Period-doubling bifurcation

A period-doubling bifurcation occurs when a fixed-point multiplier crosses $-1$ and a nearby period-two orbit is created. Under the generic nondegeneracy conditions, the orbit amplitude is proportional to the square root of the parameter displacement.

A period-doubling cascade is a succession of period-doubling bifurcations creating attracting periods $1,2,4,8,\dots$, with bifurcation parameters tending to an accumulation value. Superstable periodic orbits inside successive stability intervals give a second parameter sequence with the same limiting parameter-scaling ratio in a quadratic universality class.

// Target: dynamical-systems.bigb

For a generic quadratic period-doubling cascade, the Feigenbaum constants are the limiting parameter ratio $\delta$ and spatial ratio $\alpha$. With $s_n$ the successive superstable period-$2^n$ parameters and $d_n=f_{s_n}^{2^{n-1}}(0)$ the signed central-orbit spacing,The positive convention gives $\delta \simeq 4.669201609$ and $\alpha \simeq 2.502907875$. The orientation-reversing spatial scale at the Feigenbaum renormalization fixed point is $a=-1/\alpha$.

// Target: dynamical-systems.bigb

A generalized flip is a codimension-two period-doubling point with a simple multiplier $-1$, a vanishing cubic coefficient in its scalar centre-manifold map, and a nonzero quintic coefficient. In the odd normal form $z^{\prime }=(-1+\beta )z+c\alpha z^3+d{z}^5+\cdots$, symmetric two-cycles solve $\beta +c\alpha z^2+d{z}^4=0$. The discriminant of this quadratic in $z^2$ produces a fold of two-cycles where its positive roots collide. For $c=3$, $d=-3$, this fold is $\beta =-3{\alpha }^2/4$, $z^2=\alpha /2$, $\alpha >0$. At $\alpha =0$ the amplitude is proportional to $|\beta {|}^{1/4}$.

For $F_{\mu }(x)=\mu x+b{x}^2+a{x}^3$ near $\mu =-1$, write $\mu =-1+ϵ$. Then $F_{\mu }^2(x)-x=-2ϵx-2(a+b^2)x^3+O({ϵ}^{2}x,ϵx^2,x^4)$. The small two-cycle is attracting on $\mu <-1$ if $a+b^2>0$ and repelling on $\mu >-1$ if $a+b^2<0$. Vanishing of $a+b^2$ gives a degenerate flip requiring higher terms.

For a $C^4$ real map $f(u)=-u+c_2{u}^2+c_3{u}^3+O(u^4)$, the second iterate isIf $c_3+c_2^2>0$, sufficiently small nonzero $u$ retains its sign under $f^2$ and strictly decreases in absolute value. Its iterates therefore converge to zero, proving local asymptotic stability even though $f^{\prime }(0)=-1$. If $c_3+c_2^2<0$, the second iterate moves small positive points away from zero, proving instability. The zero-coefficient case requires higher-order terms. With strictly stable transverse directions, the centre manifold theorem for a discrete dynamical system applies this scalar test to a higher-dimensional discrete dynamical system.