Period doubling bifurcation

Logistic map

Consider the following simple dynamics: x n + 1 = r x n ( 1 − x n ) where x n , the value of x at time n , lies in the [ 0 , 1 ] interval and changes over time according to the parameter r ∈ ( 0 , 4 ] . This classic example is a simplified version of the logistic map.

For r between 1 and 3, x n converges to the stable fixed point x ∗ = ( r − 1 ) / r . Then, for r between 3 and 3.44949, x n converges to a permanent oscillation between two values x ∗ and x ∗ ′ that depend on r . As r grows larger, oscillations between 4 values, then 8, 16, 32, etc. appear. These period-doublings culminate at r ≈ 3.56995 from where more complex regimes appear. As r increases, there are some intervals where most starting values will converge to one or a small number of stable oscillations, such as near r = 3.83 . See figure.

In the interval where the period is 2 n for some positive integer n , not all the points actually have period n . 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. See Sharkovskii's theorem.

Logistical map for a modified Phillips curve

Consider the following logistical map for a modified Phillips curve:

π t = f ( u t ) + b π t e

π t + 1 = π t e + c ( π t − π t e )

f ( u ) = β 1 + β 2 e − u

b > 0 , 0 ≤ c ≤ 1 , d f d u < 0

where :

Keeping β 1 = − 2.5 ,   β 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.

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.