Period-doubling bifurcation

In mathematics, a period doubling bifurcation in a discrete dynamical system is a bifurcation in which the system switches to a new behavior with twice the period of the original system. 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

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, and $m - \pi$ 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

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