1)\) solution is \[ x_0(t,\tau )=A(\tau )+B(\tau )e^{-t}, \] where \(\tau =\epsilon t\), and identify any resonant terms at \(O(\epsilon )\). Show that the non–resonance condition for the amplitude \(A\) is \[ A_\tau =A-A^2 \] and hence determine the asymptotic behavior of \(x_0\). Comment on the domain of validity of the asymptotic expansion.

Raw LaTeX

Consider the scalar equation
\[
  \ddot{x}+\dot{x}=-\epsilon(x^2-x),\qquad 0<\epsilon\ll1
\]
Using the method of multiple scales show that the \(O(1)\) solution is
\[
  x_0(t,\tau)=A(\tau)+B(\tau)e^{-t},
\]
where \(\tau=\epsilon t\), and identify any resonant terms at \(O(\epsilon)\). Show that
the non--resonance condition for the amplitude \(A\) is
\[
  A_\tau=A-A^2
\]
and hence determine the asymptotic behavior of \(x_0\). Comment on the domain of validity
of the asymptotic expansion.

Topic Tags

Asymptotic and Exponential Stability Distributions and Weak Solutions Perturbation and Multiple Scales Methods Poincare Maps and Periodic Solutions