Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 56 / 1

The equilibria have $y=0$ and $x(\lambda +x^2)=0$. ThusAt $O$, the linearization has trace $\kappa$ and determinant $\lambda$. Hence $O$ is a saddle equilibrium when $\lambda <0$, a sink equilibrium when $\lambda >0,\kappa <0$, and a source equilibrium when $\lambda >0,\kappa >0$. The sink equilibrium/source equilibrium may be a node equilibrium or a focus equilibrium; the node-focus discriminant is not a bifurcation curve. At $P_{\pm }$, the trace and determinant are $\kappa +\lambda$ and $-2\lambda$. These points are sink equilibria for $\kappa <-\lambda$ and source equilibria for $\kappa >-\lambda$.

The local bifurcation curves are thereforeFor $\kappa \ne 0$ on the pitchfork curve, a leading centre-manifold equation is $\dot{x}=(\lambda x+x^3)/\kappa +\cdots$. For $\kappa <0$, with unfolding parameter $-\lambda$, the pitchfork is supercritical and creates two sink equilibria from a sink equilibrium that becomes a saddle equilibrium. For $\kappa >0$ it is subcritical in the centre direction; the nonzero branches are unstable, and the transverse eigenvalue is also positive. The origin of parameter space is a reflection-symmetric double-zero degeneration.

To determine Hopf criticality at the origin, use the energy $E=y^2/2+\lambda x^2/2+x^4/4$, whose derivative is $(\kappa -x^2)y^2$. For a small nearly harmonic oscillation $x=R\cos (\sqrt{\lambda }t)$, averaging gives $\dot{R}=\kappa R/2-R^3/8+\cdots$. Thus this Hopf bifurcation is supercritical and creates an attracting periodic orbit for $\kappa >0$. By the supplied opposite-criticality fact, the simultaneous Hopf bifurcations at the two nonzero equilibria are subcritical. A direct calculation confirms this: writing $x=\sqrt{-\lambda }+X$, $Y=-y/\sqrt{-2\lambda }$ puts the cubic radial coefficient at $1/4>0$, including the contribution from the quadratic terms.

The figure includes these local curves and the global curves derived below. All lines emerging from the double-zero point are local asymptotic sketches.

With primes denoting differentiation in $\tau$, the weighted scaling givesIndeed $\dot{x}={\epsilon }^2{u}^{\prime }$ and $\dot{y}={\epsilon }^3{v}^{\prime }$, so the damping terms carry one extra power of $\epsilon$. ConsequentlyAt $\epsilon =0$ this proves conservation of $H$ directly.

For $\alpha =-a_0^2<0$, the potential is a double well. Its minima are at $u=\pm a_0$, where $H=-a_0^4/4$; these are centre equilibria. The origin is a saddle equilibrium and has $H=0$. Energies $-a_0^4/4<H<0$ give two separate closed contours, one in each well. Energies $H>0$ give a single closed contour surrounding both wells. The intervening figure-eight consists of two homoclinic loops:Their outer turning points are $u=\pm \sqrt{2}{a}_0$. The positive loop, for example, is parametrized by $u=\sqrt{2}{a}_0\mathrm{sech}(a_0\tau )$, $v=u^{\prime }$.

A returning orbit must have zero net change of the energy. To first order, integrate the perturbation along either unperturbed homoclinic loop:For the positive loop, $v_h=\pm u_h\sqrt{a_0^2-u_h^2/2}$; the outward and inward halves contribute equally. ThereforeandThe integrals follow by substituting $w=a_0^2-u^2/2$; for the second one use $u^2=2(a_0^2-w)$. HenceThe equality in rescaled parameters is the leading Melnikov balance, not an assertion that the entire finite-parameter bifurcation curve is exactly straight. Reflection symmetry makes both saddle equilibrium loops occur together. The global curve lies below the nonzero-equilibrium Hopf line $\kappa =-\lambda$, as shown in the parameter sketch.

The relevant saddle equilibrium is the origin, with eigenvaluesFor $\lambda <0$, they have opposite signs. Its saddle index isnear the homoclinic curve, because that curve has $\kappa >0$. Along its leading asymptote, $\delta =1-\frac{4}{5}\sqrt{-\lambda }+o(\sqrt{-\lambda })$.

A local passage near the saddle equilibrium contributes a power $s^{\delta }$ to the return map. Because $\delta <1$, its derivative becomes large as the orbit approaches the separatrix. The periodic orbits meeting the loops are therefore unstable, and their periods diverge logarithmically. On the side between the homoclinic and subcritical Hopf curves there are two unstable single-well periodic orbits, surrounding the two sink equilibria. On crossing the double-loop curve downward, they join into one unstable outer periodic orbit surrounding both wells. The attracting outer periodic orbit is not the one that collides with the saddle equilibrium: it persists across this global event.

The unstable outer periodic orbit must subsequently meet that attracting outer periodic orbit in a fold of periodic orbits. This extra curve is required to obtain the six-region diagram. One can see its position from the same energy-balance calculation. After normalizing $a_0=1$, an outer Hamiltonian contour with $H>0$ selectsAt $H=0$, $B=4/5$. The derivative of the denominator diverges positively there, while that of the numerator stays finite, so $B$ initially decreases. Rescaling at large $H$ gives $B(H)\propto \sqrt{H}$, so it eventually increases. At its minimum the stable and unstable selected periodic orbits meet. Thus the outer-cycle fold in a weakly perturbed double-well oscillator has ${\kappa }_{\mathrm{fold}}=c_f(-\lambda )+o(|\lambda |)$ with $0<c_f<4/5$; quadrature gives $c_f\simeq 0.75226$. The decreasing portion of $B(H)$ selects the unstable periodic orbit, and the increasing portion selects the stable one, since the energy drift has sign $\beta -B(H)$.

Write $F={\kappa }_{\mathrm{fold}}(\lambda )$, $G={\kappa }_{\mathrm{hom}}(\lambda )$ and $H=-\lambda$ for $\lambda <0$, so $0<F<G<H$ near the double-zero point. The six open regions are:

The changes III–IV, IV–V and V–VI are respectively a fold of periodic orbits, the double homoclinic event, and the two subcritical Hopf bifurcations. Thus the unstable periodic orbits from the subcritical Hopf bifurcations, the expanding saddle equilibrium passage, and the attracting periodic orbit from the other Hopf bifurcation fit into one consistent portrait.