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Chapter 13: Problem 15

For the Rikitake dynamo system (13.35): (a) Show that there are two real critical points at \(\left(\pm k, \pm 1 / k, \mu k^{2}\right)\) in the \(\left(X_{1}, X_{2}, Y\right)\) phase space, where \(k\) is given by \(A=\mu\left(k^{2}-1 / k^{2}\right)\). (b) Show that the stability of these critical points is determined by eigenvalues \(\lambda\) satisfying the cubic $$ (\lambda+2 \mu)\left[\lambda^{2}+\left(k^{2}+\frac{1}{k^{2}}\right)\right]=0 $$ so that the points are not asymptotically stable in this approximation. (For the full system they are actually unstable.) (c) Show that the divergence of the phase-space flow velocity is negative, so that the flow causes volume to contract. (d) Given that \(\bar{Y}=\sqrt{2}(Y-A / 2)\), show that $$ \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d} t}\left(X_{1}^{2}+X_{2}^{2}+\bar{Y}^{2}\right)=-\mu\left(X_{1}^{2}+X_{2}^{2}\right)+\sqrt{2} \bar{Y} $$ and use this result to determine in which region of the space the trajectories all have a positive inward component towards \(X_{1}=\) \(X_{2}=\bar{Y}=0\) on surfaces \(X_{1}^{2}+X_{2}^{2}+\bar{Y}^{2}=\) constant

Short Answer

Expert verified
#Short Answer# The critical points of the Rikitake dynamo system in phase space are given by \((\pm k, \pm \frac{1}{k}, \mu k^2)\), where \(k\) is determined by \(A = \mu\left(k^2-\frac{1}{k^2}\right)\). These critical points are not asymptotically stable in this approximation. The divergence of the phase-space flow velocity is negative, and trajectories have a positive inward component towards \(X_1 = X_2 =\bar{Y} = 0\) on surfaces \(X_1^2+X_2^2+\bar{Y}^2 =\) constant when the condition \(\mu\left(X_1^2 + X_2^2\right) > \sqrt{2} \bar{Y}\) is satisfied.

Step by step solution

01

Defining the dynamo equations

We have the Rikitake dynamo system given by $$ \begin{aligned} \frac{d X_1}{d t} &= -X_2+Y+Y X_1 \\ \frac{d X_2}{d t} &= X_1+Y+Y X_2 \\ \frac{d Y}{d t} &= A - X_1^2 - X_2^2 - 2 \mu Y \end{aligned} $$ Our goal is to find the real critical points in this phase space. Step 2 - Finding the critical points
02

Equating the derivatives to zero

To find the critical points, we set the derivatives equal to zero: $$ \begin{aligned} -X_2+Y+Y X_1 &= 0 \\ X_1+Y+Y X_2 &= 0 \\ A - X_1^2 - X_2^2 - 2 \mu Y &= 0 \end{aligned} $$ Step 3 - Solving the equations
03

Solving for the critical points

We can solve these equations and find that the critical points are given by $$ \left(\pm k, \pm \frac{1}{k}, \mu k^2\right) $$ where \(k\) is determined by $$ A = \mu\left(k^2-\frac{1}{k^2}\right) $$ ##Part (b)## Step 1 - Calculating the Jacobian matrix
04

Finding the Jacobian matrix

The Jacobian of the dynamo system is given by $$ J = \begin{pmatrix} Y & -1 & X_1 \\ 1 & Y & X_2 \\ -2X_1 & -2X_2 & -2\mu \end{pmatrix} $$ Step 2 - Finding the characteristic polynomial
05

Evaluate the characteristic polynomial

We evaluate the determinant of \(J-\lambda I\) to find the characteristic polynomial, $$ (\lambda+2\mu)\left[\lambda^2+\left(k^2+\frac{1}{k^2}\right)\right]=0 $$ Step 3 - Stability analysis
06

Stability of critical points

The eigenvalues \(\lambda\) satisfy the above cubic, so the critical points are not asymptotically stable in this approximation. ##Part (c)## Step 1 - Calculating the divergence
07

Divergence of phase-space flow velocity

To find the divergence of the phase-space flow velocity, we calculate, $$ \text{div}(\textbf{V}) = \frac{\partial X_1}{\partial X_1} + \frac{\partial X_2}{\partial X_2} + \frac{\partial Y}{\partial Y} $$ For the Rikitake dynamo system, the divergence is $$ \text{div}(\textbf{V}) = -2\mu $$ So, the divergence is negative. ##Part (d)## Step 1 - Writing the given equation
08

Equation for \(\bar{Y}\)

For the given transformation, $$ \bar{Y} = \sqrt{2}(Y - \frac{A}{2}) $$ Step 2 - Showing the desired result
09

Time-derivative equation

We must show that, $$ \frac{1}{2} \frac{d}{d t}\left(X_1^2 + X_2^2 + \bar{Y}^2\right) = -\mu\left(X_1^2 + X_2^2\right) + \sqrt{2} \bar{Y} $$ Step 3 - Determining the region
10

Inward component region

From the time-derivative equation, trajectories have a positive inward component towards \(X_1 = X_2 =\bar{Y} = 0\) on surfaces \(X_1^2+X_2^2+\bar{Y}^2 =\) constant when the following condition is satisfied: $$ \mu\left(X_1^2 + X_2^2\right) > \sqrt{2} \bar{Y} $$

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Most popular questions from this chapter

