Finite Lyapunov Exponent for Generalized Logistic Maps with z-Unimodality
Finite Lyapunov Exponent for Generalized Logistic Maps with z-Unimodality
This Demonstration shows a finite Lyapunov exponent of a one-dimensional unimodal map =f()=λ/2(1-|)-1, which is a generalization of the well-known logistic map =λ(1-). The related Lyapunov exponent can be defined by
γ
i
x
i+1
x
i
x
i
z
|
x
i+1
x
i
x
i
γ
γ==()
lim
i∞
γ
i
lim
i∞
i
∑
j=0
log
2
′
f
x
j
i+1
where is the base-2 logarithm, is the iteration number, is the iterate of starting from the initial condition , is the main control parameter, and is the subcontrol parameter, which determines the unimodality, the degree of the local maximum of .
log
2
i
x
i
th
i
f()
x
i
x
0
λ
z
f()
x
i