Period-doubling bifurcation

In dynamical systems theory, a period-doubling bifurcation occurs when a slight change in a system's parameters causes a new periodic trajectory to emerge from an existing periodic trajectory—the new one having double the period of the original. With the doubled period, it takes twice as long (or, in a discrete dynamical system, twice as many iterations) for the numerical values visited by the system to repeat themselves.

A period-halving bifurcation occurs when a system switches to a new behavior with half the period of the original system.

A period-doubling cascade is an infinite sequence of period-doubling bifurcations. Such cascades are a common route by which dynamical systems develop chaos. In hydrodynamics, they are one of the possible routes to turbulence. Period-halving bifurcations (L) leading to order, followed by period-doubling bifurcations (R) leading to chaos.

Examples Bifurcation diagram for the logistic map. It shows the attractor values, like $x_{*}$ and $x'_{*}$ , as a function of the parameter $r$ .

Logistic map

The logistic map is

$x_{n+1}=rx_{n}(1-x_{n})$ where $x_{n}$ is a function of the (discrete) time $n=0,1,2,\ldots$ . The parameter $r$ is assumed to lie in the interval $(0,4]$ , in which case $x_{n}$ is bounded on $[0,1]$ .

For $r$ between 1 and 3, $x_{n}$ converges to the stable fixed point $x_{*}=(r-1)/r$ . Then, for $r$ between 3 and 3.44949, $x_{n}$ converges to a permanent oscillation between two values $x_{*}$ and $x'_{*}$ that depend on $r$ . As $r$ grows larger, oscillations between 4 values, then 8, 16, 32, etc. appear. These period doublings culminate at $r\approx 3.56995$ , beyond which more complex regimes appear. As $r$ increases, there are some intervals where most starting values will converge to one or a small number of stable oscillations, such as near $r=3.83$ .

In the interval where the period is $2^{n}$ for some positive integer $n$ , not all the points actually have period $2^{n}$ . These are single points, rather than intervals. These points are said to be in unstable orbits, since nearby points do not approach the same orbit as them.

Kuramoto–Sivashinsky equation Period doubling in the Kuramoto–Sivashinsky equation with periodic boundary conditions. The curves depict solutions of the Kuramoto–Sivashinsky equation projected onto the energy phase plane (E, dE/dt), where E is the L2-norm of the solution. For ν = 0.056, there exists a periodic orbit with period T ≈ 1.1759. Near ν ≈ 0.0558, this solution splits into 2 orbits, which further separate as ν is decreased. Exactly at the transitional value of ν, the new orbit (red-dashed) has double the period of the original. (However, as ν increases further, the ratio of periods deviates from exactly 2.)

The Kuramoto–Sivashinsky equation is an example of a spatiotemporally continuous dynamical system that exhibits period doubling. It is one of the most well-studied nonlinear partial differential equations, originally introduced as a model of flame front propagation.

The one-dimensional Kuramoto–Sivashinsky equation is

$u_{t}+uu_{x}+u_{xx}+\nu \,u_{xxxx}=0$ A common choice for boundary conditions is spatial periodicity: $u(x+2\pi ,t)=u(x,t)$ .

For large values of $\nu$ , $u(x,t)$ evolves toward steady (time-independent) solutions or simple periodic orbits. As $\nu$ is decreased, the dynamics eventually develops chaos. The transition from order to chaos occurs via a cascade of period-doubling bifurcations, one of which is illustrated in the figure.

Logistic map for a modified Phillips curve

Consider the following logistical map for a modified Phillips curve:

$\pi _{t}=f(u_{t})+b\pi _{t}^{e}$ $\pi _{t+1}=\pi _{t}^{e}+c(\pi _{t}-\pi _{t}^{e})$ $f(u)=\beta _{1}+\beta _{2}e^{-u}\,$ $b>0,0\leq c\leq 1,{\frac {df}{du}}<0$ where :

• $\pi$ is the actual inflation
• $\pi ^{e}$ is the expected inflation,
• u is the level of unemployment,
• $m-\pi$ is the money supply growth rate.

Keeping $\beta _{1}=-2.5,\ \beta _{2}=20,\ c=0.75$ and varying $b$ , the system undergoes period-doubling bifurcations and ultimately becomes chaotic.[citation needed]

Experimental observation

Period doubling has been observed in a number of experimental systems. There is also experimental evidence of period-doubling cascades. For example, sequences of 4 period doublings have been observed in the dynamics of convection rolls in water and mercury. Similarly, 4-5 doublings have been observed in certain nonlinear electronic circuits. However, the experimental precision required to detect the ith doubling event in a cascade increases exponentially with i, making it difficult to observe more than 5 doubling events in a cascade.