Perturbation and Multiple Scales Methods

Consider the van der Pol equation \[ \ddot {x}+x+\epsilon (x^2-1)\dot {x}=\Gamma \cos (\omega t), \qquad 0<\epsilon \ll 1 \] with \(\Gamma =O(1)\) and \(\omega \ne 1/3,1,3\). Using the method of multiple scales show that the solution is attracted to \[ x(t)=\frac {\Gamma }{1-\omega ^2}\cos \omega t+O(\epsilon ) \] when \(\Gamma ^2\ge 2(1-\omega ^2)^2\) and \[ x(t)=2\left [1-\frac {\Gamma ^2}{2(1-\omega ^2)^2}\right ]^{1/2}\cos t +\frac {\Gamma }{1-\omega ^2}\cos \omega t+O(\epsilon ) \] when \(\Gamma ^2<2(1-\omega ^2)^2\). Explain why this result breaks down when \(\omega =1/3,1,3\).

Consider Mathieu’s equation for a parametric oscillator: \[ \ddot {x}+(\alpha +\beta \cos t)x=0. \]

1. Suppose that \(\alpha \approx 1\), \(\beta \approx 0\). Use a perturbation expansion in \(\beta \) to show that the transition curves for Mathieu’s equation are given approximately by \[ \alpha =1-\frac {\beta ^2}{12},\qquad \alpha =1+\frac {5}{12}\beta ^2. \]
2. Now suppose that \(\alpha \approx 1/4+\alpha _1\beta \), \(\beta \approx 0\). In the unstable region near \(\alpha =1/4\), solutions of Mathieu’s equation are of the form \[ c_1e^{\sigma t}q_1(t)+c_1e^{\sigma t}q_1(t) \] where \(\sigma \) is real and positive, and \(q_1,q_2\) are \(4\pi \)-periodic. Derive the second order equation for \(q_1,q_2\) and perform a power series expansion in \(\beta \) to show that \(\sigma \approx \pm \beta \sqrt {1/4-\alpha _1^2}\).
3. Use part (b) to deduce that solutions of the damped Mathieu equation \[ \ddot {x}+\kappa \dot {x}+(\alpha +\beta \cos t)x=0, \] where \(\kappa =\kappa _1\beta +O(\beta ^2)\), are stable if to first order in \(\beta \), \[ \alpha <\frac {1}{4}-\frac {\beta }{2}\sqrt {1-\kappa _1^2} \quad \text {or}\quad \alpha >\frac {1}{4}+\frac {\beta }{2}\sqrt {1-\kappa _1^2}. \]

Consider the nonlinear equation \[ \ddot {x}+\epsilon \dot {x}(x^2-1)+x=0,\qquad 0<\epsilon \ll 1 \] and choose initial conditions \(x(0)=1\), \(\dot {x}(0)=0\).

1. Using the method of multiple scales show that this has an asymptotic series solution of the form \[ x(t)\sim R(\epsilon t)\cos (t+\theta (\epsilon t))+\mathcal {O}(\epsilon ) \] with \[ \theta _\tau =0,\qquad R_\tau =\frac {1}{8}R(4-R^2) \] where \(\tau =\epsilon t\).
2. Derive the solution \[ R(\tau )=\frac {2}{(1+a_0e^{-\epsilon t})^{1/2}} \] and determine \(a_0\) from the initial conditions. Hence, establish that there exits a stable periodic orbit. [HINT: perform the change of variable \(y=R^2\).]

Consider the scalar equation \[ \ddot {u}+\epsilon (u^2-1)\dot {u}+u=0,\qquad 0<\epsilon \ll 1. \] Apply the method of multiple scales, with variables \(t\) and \(\eta =\epsilon t\) to show that if \(u(0)=a>0\) and \(\dot {u}(0)=0\) then for \(t=O(\epsilon ^{-1})\), \[ u=\frac {2\cos t}{\sqrt {1+\left (\frac {4}{a^2}-1\right )e^{-\epsilon t}}} +O(\epsilon ). \]

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.

Consider the nonlinear equation \[ \ddot {x}+\varepsilon \left (\frac {1}{3}\bigl (\dot {x}\bigr )^3-\dot {x}\right )+x=0, \quad 0<\varepsilon \ll 1 \] and choose initial conditions \(x(0)=a\), \(\dot {x}(0)=0\).

1. Using the method of multiple scales show that this has an asymptotic series solution of the form \[ x(t)\sim 2R(\varepsilon t)\cos (t+\theta (\varepsilon t))+O(\varepsilon ) \] with \[ \theta _\tau =0,\quad R_\tau =\frac {1}{2}R(1-R^2) \] where \(\tau =\varepsilon t\).
2. Derive the solution \[ R(\tau )=\left (1+a_0e^{-\varepsilon t}\right )^{-1/2} \] and determine \(a_0\) from the initial conditions. Hence, establish that there exits a stable periodic orbit.

1. Consider the van der Pol oscillator \[ \ddot {x}+\varepsilon \dot {x}(x^2-1)+x=0,\quad 0<\varepsilon \ll 1 \] and choose initial conditions \(x(0)=1\), \(\dot {x}=0\). Using the method of multiple scales show that this has an asymptotic series solution of the form \[ x(t)\sim R(\varepsilon t)\cos (t+\theta (\varepsilon t))+O(\varepsilon ) \] with \[ \theta _\tau =0,\quad R_\tau =\frac {1}{8}R(4-R^2) \] where \(\tau =\varepsilon t\), \(\theta (0)=0\) and \(R(0)=1\). Hence, establish that there exits a stable periodic orbit.
2. Consider the Duffing equation \[ \ddot {x}+x=-\varepsilon x^3,\quad 0<\varepsilon \ll 1. \] Using the method of multiple scales show that the leading order solution is of the form \[ x(t)\sim R_0\cos \left (t+t_0+\frac {3}{8}R_0^2\varepsilon t\right )+O(\varepsilon ). \]