35 Problem set 5 solutions
Q1. Sensitivity of the climate
- Consider a perturbation of the solar radiation, say \(Q = Q_0 + \delta\) where \(\delta\) is small in comparison to \(Q_0\). Expand now the temperature into a series: \[ T = T_0 + \delta T_1 + \ldots \] Show that at \(O(\delta)\), the perturbation is governed by \[ C \frac{\mathrm{d}T_1}{\mathrm{d}t} = (1 - a(T_0)) - B(T_0) T_1 - A'(T_0) T_1 - T_0 B'(T_0) T_1 - Q_0 a'(T_0) T_1. \]
- Consequently, show that the temperature perturbation can be written as \[ B(T_0) \tau \frac{\mathrm{\partial}T_1}{\mathrm{\partial}t} = [1 - a(T_0)] - \frac{B(T_0)}{g} T_1. \]
- Consider (Equation 28.3) at steady state, so therefore the perturbed equilibrium temperature is equal to \[ \delta T_1 = \frac{[1 - a(T_0)] \delta g}{B(T_0)}. \] If the CO2 level in the atmosphere doubles, then the radiative forcing might be adjusted as: \[ (1 - a(T_0)) \delta = 3.7 \, \mathrm{W} \cdot \mathrm{m}^{-2}. \] Assuming that the climate gain is \(g = 3\) and \(B(T_0) = 1.9 \mathrm{W} \cdot \mathrm{m}^{-2} \cdot ({}^\circ \mathrm{C})^{-1}\), what is the expected increase in temperature?
Q2. A van der Pol equation
- Determine \(f(x)\) so that this equation can be written as a Liénard phase plane system in the form
\[\begin{align} \epsilon \dot{x} &= f(x) + 4y, \\ \dot{y} &= -x. \end{align}\]
- For fixed \(\epsilon > 0\), find the equilibrium point(s) in the phase plane, find their eigenvalues, and classify their linear stability.
- Use the expansions \(x(t) = x_0(t) + \epsilon x_1(t) + O(\epsilon^2)\) to determine the equations for the leading-order slow solution. Sketch the slow manifold, indicate the directions motion on each part, and identify the two attracting points on the curve.
As usual expand the solutions into a series in powers of \(\epsilon\). The slow manifold is \(y_0(x_0) = S(x_0) = (x_0^3 - 3 x_0^2 - 9 x_0)/4\). In image of the slow manifold is shown below in blue. Notice that for \(x < 0\), \(y\) is increasing and for \(x > 0\), \(y\) is decreasing (due to the second differential equation).
- Use the expansions \(x(t) = X_0(T) + \epsilon X_1(T) + O(\epsilon^2)\) and \(y(t) = Y_0(T) + \epsilon Y_1(T) + O(\epsilon^2)\) with \(T = t/\epsilon\) to obtain the leading-order fast solution.
- Use the phase plane to determine the maximum and minimum values of \(x(t)\) during an oscillation. Sketch \(x(t)\) as a function of time.
By checking the derivatives, we can see that the slow manifold has local extrema at \((-1, 5/4)\) and \((3, 27/4)\). In the fast evolution stages, \(y\) is constant and \(x\) evolves from one extrema value to the other on the slow manifold, satisfying \(y = S(x)\).
Here is a plot of the key points.
- Using the time required for the slow motions in part c (neglecting the short times for the fast solutions in part d), determine the leading-order estimate for the period \(P\) of oscillation for the limit cycle.
Q3. Fast-slow dynamics with three variables
- For the system \[ \begin{aligned}[c] \dot{x} &= 2 - y, \\ \dot{y} &= x-z, \\ \epsilon\dot{z} &= y - y^2 + \frac{1}{3}y^3 - z, \end{aligned} \] with the initial conditions of \(x(0) = 1\), \(y(0) = 3\), and \(z(0) = 0\).
Identify the surface \(z = S(x, y)\) that defines the slow manifold. Find the equilibrium point of the leading=order slow phase plane system and show that it is asymptotically stable for \(t \to \infty\). Also determine the form of the initial layer that describes the transition from the initial conditions to the slow manifold.
- For the system \[ \begin{aligned}[c] \dot{x} &= 2 - y, \\ \epsilon\dot{y} &= x-z, \\ \dot{z} &= y - y^2 + \frac{1}{3}y^3 - z, \end{aligned} \] with the initial conditions of \(x(0) = 0\), \(y(0) = 3\), and \(z(0) = 1\).
Show that the slow manifold reduces to a curve that could be written in parametric form as \(x = x(z), y = y(z), z = z\). Determine the asymptotic solution for \(t \to \infty\). Also determine the form of the initial layer that describes the transition from initial conditions to the slow manifold.