Recap

1
Recap

• Discussed simple model for population dynamics
• New population xn1 gotten from old xn via
  logistic function
• xn1axn(1-xn)
• a gives rate of reproduction
• simple but nonlinear
• x given by iteration

2
The Logistic Map

• xn1axn(1-xn)
• After many iterations x reaches some value(s)
  independent of its starting value
• 3 regimes
• alt1 x0 for large n
• 1ltalt3 xconstant for large n
• 3ltaltac cyclic behavior
• aclta mostly chaotic
• ac3.69.. approximately

3
BehaviorsPeriod 1 and 2
xn
alt3.0
n
xn
a3.2 - 2-cycle
n
4
Behaviors Period 4
xn
n
a3.53 - 4-cycle
5
BehaviorsPeriod 8
xn
n
a3.55 - 8 -cycle
6
Period Doubling

• As a is increased beyond 3.0 the system first
  shows 2-cycle behavior then 4-cycle, then 8
• The period keeps doubling
• Beyond some value ac3.7.. motion is irregular
  (chaotic)
• This period doubling route to chaos is seen
  frequently

7
Why is special about the points on a cycle?

• Consider fixed points. Under iteration the new
  value f(x) must equal the old value x
• xf(x)
• For a 2-cycle it must come back to x after 2
  iterations
• yf(x)
• xf(y)f(f(x))
• but f(x)ax(1-x)
• so f(f(x))ax(1-x)1-ax(1-x)

8
Graphical solution

• Fixed-pts
• Solution xf(x) corresponds to intersection of
  the graph yx with yf(x) where f(x)ax(1-x)
• Similarly, the 2 points on a 2-cycle are
  intersections of yx with yf(f(x))

9
Why do we see only these points ?

• We now understand how to find these special
  cycles
• But why should all motions end up on one of these
  cycles ?
• Answer they are attractors
• if we start out with some x close to some special
  cycle point it will end up after many iterations
  at the special point

10
Example

• Imagine a2.5
• x2.5x(1-x)
• -gt x0.6
• Iterate with x0.5 -gt 0.6
• Iterate with x0.7 -gt 0.6 !
• Property of the logistic map for this value of a.

11
Doubling ?

• For a2.5 see that twice iterated map xf(f(x))
  has just one intersection - see fixed point
  behavior.
• But for a3.3 the twice iterated map xf(f(x))
  has now 2 intersections - a 2-cycle.

12
Bifurcation Diagram
x
1
2
4
8
a
13
Convergence

• First bifurcation at a13.0
• Second at a23.449
• Third at a33.544
• Fourth at a43.567
• Gap is getting smaller
• Define d(an1-an)/(an2-an1)
• Large n d is constant
• d4.6692 Feigenbaum constant

14
Summary

• Logistic map can show variety of behaviors
  depending on a
• regular cycles or chaos
• Values on these cycles can be found by drawing
  graphs
• These cycles are attractive
• As chaotic regime is approached the period keeps
  doubling - infinite cycle is chaotic !