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.
Examples.
Logistic map.
The logistic map is 
where formula_2 is a function of the (discrete) time formula_3. The parameter formula_4 is assumed to lie in the interval formula_5, in which case formula_2 is bounded on formula_7.
For formula_4 between 1 and 3, formula_2 converges to the stable fixed point formula_10. Then, for formula_4 between 3 and 3.44949, formula_2 converges to a permanent oscillation between two values formula_13 and formula_14 that depend on formula_4. As formula_4 grows larger, oscillations between 4 values, then 8, 16, 32, etc. appear. These period doublings culminate at formula_17, beyond which more complex regimes appear. As formula_4 increases, there are some intervals where most starting values will converge to one or a small number of stable oscillations, such as near formula_19.
In the interval where the period is formula_20 for some positive integer formula_21, not all the points actually have period formula_20. 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
A common choice for boundary conditions is spatial periodicity: formula_24.
For large values of formula_25, the solution formula_26 evolves toward steady (time-dependent) solutions or simple periodic orbits. As formula_25 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: 
formula_28
formula_29
formula_30
formula_31
where :
Keeping formula_35 and varying formula_36, 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 "i"th doubling event in a cascade increases exponentially with "i", making it difficult to observe more than 5 doubling events in a cascade.