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Recap. Discussed simple model for population dynamics New population x n+1 gotten from old x n via logistic function x n+1 =ax n (1-x n ) a gives rate of reproduction simple but nonlinear x given by iteration. The Logistic Map. x n+1 =ax n (1-x n )
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Recap • Discussed simple model for population dynamics • New population xn+1 gotten from old xn via logistic function xn+1=axn(1-xn) • a gives rate of reproduction • simple but nonlinear • x given by iteration
The Logistic Map xn+1=axn(1-xn) • After many iterations x reaches some value(s) independent of its starting value • 3 regimes: • a<1: x=0 for large n • 1<a<3: x=constant for large n • 3<a<ac: cyclic behavior • ac<a: mostly chaotic • ac=3.69.. approximately
BehaviorsPeriod 1 and 2 xn a<3.0 n xn a=3.2 - 2-cycle n
Behaviors Period 4 xn n a=3.53 - 4-cycle
BehaviorsPeriod 8 xn n a=3.55 - 8 -cycle
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 ac=3.7.. motion is irregular (chaotic) • This period doubling route to chaos is seen frequently
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 x=f(x) • For a 2-cycle it must come back to x after 2 iterations y=f(x) x=f(y)=f(f(x)) • but f(x)=ax(1-x) so f(f(x))=ax(1-x)[1-ax(1-x)]
Graphical solution • Fixed-pts • Solution x=f(x) corresponds to intersection of the graph y=x with y=f(x) where f(x)=ax(1-x) • Similarly, the 2 points on a 2-cycle are intersections of y=x with y=f(f(x))
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
Example • Imagine a=2.5 x=2.5x(1-x) -> x=0.6 • Iterate with x=0.5 -> 0.6 • Iterate with x=0.7 -> 0.6 ! • Property of the logistic map for this value of a.
Doubling ? • For a=2.5 see that twice iterated map x=f(f(x)) has just one intersection - see fixed point behavior. • But for a=3.3 the twice iterated map x=f(f(x)) has now 2 intersections - a 2-cycle.
Bifurcation Diagram x 1 2 4 8 a
Convergence • First bifurcation at a1=3.0 • Second at a2=3.449 • Third at a3=3.544 • Fourth at a4=3.567 • Gap is getting smaller … • Define d=(an+1-an)/(an+2-an+1) • Large n: d is constant • d=4.6692… Feigenbaum constant
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 !
