Chapter 13

For the pendulum equation \(\ddot{\theta}+(g / l) \sin \theta=0\) find the equations of the trajectories in the phase plane and sketch the phase portrait of the system. Show that on the separatrices $$ \theta(t)=(2 n+1) \pi \pm 4 \arctan [\exp (\omega t+\alpha)] $$ where \(n\) is an integer, \(\alpha=\) constant and \(\omega=\sqrt{g / l}\). [Hint: Use the substitution \(\left.u=\tan \frac{1}{4}\\{\theta-(2 n+1) \pi\\} .\right]\)

The relativistic equivalent of the simple harmonic oscillator equation for a spring with constant \(k\) and a rest mass \(m_{0}\) attached is $$ \frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{m_{0} y}{\sqrt{1-y^{2} / c^{2}}}\right)+k x=0 \quad \text { with } \quad \dot{x}=y $$ where \(c\) is the speed of light. Show that the phase trajectories are given by $$ m_{0} c^{2} / \sqrt{1-y^{2} / c^{2}}+\frac{1}{2} k x^{2}=\text { constant } $$ and sketch the phase portrait for this system.

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 \(\quad \mu=2 \omega_{0}\)

Consider the gradient system (13.17) in the case \(U(x, y)=x^{2}(x-1)^{2}+y^{2}\). Find the critical points and their character. Sketch the phase portrait for the system.

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).

For the Arms-Race model system (13.26) with all parameters positive show that there is an asymptotically stable coexistence or a runaway escalation according as \(c_{1} c_{2}>a_{1} a_{2}\) or \(c_{1} c_{2}

A simple model for the dynamics of malaria due to Ross (1911) and Macdonald (1952) is $$ \begin{aligned} &\dot{x}=\left(\frac{a b M}{N}\right) y(1-x)-r x \\ &\dot{y}=a x(1-y)-\mu y \end{aligned} $$ where: \(x, y\) are the infected proportions of the human host, female mosquito populations, \(N, M\) are the numerical sizes of the human, female mosquito populations, \(a\) is the biting rate by a single mosquito, \(b\) is the proportion of infected bites that result in infection, \(r, \mu\) are per capita rates of recovery, mortality for humans, mosquitoes, respectively. Show that the disease can maintain itself within these populations or must die out according as $$ R=\frac{M}{N} \frac{a^{2} b}{\mu r}>1 \text { or }<1. $$

Show that Rayleigh's equation in the form (13.28) has a single critical point at \((0,0)\) and that this is always unstable. Making use of substitution \(\dot{x}=v / \sqrt{3}\) show that \(v\) satisfies the Van der Pol equation (13.29).

Show that the origin is the only critical point for the system $$ \begin{aligned} &\dot{x}=-y+\alpha x\left(\beta-x^{2}-y^{2}\right) \\ &\dot{y}=x+\alpha y\left(\beta-x^{2}-y^{2}\right) \end{aligned} $$ where \(\alpha, \beta\) are real parameters, with \(\alpha\) fixed and positive and \(\beta\) allowed to take different values. Show that the character of the critical point and the existence of a limit cycle depend on the parameter \(\beta\), so that the system undergoes a supercritical Hopf bifurcation at \(\beta=0\). (Hint: Make the change from Cartesian co-ordinates to plane polars.)

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 \((1926 / 27)]\).

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).

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

In contrast to Problem 16 , consider the perfectly elastic bouncing of a ball vertically under gravity above the plane \(x=0\). We have \(\dot{x}=y, \dot{y}=\) \(-g\) and we can solve for \(x\left(x_{0}, y_{0}, t\right), y\left(x_{0}, y_{0}, t\right)\) in terms of the initial data \(\left(x_{0}, y_{0}\right)\) at \(t=0\). Show that in this case the resulting perturbations \(\Delta x, \Delta y\) essentially grow linearly with time \(t\) along the trajectory when we make perturbations \(\Delta x_{0}, \Delta y_{0}\) in the initial data. That is to say the distance along the trajectory \(d \equiv \sqrt{(\Delta x)^{2}+(\Delta y)^{2}} \sim \kappa t\) when \(t\) is large and \(\kappa\) is a suitable constant.

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.)