Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 32B (Dynamical Systems)

A horseshoe consists of two disjoint intervals each mapped across their union, producing full two-symbol itinerary dynamics.

Glendinning chaos means there is an invariant set on which an iterate is semiconjugate to the full shift, giving sensitive dependence and periodic points of every period. For a 3-cycle ordered $x_1<x_2<x_3$, the intermediate-value theorem applied to the images of the intervening intervals constructs a horseshoe for an iterate; hence $F$ is chaotic.

At zero, $F^{\prime }(0)=-(1+\mu )$, so it is stable for $-2<\mu <0$, loses stability through multiplier $+1$ at $-2$ and $-1$ at $0$. Nonzero fixed points areborn in a saddle-node at $\mu =-2-q^2/(4p)$. Their stability follows from $F^{\prime }(x_{\ast })=3+\mu +p{x}_{\ast }^2$; $|F^{\prime }|<1$ marks solid branches. Thus the three bifurcations are the saddle-node, the exchange at $-2$, and the flip at $0$.

With $q=0$, expand $F(F(x))$ for $x=O(\sqrt{\mu })$ through $O({\mu }^{3/2})$. Besides zero, $F^2(x)=x$ gives $x^2=\mu /p+O({\mu }^2)$, and the two points are exchanged by $F$. The derivative product is $1-4\mu +O({\mu }^2)$, so the period-two orbit $|x|=\sqrt{\mu /p}+O({\mu }^{3/2})$ is stable.