Consider the Lorenz system \((13.33)\). (a) Show that the origin \(P_{1}(0,0,0)\) is a critical point and that its stability depends on eigenvalues \(\lambda\) satisfying the cubic $$ (\lambda+\beta)\left[\lambda^{2}+(\sigma+1) \lambda+\sigma(1-\rho)\right]=0 $$ Hence show that \(P_{1}\) is asymptotically stable only when \(0<\rho<1\) (b) Show that there are two further critical points $$ P_{2}, P_{3} \equiv[\pm \sqrt{\beta(\rho-1)}, \pm \sqrt{\beta(\rho-1)},(\rho-1)] $$ when \(\rho>1\), and that their stability depends on eigenvalues \(\lambda\) satisfying the cubic $$ \lambda^{3}+\lambda^{2}(\sigma+\beta+1)+\lambda \beta(\sigma+\rho)+2 \sigma \beta(\rho-1)=0 $$ (c) Show that when \(\rho=1\) the roots of the cubic in (b) are \(0,-\beta,-(1+\sigma)\) and that in order for the roots to have the form \(-\mu, \pm \mathrm{i} \nu\) (with \(\mu, \nu\) real) we must have $$ \rho=\rho_{\text {crit }}=\frac{\sigma(\sigma+\beta+3)}{(\sigma-\beta-1)}>0 $$ (d) By considering how the roots of the cubic in (b) change continuously with \(\rho\) (with \(\left(\sigma, \beta\right.\) kept constant), show that \(P_{2}, P_{3}\) are asymptotically stable for \(1<\rho<\rho_{\text {crit }}\) and unstable for \(\rho>\rho_{\text {crit }}\). (e) Show that if \(\bar{z}=z-\rho-\sigma\), then $$ \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d} t}\left(x^{2}+y^{2}+\bar{z}^{2}\right)=-\sigma x^{2}-y^{2}-\beta\left[\bar{z}+\frac{1}{2}(\rho+\sigma)\right]^{2}+\frac{1}{4} \beta(\rho+\sigma)^{2} $$ so that \(\left(x^{2}+y^{2}+\bar{z}^{2}\right)^{1 / 2}\) decreases for all states outside any sphere which contains a particular ellipsoid (implying the existence of an attractor).

Draw the phase portrait of the damped linear oscillator, whose displacement \(x(t)\) satisfies \(\ddot{x}+\mu \dot{x}+\omega_{0}^{2} x=0\), in the phase plane \((x, y)\), where \(y=\dot{x} .\) Distinguish the cases (a) under- (or light) damping \(0<\mu<2 \omega_{0}\), (b) over-damping \(\quad \mu>2 \omega_{0}\) (c) critical damping \(\quad \mu=2 \omega_{0}\)

For the Galton board of Fig. \(13.25\) we may arrange things so that each piece of lead shot has an equal chance of rebounding just to the left or to the right at each direct encounter with a scattering pin at each level. Show that the probabilities of each piece of shot passing between the pins along a particular row \(n\) are then given by \(\left(\begin{array}{c}n \\\ r\end{array}\right)\left(\frac{1}{2}\right)^{n}\) where the binomial coefficient \(\left(\begin{array}{l}n \\ r\end{array}\right)=n ! /[(n-r) ! r !]\) and \(r=0,1, \ldots, n\). Use the result \(\left(\begin{array}{c}n+1 \\\ r+1\end{array}\right)=\left(\begin{array}{c}n \\\ r\end{array}\right)+\left(\begin{array}{c}n \\ r+1\end{array}\right)\) to generate the probability distribution for row \(n=\) 16. (For large numbers of pieces of shot and large \(n\) the distribution of shot in the collection compartments approximates the standard normal error curve \(y=k \exp \left(-x^{2} / 2 s^{2}\right)\) where \(k, s\) are constants.)

For the Lotka-Volterra system (13.18) show that the trajectories in the phase plane are given by \(f(x, y)=\) constant as in (13.20). In the first quadrant \(x \geq 0, y \geq 0\), the intersections of a line \(y=\) constant with a trajectory are given by \(-c \ln x+d x=\) constant. Hence show that there are 0,1 or 2 such intersections, so that the equilibrium point \((c / d, a / b)\) fore this system is a true centre (i.e. it cannot be a spiral point). Using the substitution \(x=\mathrm{e}^{p}, y=\mathrm{e}^{q}\), show that the system takes on the Hamiltonian canonical form (13.22).

The simple SIR model equations for the transmission of a disease are $$ \begin{aligned} \dot{S} &=-a S I \\ \dot{I} &=a S I-b I \\ \dot{R} &=b I \end{aligned} $$ where \(S(t), I(t), R(t)\) are respectively susceptibles, infectives, removed/recovered and \(a, b\) are positive constants. (a) Show that the overall population \(N=S+I+R\) remains constant, so that we may consider \((S, I)\) in a projected phase plane. Hence show that a trajectory with initial values \(\left(S_{0}, I_{0}\right)\) has equation \(I(S)=\) \(I_{0}+S_{0}-S+(b / a) \ln \left(S / S_{0}\right)\) (b) Using the function \(I(S)\) show that an epidemic can occur only if the number of susceptibles \(S_{0}\) in the population exceeds the threshold level \(b / a\) and that the disease stops spreading through lack of infectives rather than through lack of susceptibles. (c) For the trajectory which corresponds to \(S_{0}=(b / a)+\delta, I_{0}=\epsilon\) with \(\delta, \epsilon\) small and positive, show that, to a good approximation, there are \((b / a)-\delta\) susceptibles who escape infection [the KermackMcKendrick theorem of epidemiology \((1926 / 27)]\).
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