# Trajectory-Scaling Functions for Generalized Logistic Maps with z-Unimodality

Trajectory-Scaling Functions for Generalized Logistic Maps with z-Unimodality

This Demonstration shows Feigenbaum's trajectory scaling function (TSF) [1–6],

"TSF1": ,

σ(m)=

n

d(m)

n+1

d(m)

n

of a unimodal map as a step-like function. Here

x=f(λ,z,x)

i+1

i

d(m)=f(λ,x)-f(λ,x)

n

m

n

c

m+2

n-1

n

c

where is the iteration number, is the iterate of starting from the initial condition (i.e. ), is the control parameter, is the point in the domain of the function where , is the period-doubling bifurcation order starting from ( and are not defined for because becomes ), is the superstable parameter value for each bifurcation order (e.g. for period 2, for period 4, for period 8, etc.), and is the integer that relates all superstable orbits between and periods (i.e. the domain of the scaling function is ). By introducing a new variable , the TSF can also be defined as

i

x

i

i

th

f(λ,x)

i

x

0

x=f(λ,x)

i

i

0

λ

x

c

f(x)

f(x)=0

′

c

n

n=1

d(m)

0

σ(m)

0

n=0

2

n-1

1/2

λ

n

λ

1

λ

2

λ

3

m

2

n

2

n+1

σ(m)

n

1≤m≤2

n+1

t=m/2

n+1

"TSF2": ,

σ(t)=

n

d(t)

n+1

d(t)

n

where .

2≤t≤1

-(n+1)