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. That is, there exists two points such that applying the dynamics to each of the points yields the other point. 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,
• $m - \pi$ 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

References

• Kuznetsov, Yuri A. (2004). Elements of Applied Bifurcation Theory. Applied Mathematical Sciences 112 (3rd ed.). Springer-Verlag. ISBN 0-387-21906-4. Zbl 1082.37002. 

External links

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