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. X x1x1 f x2x2 f f f x0x0 x3x3 x4x4

3. Control Parameter  Define a map based on a quadratic function. f : R 1  R 1f : R 1  R 1 x  rx(1  x)x  rx(1  x)  The parameter r is the control parameter. Range r from 0 to 4Range r from 0 to 4  Initial condition x 0 = 0.5. Range x from 0 to 1Range x from 0 to 1

4. Graphical Analysis  Fixed points occur when the function intersects y = x.  Solutions are q = 0, p = 1–1/r.  Stable p for r > 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 < r < 1: p is negativep is negative q attracts, p repelsq attracts, p repels  1 < r < 3: p is positive and attractsp is positive and attracts q repelsq repels For r > 2 convergence alternates around pFor r > 2 convergence alternates around p

7. Period Doubling  At r = 3 the derivative is -1 Neutral between attracting and repelling  For 3 < r < 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 valuesNeglect the initial transient values next diagram from Mathworld