Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 4 / 31A (Dynamical Systems)

The fixed point zero has multiplier $F^{\prime }(0)=\mu$, so it is attracting for $|\mu |<1$ and repelling for $|\mu |>1$. Other real fixed points satisfyTheir multiplier is $M=1+bx+2a{x}^2$, and they are attracting exactly when $-2<bx+2a{x}^2<0$. At equality $M=\pm 1$ the linear test is inconclusive. For $b\ne 0$ two branches meet at $x_s=-b/(2a)$, ${\mu }_s=1+b^2/(4a)$, a generic saddle-node bifurcation. At $\mu =1$, the nonzero branch crossing zero gives a transcritical bifurcation when $b\ne 0$.

The three sketches use $(a,b)=(1,0),(1,1/2),(-1,-1/2)$ respectively; case (iii) therefore illustrates the subcritical-flip choice.

For $b=0,a>0$, the nonzero branches $x=\pm \sqrt{(1-\mu )/a}$ exist for $\mu <1$ and have multiplier $3-2\mu >1$, so both repel. Zero changes from attracting below one to repelling above one: the bifurcation at one is a subcritical pitchfork bifurcation. At minus one its multiplier crosses $-1$, producing a supercritical period-doubling bifurcation; the newly stable small two-cycle lies on the side $\mu <-1$.

For $a,b>0$, ${\mu }_s>1$. The negative branch segment $-b/(2a)<x<0$ has $M<1$ and is attracting wherever $M>-1$; all other nonzero segments repel or undergo further flips according to the full multiplier criterion. Zero and the crossing branch exchange stability at one, and the two negative branches meet at the saddle-node. The origin's flip at minus one is supercritical.

For $a,b<0$, ${\mu }_s<1$. The local attracting nonzero segments have $x<-b/(2a)$ or $x>0$, again subject to $M>-1$; the segment between the saddle-node and zero repels. The crossings at ${\mu }_s$ and one are saddle-node and transcritical. The quadratic-cubic flip criticality at the origin requires additional information: writing $\mu =-1+ϵ$ givesIt is supercritical if $a+b^2>0$ and subcritical if $a+b^2<0$, with a degenerate flip if $a+b^2=0$. The sign assumptions in case (iii) do not determine this criticality. The original “no further bifurcations in the region sketched” qualification is local; it does not remove this possible degeneracy or the additional $M=-1$ crossings farther along fixed branches.

For $F(x)=x(\mu -x^2)$, the fixed points are zero for all $\mu$ and $\pm \sqrt{\mu -1}$ for $\mu >1$. Their multipliers are respectively $\mu$ and $3-2\mu$, giving attraction for $-1<\mu <1$ and $1<\mu <2$ respectively.

Remove these fixed points from the roots of $F^2(x)=x$. The factor $\mu +1-x^2$ gives the genuine two-cycleIts multiplier is $(2\mu +3{)}^2>1$, so it is always unstable. It collapses to zero at $\mu =-1$, the subcritical period-doubling bifurcation of zero.

The remaining equation is $x^4-\mu x^2+1=0$, whose positive roots in $x^2$ exist only for $\mu \ge 2$. At $\mu =2$ they give the fixed points $\pm 1$, not genuine two-cycles. For $\mu >2$, put $r=\sqrt{(\mu +\sqrt{{\mu }^2-4})/2}>1$. Since $\mu =r^2+r^{-2}$, $F(r)=r^{-1}$ and $F(r^{-1})=r$. There are exactly two further cycles,Their common multiplier is $(\mu -3r^2)(\mu -3r^{-2})=9-2{\mu }^2$. They are attracting for $2<\mu <\sqrt{5}$ and repelling for $\mu >\sqrt{5}$, with marginal multiplier $-1$ at $\sqrt{5}$. Thus there are no cycles for $\mu \le -1$, one for $-1<\mu \le 2$, and three for $\mu >2$.

The nonzero fixed points emerge in a supercritical pitchfork bifurcation at one and each flips to an attracting two-cycle at two. Beyond $\sqrt{5}$ further period doubling to four-cycles is expected, followed by additional bifurcations and potentially chaotic dynamics. This local expectation does not assert chaos for every larger parameter; periodic windows and escape are possible.