Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 60 / 4

After one iteration $y_n\in \{1,-1\}$, so work on that invariant sign sheet. For $x_n\ne 0$,since $z_n=-x_n{y}_n$ and $|y_n|=1$. This is the signed gluing-map reduction. If an arbitrary initial value of $y$ is admitted, the formula applies after that first iterate, not before it. The point $z=0$ represents a separatrix at which the original flow return is singular, even though the scalar map can be extended continuously there.

Let a scalar fixed point have magnitude $r>0$. If $z=-r<0$, then $x$ and $y$ have the same sign; the two lifts are the fixed points $(r,1)$ and $(-r,-1)$. If $z=r>0$, their signs are opposite, and the lift is the period-two orbit $(r,-1)\leftrightarrow (-r,1)$. The scalar fixed-point multiplier isThe two-return fixed-point multiplier of the period-two lift is $m^2$, so in either case dynamical stability requires $m<1$. The negative fixed point represents one of two mirror-related single-lobe periodic orbits in the flow, whereas the positive fixed point represents a double-lobe periodic orbit.

For $\delta >1$, the nearby fixed point is $z=\mu +o(\mu )$ and its fixed-point multiplier tends to zero. Thus two stable single-lobe limit cycles for $\mu <0$ glue into one stable double-lobe limit cycle for $\mu >0$. Their periods diverge as the separatrix is approached; $\mu =0,z=0$ is not a finite-period limit cycle.

For $0<\delta <1$, the small branch instead satisfies $z\sim -\mathrm{sgn}(\mu )(|\mu |/A{)}^{1/\delta }$ and its fixed-point multiplier tends to infinity. Hence one unstable double-lobe limit cycle on $\mu <0$ splits into two unstable single-lobe limit cycles on $\mu >0$. Additional finite-size stable branches depend on the surrounding map; near the resonance $\delta =1$ with $A<1$ they are resolved by the saddle-node bifurcations in part (c).

For a fixed point write $\mu =h(z)=z-A\mathrm{sgn}(z)|z{|}^{\delta }$. A saddle-node bifurcation requires both $f(z)=z$ and $f^{\prime }(z)=1$. For $\delta \ne 1$ these conditions giveThe second derivative is nonzero at either nonzero saddle-node bifurcation, making it an ordinary saddle-node bifurcation. For $\delta <1$, defineThe saddle-node bifurcations occur at $(\mu ,z)=(-C,r)$ and $(C,-r)$, and three scalar fixed points coexist for $|\mu |<C$. The two outer branches have fixed-point multiplier below one and are stable, while the middle branch is unstable. Negative points lift to two symmetry-related limit cycles, so these scalar branch counts need not equal the number of flow limit cycles. The cusp records coexistence of distinct return-map branches; even outside it a negative branch represents a symmetry-related pair of flow cycles.

If $\delta =1-\eta$, $\eta \downarrow 0$, thenThis exponentially narrow region ends in a gluing-map cusp at $(\mu ,\delta )=(0,1)$. At its boundaries the graph of $f$ is tangent to the diagonal at $z=+r$ or $-r$, as shown below. At exactly $\delta =1$, $f=\mu +Az$ and $A<1$ gives one stable fixed point for nonzero $\mu$.

For $\delta =1+\eta$, the formal saddle-node bifurcation radius is $(A\delta {)}^{-1/\eta }\to \infty$, and the corresponding saddle-node bifurcation parameters also diverge. Thus the near-origin gluing-map cusp lies on the $\delta <1$ side; the formal large saddle-node bifurcations of the power-law formula on the other side are outside the local gluing-map approximation. They must not be drawn as a second small gluing-map cusp near the origin.

For $\delta =1+\epsilon >1$, the graph of $h(z)$ has a positive maximum at $z=r$ and a negative minimum at $z=-r$. The central branch $|z|<r$ is stable, while the formal outer branches $|z|>r$ are unstable. Near the gluing point only the stable central branch is relevant: it crosses from negative $z$ to positive $z$, representing the stable two-to-one gluing described in part (b). The formal full power-law diagram has three fixed points between its two large saddle-node bifurcations.

For $\delta =1+\epsilon <1$, the positive branch has its minimum at $(\mu ,z)=(-C,r)$, while the negative branch has its maximum at $(C,-r)$. The inner pieces $|z|<r$ are unstable and outer pieces stable. In the gluing-map cusp interval there is a stable negative branch, a stable positive branch, and an unstable central branch; the unstable branch passes through the singular gluing point with opposite sign to $\mu$. Outside the gluing-map cusp just one stable outer branch remains. The lower panels of the figure plot these diagrams in fold-scaled variables; dashed lines denote instability and solid lines dynamical stability. Neither diagram has a flip bifurcation, since the scalar derivative is positive wherever defined.