Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 65 / 4

For $\lambda >0$ put $s=\sqrt{\lambda }$. The equilibrium points are $(s,0)$ and $(-s,0)$, withThe positive branch is always a saddle. The negative branch is a sink for $\mu <s$ and a source for $\mu >s$; it is a node or focus according as $(\mu -s{)}^2-8s$ is positive or negative. At $\lambda =0$ the two branches meet in a saddle-node bifurcation when $\mu \ne 0$; at $(\lambda ,\mu )=(0,0)$ the double zero eigenvalue with a nontrivial Jordan block gives a Bogdanov–Takens bifurcation. There are no real equilibrium points for $\lambda <0$.

The Hopf bifurcation curve is $\mu =s>0$, equivalently $\lambda ={\mu }^2$ on its positive-$\mu$ branch. Its frequency is $\omega =\sqrt{2s}$. To determine the direction, shift $u=x+s$, $v=y$ at this threshold; thenSet $X=u$, $Y=-v/\omega$ to obtain $\dot{X}=-\omega Y$, $\dot{Y}=\omega X-X^2/\omega +XY$. A quadratic near-identity transformation, or harmonic averaging with its quadratic correction retained, gives the radial cubic coefficientwhere $g=-X^2/\omega +XY$. The radial equation is $\dot{R}=(\mu -s)R/2+{\ell }_H{R}^3+\cdots$. Hence the Hopf bifurcation is subcritical, with an unstable small periodic orbit on the stable-focus side $\mu <s$.

The saddle's outgoing separatrix can return to its incoming separatrix, creating a homoclinic orbit. This global bifurcation cannot be detected from the equilibrium point's Jacobian matrix alone. The weakly Hamiltonian scaling below locates its curve and shows how the unstable Hopf cycle grows into the saddle loop. The final diagram includes the local saddle-node and Hopf curves and this global curve; it describes the neighborhood of the double-zero point.

Set $\tau =\epsilon t$ and use the stated weighted spatial and parameter scales. Dividing the two equations by ${\epsilon }^3$ and ${\epsilon }^4$ respectively giveswhere prime now means differentiation with respect to $\tau$. The Hamiltonian function satisfiesThus $H=v^2/2+\alpha u-u^3/3$ is conserved for $\epsilon =0$.

For $\alpha >0$, let $s=\sqrt{\alpha }$. The conservative system has a center at $(-s,0)$ with $H_c=-2{\alpha }^{3/2}/3$ and a saddle at $(s,0)$ withClosed contours in the left potential well have $H_c<H<H_h$. At $H_h$ the left-hand separatrix is a homoclinic orbit, sinceIt leaves and returns to $(s,0)$, with its left turning point at $u=-2s$. The right separatrix branches are unbounded. The energy levels above the saddle value are open, as are the separate right-hand components of lower energy levels. For $\alpha \le 0$ there is no nondegenerate potential well and no such saddle loop.

The displayed orbit has the normalization $\alpha =1$. To verify it put $z=\tanh (\tau /\sqrt{2})$ and $f=1-z^2={\mathrm{sech}}^2(\tau /\sqrt{2})$. Then $z^{\prime }=f/\sqrt{2}$, $f^{\prime }=-\sqrt{2}fz$, soAlso $u=-2+3z^2$, and $(u^2-1)=3(1-z^2)(1-3z^2)$, exactly the same expression as $v^{\prime }$. Therefore the conservative equations hold. As $\tau \to \pm \infty$, $(u,v)\to (1,0)$; at $\tau =0$ the orbit reaches $(-2,0)$. Its conserved energy is $2/3$, so this is the required homoclinic orbit.

For a general positive $\alpha$ set $s=\sqrt{\alpha }$. Rescaling the verified solution givesDirect differentiation gives $u^{\prime }=v$ and $v^{\prime }=u^2-\alpha$, and the endpoints are $(s,0)$. This restores the parameter dependence implicit in the normalized printed orbit.

A persistent saddle loop must return with the same energy. The Melnikov energy-balance method therefore requires, at first order,For the normalized orbit of part (c), put $z=\tanh (\tau /\sqrt{2})$. Since $u_0=-2+3z^2$, $v_0^2=18(1-z^2{)}^2{z}^2$ and $d\tau =\sqrt{2}dz/(1-z^2)$,For $s=\sqrt{\alpha }$ the general orbit has $I_0(s)=s^{5/2}{I}_0$ and $I_1(s)=s^{7/2}{I}_1$. The energy mismatch is therefore proportional to $I_0(s)[\beta -(5/7)s]$, whose derivative with respect to $\beta$ is nonzero. This simple zero selects the saddle-loop curve:The remainder can be sharpened. Reversal of $\tau$ and $v$ interchanges $\epsilon$ and $-\epsilon$ without changing the saddle-loop condition. Smoothness of the homoclinic splitting thus makes the selected $\beta$ an even function of $\epsilon$, givingThe leading coefficient alone is enough to separate it from the Hopf curve ${\mu }_H=\sqrt{\lambda }$. The integral hint in the PDF equals $2/(k+1)$ for nonnegative even integers $k$; for odd $k$ its integrand is odd and the integral is zero. Only even powers enter the calculation above.

Near the double-zero point, and with the stipulated absence of further bifurcations, the distinct open regions are:

On the homoclinic curve the cycle becomes a saddle loop; its positive saddle trace $\mu +\sqrt{\lambda }$ makes the loop repelling on the cycle side. At the Hopf curve the unstable cycle shrinks into the negative equilibrium point. The node–focus discriminant curves merely change the spiral appearance and are not additional bifurcations. The impossibility of a periodic orbit for $\lambda <0$ can also be seen without a picture: at a local minimum of periodic $x(t)$ one would have $y=0$ and $\ddot{x}=-\lambda +x^2>0$, but at its maximum the same strictly positive expression would be required to be nonpositive.