Choose iteration map f
xk+1 = f ( xk ), x0 ∈ [0,1]

1)  f(x) = r·x( 1−x )
2)  f(x) = r·(1−|2x−1|s)
3)  f(x) = r·x( 1−x )·e−3(x−s)2
4)  f(x) = r·sin(πx)
1) Logistic 0≤r≤4
2) Power ½≤s≤4 0≤r≤1
3) Skew 0≤s≤1 0≤r≤4
4) Sine 0≤r≤1
r:

Intervals of iteration (k)

Initiation: 0 →
Attraction: →
Iterate in one go Set x0
Iterate gradually

Points of interest

in logistic map

r ∈ [0,1] → x ∞ = 0       r ∈ [1,3] → x∞= (r-1)/r

Bifurcation points with branching 2n→2n+1
3(n=0)  3.4495(1)  3.5441(2)  →  3.56995(∞)

Limiting approach: rn → r∞− C · δ -n
with Feigenbaums constant δ=4.669201...

Bifurcations from period 3→6→12→...→∞
3.8284→...→3.8415 same δ but another C

All bifurcation series have the same δ=δ2
Trifurcations etc. occur with δ3, δ4, δ5 etc.

Maps with a quadratic max have the same δ2

Choose map f

1)  rx( 1−x )
2)  r(1−|2x−1|s)
3)  rx( 1−x )·e−3(x−s)2
4)  rsin(πx)
s r
1) Logistic 0→4
2) Power ½→4 0→1
3) Skew 0→1 0→4
4) Sine 0→1
r:

k-Intervals

I: 0 →
A: →
In one go
Gradual

Iterations
Attractors
Zoom window
rMin → rMax
xMin → xMax
Iteration map and web plot
f2 f4 Maps f3 f5

Compositions

f 2(x) = (f ∘ f)(x) = f( f(x))
f k = f k-1 ∘ f
f 2 f 4 f 8
f 3 f 5 f 6

Lyapunov exponent


The program iteratates xk+1 = r·f(xk) for various maps f.
It shows how xk evolves and its limiting behaviour.
The limit depends on r with areas of periodic attractors,
followed by bifurcations and areas of chaos.

The Lyapunv exponent measures the separation rate
of two initially close start values: |δxn| → eλn|δx0|.
Lyapunov exponent formula
The program sums over k for the 'attracting' phase.

λ<0: convergence towards a periodic attractor.
λ>0: indication of chaos and a strange attractor.

Alt+1Logistic function
Alt+2'Power' function where shape of max is set by parameter s
Alt+3'Skewed' function with maximum shifted by parameter s
Alt+4Sine function
Alt+RSet parameter r in the chosen iteration map
Alt+SSet parameter s for iteration map 2 and 3
Alt+ISet the number of iterations for the settling-in phase
Alt+ASet the number of iterations for long-term bahavior
Alt+CIterate the map for the current value of r
Alt+VIterate the map for r in a given interval
Click-drag a rectangle too zoom in on the attractor diagram.
Click function type to clear screen and reset to default values.
Click during gradual display to stop it and make a reset.