Motivation • As we’ve seen, chaos in nonlinear oscillator systems, such as

the driven damped pendulum discussed last time is very complicated! – The nonlinear oscillator problem (& its differential equation) is

complex!! As we’ve seen, chaos may happen, but its not easily understood, because a computer solution to the problem is always needed.

To try to understand chaos further, instead of a nonlinear oscillator, we now investigate a system with simpler math, but which contains some of the same qualitative behavior (Chaos) as the nonlinear oscillator.

We investigate a discrete system which obeys a

“nonlinear mapping”.

Mapping• Notation & terminology:

x = A physical observable of the system.

n = The time sequence of the system For example, the time progression of nonlinear oscillator system can be found by investigating how (n+1)th state depends on nth state.

– Instead of the nonlinear oscillator, we investigate a system with simpler math, but which contains Chaos!

Consider a discrete system which has a nonlinear mapping & is called the “Logistic Map”.

• A simple example of a discrete nonlinear system (“map”) is the difference eqtn: xn+1 = (2xn+3)2

• More generally, a nonlinear mapping is written: xn+1 = f(xn), where f(xn) is a specified function.

• Poincaré Sections (discussed last time) are 2 dimensional nonlinear maps for the driven, damped pendulum.

• Here, we illustrate nonlinear mapping using a simple nonlinear difference equation which contains Chaos

• A very general nonlinear map is the equation

xn+1 = f(α,xn) where 0 xn 1. The specified function f(α,xn) generates xn+1 from xn in a specified manner. α is a parameter characteristic of the system.– The collection of points generated is called a “Map” of

f(α,xn). Generating these points is called “Mapping” f(α,xn).

– If f(α,xn) is nonlinear, we often need to solve

xn+1 = f(α,xn) numerically by iteration. – We consider only one dimen. here. Generalization to higher

dimensions is straightforward, but tedious.

• To be specific, consider the discrete, nonlinear map:

f(α,x) = αx(1-x). This results in the iterative difference equation: xn+1

= αxn(1-xn) (1)Obviously, if x were continuous, (1) is a trivial quadratic equation for x!

• When x is discrete, (1) is called the “Logistic Equation”– Applications to physics????

– Application to biology: Studying the population growth of fish in an isolated pond.

x1 = # fish in the pond at beginning of the 1st year (normalized to 0 x1

1). xn = (relative) # fish in the pond at beginning of nth year (normalized to 0 xn 1)

– If x1 is small, the population may grow rapidly for small n, but as n increases, xn may decrease because of overpopulation!

• The “Logistic Equation”: xn+1 = αxn(1-xn)

– The xn are scaled so that 0 xn 1.

α is a model-dependent parameter. In some sense, it represents the average effects of environmental factors on the fish population.

– Experience shows that 0 α 4. This prevents negative or infinite populations!

• The “Logistic Equation”: f(α,xn) = xn+1 = αxn(1-xn)

A way to illustrate the numerical iteration solution schematically is by a

graph, called the Logistic Map. It plots xn+1 vs. xn. Shown here for α = 2.0.

• Procedure: xn+1 = 2xn(1-xn)

Start with an initial value x1 on the

horizontal axis. Move up vertically

until the curve is intersected. Then

move to the left to find x2 on the

vertical axis.Then, start with this

value of x2 on horizontal axis &

Repeat the procedure. Do this for

several iterations. This converges to x = 0.5. The fish population

stabilizes at half its maximum.This result is independent of the initial

choice of x1 as long as its not 0 or 1. An obvious & not surprising result!

• An Easier Procedure! xn+1 = 2xn(1-xn)

– Add the 45º line xn+1 = xn to the graph, as in the figure.

– After intersecting

the curve vertically

from x1, move

horizontally to

intersect the 45º

line to find x2, &

move up vertically to find x3, etc.

– This gets the same result as before, but it does so faster. Convergence to xn = 0.5 is faster!

• The “Logistic Equation”: f(α,xn) = xn+1 = αxn(1-xn)

– In practice, biologists study the behavior of this system as the parameter α is varied.

– Naively (thinking “linearly”)

one might think that the

solutions would vary

smoothly & continuously

with changing α.

