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 one route by which dynamical systems can develop chaos. In hydrodynamics, they are one of the possible routes to turbulence.
Examples
Logistic map The logistic map is
x n + 1 = r x n ( 1 − x n ) {\displaystyle x_{n+1}=rx_{n}(1-x_{n})}
where x n {\displaystyle x_{n}} is a function of the (discrete) time n = 0 , 1 , 2 , … {\displaystyle n=0,1,2,\ldots } . The parameter r {\displaystyle r} is assumed to lie in the interval [ 0 , 4 ] {\displaystyle [0,4]} , in which case x n {\displaystyle x_{n}} is bounded on [ 0 , 1 ] {\displaystyle [0,1]} . For r {\displaystyle r} between 1 and 3, x n {\displaystyle x_{n}} converges to the stable fixed point x ∗ = ( r − 1 ) / r {\displaystyle x_{*}=(r-1)/r} . Then, for r {\displaystyle r} between 3 and 3.44949, x n {\displaystyle x_{n}} converges to a permanent oscillation between two values x ∗ {\displaystyle x_{*}} and x ∗ ′ {\displaystyle x'_{*}} that depend on r {\displaystyle r} . As r {\displaystyle r} grows larger, oscillations between 4 values, then 8, 16, 32, etc. appear. These period doublings culminate at r ≈ 3.56995 {\displaystyle r\approx 3.56995} , beyond which more complex regimes appear. As r {\displaystyle 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 {\displaystyle r=3.83} , where there is a stable period-three solution. In the interval where the period is 2 n {\displaystyle 2^{n}} for some positive integer n {\displaystyle n} , not all the points actually have period 2 n {\displaystyle 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
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 + u u x + u x x + ν u x x x x = 0 {\displaystyle u_{t}+uu_{x}+u_{xx}+\nu \,u_{xxxx}=0}
A common choice for boundary conditions is spatial periodicity: u ( x + 2 π , t ) = u ( x , t ) {\displaystyle u(x+2\pi ,t)=u(x,t)} . For large values of ν {\displaystyle \nu } , u ( x , t ) {\displaystyle u(x,t)} evolves toward steady (time-independent) solutions or simple periodic orbits. As ν {\displaystyle \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:
π t = f ( u t ) + b π t e {\displaystyle \pi _{t}=f(u_{t})+b\pi _{t}^{e}}
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