Past exam of the topics in applied mathematics course of the University of Oxford / 2007 / B / 5

A saddle-node bifurcation creates or destroys two equilibria when a real eigenvalue passes through zero with a nonzero quadratic normal-form term. With a parameter orientation chosen so the pair appears for $\mu >0$, useThere are no equilibria for $\mu <0$, a double nonhyperbolic equilibrium at zero, and equilibria $q=\pm \sqrt{\mu }$ for $\mu >0$. The positive branch is stable and the negative branch unstable in the active direction; additional hyperbolic directions can make them a node and a saddle equilibrium. “Supercritical” for a fold depends on this chosen parameter orientation; unlike the pitchfork and Hopf cases it is not an intrinsic distinction between the stability of two possible cubic branches.

The supercritical pitchfork normal form of a dynamical system isThe symmetric equilibrium $q=0$ is stable for $\mu <0$ and unstable for $\mu >0$. Two new stable equilibria $q=\pm \sqrt{\mu }$ emerge on the unstable side of the original branch. Reflection symmetry $q\mapsto -q$ preserves the pair. The new equilibria approach zero continuously as the parameter reaches the threshold.

A Hopf bifurcation occurs when a simple pair of complex conjugate eigenvalues crosses the imaginary axis at nonzero frequency. A supercritical normal form of a dynamical system isWriting $z=r{e}^{i\theta }$ gives $\dot{r}=\mu r-r^3$ and $\dot{\theta }={\omega }_0$. A stable periodic orbit of radius $\sqrt{\mu }$ is born for $\mu >0$ as the equilibrium loses stability. In contrast, a subcritical Hopf has an unstable small periodic orbit on the side where the equilibrium is still stable.

The Lorenz equilibria satisfy $y=x$, $z=x^2/b$ and $x(r-1-z)=0$. ThusAt the origin the third eigenvalue is $-b$ and the other two solve ${\lambda }^2+(\sigma +1)\lambda +\sigma (1-r)=0$. Hence the origin is stable for $r<1$, has one zero eigenvalue at $r=1$, and becomes a saddle equilibrium with one unstable direction for $r>1$. The symmetry $(x,y,z)\mapsto (-x,-y,z)$ interchanges the new branches.

The nonzero-equilibrium characteristic polynomial isNear $r=1$ its Routh-Hurwitz inequalities are strict, so both new branches are stable. More explicitly, center-manifold projection at $r=1$ givesThe linear coefficient crosses positively and the cubic coefficient is negative. Therefore the equilibrium bifurcation is a supercritical pitchfork at $r=1$, not a saddle equilibrium-node.

For completeness, the later stability boundary satisfiesIf $\sigma \le b+1$, the nonzero branches remain linearly stable for all $r>1$. If $\sigma >b+1$, they lose stability atAt that point $P=(\lambda +\sigma +b+1)({\lambda }^2+{\omega }_H^2)$ andThis establishes a genuine transverse Hopf crossing. It is subcritical for these positive parameters, so it is not the supercritical Hopf normal form of a dynamical system from (c). One way to check its nonlinear sign is the cubic Hopf formula. At $C_{+}$ write $s=\sqrt{b(r_H-1)}$ andFor right and adjoint eigenvectors $Aq=i{\omega }_{H}q$, $A^{\ast }p=-i{\omega }_{H}p$, $⟨p,q⟩=1$, the cubic coefficient has the sign ofSubstitution and clearing positive factors reduces its sign to that of the following polynomial in $h=\sigma -b-1>0$:Every term is positive. The radial cubic is therefore positive, giving the unstable small cycle on the stable-equilibrium side. There is no saddle equilibrium-node among the equilibrium branches for $\sigma ,b,r>0$: the only zero-eigenvalue branch intersection is the symmetry-forced pitchfork. This does not rule out separate global bifurcations of periodic orbits.