– In fact, this has been found

to be true for all α < 3.0. This means that for α < 3.0, stable fish populations result. Shown is the schematic iteration procedure for α = 2.9. The numerical iterative solutions follow the square, spiral path to a converged result.

• “Logistic Equation”: f(α,xn) = xn+1 = αxn(1-xn)– However, surprisingly

(if you think “linearly”!),

it has been found that,

for α just > 3.0, more

than one solution for

the fish population

exists! Shown is α = 3.1.

Schematically, the numerical iterative solutions follow the square, spiral path, but they never converge to one point! Instead, the iteration alternates back & forth between 2 solutions!

Solution 1

Solution 2

Bifurcation• “Logistic Equation”: f(α,xn) = xn+1 = αxn(1-xn)

– For α just < 3.0, One solution!– For α just > 3.0, Two solutions!

• Again, this is very “weird” for linear systems. However, its not unusual at all for nonlinear systems!

• Generally, a sudden change in the number of solutions to a nonlinear equation when a single parameter (such as α) is changed only slightly is called a BIFURCATION.

• “Logistic Equation”: f(α,xn) = xn+1 = αxn(1-xn)

• We can obtain a general view of this eqtn & its solutions by plotting a BIFURCATION DIAGRAM

This is a plot

of the converged

xn, after many

iterations, as a

function of the

parameter α.

This is shown here for 2.8 < α < 4.0 α

“Periodicity”

Chaos

• “Logistic Equation”: f(α,xn) = xn+1 = αxn(1-xn)

• The bifurcation diagram shows many new & interesting effects (which are totally weird if you think linearly!)– There are regions & “windows” of stability.– There are regions of Chaos!For α = 2.9, after a few iterations,

the iterations converge to a stable

solution x = 0.655.

Definition: N Cycle A solution

that returns to its initial value after N iterations. That is xN+i = xi

– For α = 2.9, there is 1 solution. The “period” = 1 “cycle”.– For α = 3.1, after a few iterations, get 2 solutions which are alternately

(oscillating between) x = 0.558 and x = 0.765.

For α = 3.1, the “period” = 2 “cycle”.

1 cycle

2 cycle

4 cycle

• “Logistic Equation”: f(α,xn) = xn+1 = αxn(1-xn)

• The bifurcation occurring at α = 3.0 is called a Pitchfork Bifurcation

because of the shape of

bifurcation diagram there.• At α = 3.1, the period is doubled

(2 cycle), so the solutions have

the form x2+i = xi. At α = 3.45, this 2 cycle bifurcation bifurcates again, to a 4 cycle (4 solutions to the eqtn!)!!!

• This period doubling continues over & over again as α is increased & the intervals between the doublings decrease. This continues up to an # of cycles (CHAOS!) at α near 3.57.

α = 3.0

α = 3.45

α = 3.57

• “Logistic Equation”: f(α,xn) = xn+1 = αxn(1-xn)

• Chaos occurs for many α values between 3.57 & 4.0. There are still windows of periodicity. A wide window of this occurs

around α = 3.84.• Interesting behavior occurs

for α = 3.82831. A 3 cycle

occurs for several periods

& appears stable. Then, it suddenly & violently changes for a few cycles & then returns to the 3 year cycle.

• This intermittent behavior (stability & instability; chaos & back again) obviously could be devastating to a biological system (the fish population)!

α = 3.82831 α = 3.84

Example• “Logistic Equation”: f(α,xn) = xn+1 = αxn(1-xn)

Let Δαn αn - αn+1 be defined as the width (in α) between successive period doubling bifurcations. From the figure, let α1 = 3.0 = the α value where the first bifurcation occurs & α2 = 3.449490 = the value where the second one occurs.

Also define:

δn (Δαn)/(Δα n+1)

Let δn δ as n

Find (numerically) δn for the first few bifurcations & also find the limit δ

• Solution is in the Table: As αn the limit of 3.5699456, the number of doublings

and δn δ = 4.669202. This value of δ has been found to be a

universal

property of the

period doubling

route to chaos

when the function

being considered has a quadratic maximum. Not confined to 1 dimension! Also true for 2 dimensions!

4.669202 “Feigenbaum’s number”