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.
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[edit] Examples
- Logistic map
- Logistical map for a modified Phillips curve
Consider the following logistical map for a modified Phillips curve:




where
is the actual inflation,
is the expected inflation, u is the level of unemployment, and
is the money supply growth rate. Keeping
and varying
, the system undergoes period doubling bifurcations, and after a point becomes chaotic, as illustrated in the bifurcation diagram on the right.
[edit] Period-halving bifurcation
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.
[edit] See also
[edit] External links
- The Flip (Period Doubling) Bifurcation in Discrete Time, Dynamic Processes