Period-doubling bifurcation

In mathematics, a period doubling bifurcation in a discrete dynamical system is a bifurcation in which a slight change in a parameter value in the system's equations leads to the system switching to a new behavior with twice the period of the original system. With the doubled period, it takes twice as many iterations as before for the numerical values visited by the system to repeat themselves.

A period doubling cascade is a sequence of doublings and further doublings of the repeating period, as the parameter is adjusted further and further.

Period doubling bifurcations can also occur in continuous dynamical systems, namely when a new limit cycle emerges from an existing limit cycle, and the period of the new limit cycle is twice that of the old one.

Examples

• Logistic map
• Logistical map for a modified Phillips curve

File:Bifurc pi empl.jpg

Bifurcation diagram for the modified Phillips curve.

Consider the following logistical map for a modified Phillips curve:

$\pi_{t} = f(u_{t}) + a \pi_{t}^e$

$\pi_{t+1} = \pi_{t}^e + c (\pi_{t} - \pi_{t}^e)$

$f(u) = \beta_{1} + \beta_{2} e^{-u} \,$

$b > 0, 0 \leq c \leq 1, \frac {df} {du} < 0$

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 after a point becomes chaotic, as illustrated in the bifurcation diagram on the right.

• Complex quadratic map

File:Bifurcation1-2.png

Bifurcation from period 1 to 2 for complex quadratic map

Period-halving bifurcation

Period-halving bifurcations (L) leading to order, followed by period doubling bifurcations (R) leading to chaos.

A period halving bifurcation in a dynamical system is a bifurcation in which the system switches to a new behavior with half the period of the original system. A series of period-halving bifurcations leads the system from chaos to order.

See also

• Feigenbaum constants

External links

• The Flip (Period Doubling) Bifurcation in Discrete Time, Dynamic Processes