# 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 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
    $x_{n+1}=rx_{n}(1-x_{n})$
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.

### Quadratic map

Real version of complex quadratic map is related with real slice of the Mandelbrot set.
* Period-doubling cascade in an exponential mapping of the Mandelbrot set
* 1D version with an exponential mapping
* period doubling bifurcation

### 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
    $u_{t}+uu_{x}+u_{xx}+\nu \,u_{xxxx}=0$
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.

## 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.