Past exam of the nonlinear systems course of the University of Oxford / 2023 / 1

At a fixed point, the first equation gives $y=0$, the second then gives $x=0$, and the third gives $z=-\mu$. Thus there is exactly one equilibrium for every real $\mu$:The Jacobian matrix there isOne eigenvalue is $-1$. The other two have product one and sum $5ϵ\mu$. For $\mu <0$ their real parts are negative, so the equilibrium is asymptotically stable; for $\mu >0$ both real parts are positive, leaving two unstable directions and one stable direction. At $\mu =0$ the eigenvalues are $-1,i,-i$, withTherefore the only local bifurcation of this equilibrium is a Hopf bifurcation at $\mu =0$. A repeated real eigenvalue away from zero does not itself cause a bifurcation: the equilibrium remains hyperbolic there.

The cubic centre manifold calculation below gives the Hopf normal form radial coefficient $-9ϵ/8<0$, so this is a supercritical Hopf bifurcation. Small attracting periodic orbits occur on the $\mu >0$ side. In particular, at $\mu =0$ the leading cubic radial drift is inward, although linear stability alone cannot decide that boundary case.

Adjoin $\dot{\mu }=0$ and put $w=z+\mu$. The extended centre manifold is a graph $w=H(x,y,\mu )$ tangent to $w=0$ at the origin. The equations becomeHere $ϵ>0$ is fixed. The centre-manifold invariance equation, obtained by the chain rule, isWrite $H=H_2+O({\rho }^3)$, where $\rho =|x|+|y|+|\mu |$ and $H_2$ is a homogeneous quadratic polynomial. To quadratic order the centre-manifold invariance equation isThe terms $\mu x,\mu y,{\mu }^2$ have zero coefficients: their equations are homogeneous, and the operator $1+y{\partial }_x-x{\partial }_y$ is invertible on these quadratic polynomials. For $H_2=a{x}^2+bxy+c{y}^2$, comparison of coefficients givesHence $a=3/5$, $b=-2/5$ and $c=2/5$, givingSubstitution into the $y$ equation givesThe constant in the big O notation may depend on fixed $ϵ$. Thus the equations printed in part (b) are the retained terms of the local Taylor series, rather than an exact identity for the full reduced vector field.

For the Hopf bifurcation classification in part (a), set $\mu =0$ and use $x=r\cos \theta$, $y=-r\sin \theta$. The radial derivative is $\dot{r}=(x\dot{x}+y\dot{y})/r$. Its cubic angular mean isHere $⟨\cdot ⟩$ denotes the mean over one revolution. Removing the oscillatory cubic terms by a near-identity transformation gives this same cubic radial coefficient in the Hopf normal form. The linear radial coefficient is $5ϵ\mu /2$.

Apply the Poincaré-Lindstedt method to the truncated system from part (b). Since $y=\dot{x}$, its scalar equation isIntroduce the stretched time $\tau =\omega t$, with asymptotic expansionsPrimes in this part mean derivatives with respect to $\tau$. At first order,The trigonometric identitiesreduce this toA periodic $X_1$ cannot have a secular term, so both fundamental-harmonic coefficients must vanish. Taking $A>0$ givesIn particular $\mu >0$ is necessary. Once these conditions hold, a bounded particular correction isadding fundamental sine and cosine terms adjusts the phase and the first correction to the amplitude. Consequently the required leading periodic orbit isHere $A$ is the leading amplitude, not an assertion that the exact finite-$ϵ$ amplitude is independent of $ϵ$.

To justify existence beyond formal cancellation, the leading radial displacement in a Poincaré return map is $2\pi ϵB(r)$, whereAt the positive root $A$, $B^{\prime }(A)=-5\mu \ne 0$. The implicit function theorem, applied after dividing the displacement by $ϵ$, continues this root to a periodic orbit for sufficiently small positive $ϵ$. Its radial Floquet multiplier is $1-10\pi \mu ϵ+O({ϵ}^2)$, so it is attracting. These statements apply to the truncated system with fixed $\mu >0$; its interpretation as a local approximation to the original three-dimensional system also requires $\mu$ small, so that $A=O(\sqrt{\mu })$ stays near the Hopf bifurcation.