Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 58 / 1 / iii / Solution

Put $a=a_1>0$, so $a_2=4a/5$ and $a_3=8a/5$. The second equilibrium equation factors as $B({\lambda }_2+a_{3}A)=0$. The resulting equilibria areThe mixed pair exists strictly when ${\lambda }_2({\lambda }_1-5{\lambda }_2/8)>0$; equality describes its merger with an axial equilibrium. At $O$ the eigenvalues of the Jacobian matrix are $({\lambda }_1,{\lambda }_2)$, and at $P$ they are $(-{\lambda }_1,{\lambda }_2-8{\lambda }_1/5)$. At a nonzero mixed equilibrium,Thus both mixed equilibria are saddle equilibria. The axial equilibria are nodes when their two real eigenvalues have the same sign and saddles when the signs differ. In particular,There are exactly three local bifurcation lines away from their common codimension-two intersection. On ${\lambda }_1=0$, $O$ and $P$ meet in a transcritical bifurcation, visible in $\dot{A}={\lambda }_{1}A+a{A}^2$ on the invariant axis $B=0$. On ${\lambda }_2=0$, the mixed pair meets $O$ in a pitchfork bifurcation. Eliminating $A$ near $O$, for fixed nonzero ${\lambda }_1$, gives $\dot{B}={\lambda }_{2}B-(a_2{a}_3/{\lambda }_1)B^3+\cdots$: this pitchfork is supercritical for ${\lambda }_1>0$ and subcritical for ${\lambda }_1<0$. On ${\lambda }_2=8{\lambda }_1/5$, the mixed pair meets $P$. Writing $A=-{\lambda }_1/a+u$ and $\eta ={\lambda }_2-8{\lambda }_1/5$, elimination gives $\dot{B}=\eta B+(a_2{a}_3/{\lambda }_1)B^3+\cdots$. This pitchfork is subcritical for ${\lambda }_1>0$ and supercritical for ${\lambda }_1<0$. A supercritical pitchfork here need not produce a stable mixed state: its other eigenvalue can already be positive. The negative mixed Jacobian determinant settles the full stability question. There is no Hopf bifurcation in this quadratic truncation.

Along ${\lambda }_2={\lambda }_1-\Delta$, the axial curves are $A=0$ and $A=-{\lambda }_1/a$. The mixed pair has the same plotted amplitude for its two signs of $B$:Consequently the mixed curves exist outside the interval whose endpoints are $\Delta$ and $-5\Delta /3$. The stable portions of the axial curves areFor $\Delta >0$, stability passes directly from $O$ to $P$ at the transcritical bifurcation ${\lambda }_1=0$; the two pitchfork bifurcations lie at $-5\Delta /3$ and $\Delta$. For $\Delta <0$, $O$ loses stability at ${\lambda }_1=\Delta$ and $P$ becomes stable only at ${\lambda }_1=-5\Delta /3$. Between these points none of the equilibria is stable. This local statement does not prescribe a bounded attractor of the quadratic truncation. All mixed portions are unstable.