Logistic Map

1
Logistic Map
2
Discrete Map

• A map f is defined on a metric space X.
• Repeated application of f forms a sequence.
• Discrete set of points
• A sequence of points forms an orbit.

x2
x1
x3
f
f
x4
f
f
x0
X
3
Control Parameter

• Define a map based on a quadratic function.
• f R1 ? R1
• x ? rx(1-x)
• The parameter r is the control parameter.
• Range r from 0 to 4
• Initial condition x0 0.5.
• Range x from 0 to 1

4
Graphical Analysis

• Fixed points occur when the function intersects y
  x.
• Solutions are q 0, p 11/r.
• Stable p for r gt 2.

5
Graphical Sequence

• Treat each point in the sequence as a pair (x,
  x).
• Find the next point.
• Move vertically to the curve
• Move horizontally to the line

6
1-Cycles

• Values of r from 0 to 3 create a single
  attracting fixed point
• 0 lt r lt 1
• p is negative
• q attracts, p repels
• 1 lt r lt 3
• p is positive and attracts
• q repels
• For r gt 2 convergence alternates around p

7
Period Doubling

• At r 3 the derivative is -1
• Neutral between attracting and repelling
• For 3 lt r lt 3.4 there are two new stable points.
• Limit cycle is period 2
• Period continues to double for higher r.
• For r above 3.5699, the motion is chaotic.

8
Bifurcation Diagram

• For each value of the control parameter there is
  a set of points.
• Neglect the initial transient values

diagram from Mathworld
next