1

GEM2505M

Frederick H. Willeboordsefrederik@chaos.nus.edu.sg

Taming Chaos

2

Quantifying the Dynamics

Lecture 10

3

GEM2505M

Important Notice!

Special Q&A session.

See announcement on web!

4

GEM2505M

Today’s Lecture

Lyapunov Exponents

Homoclinic Points

Intermittency

Fractals & The Logistic Map

The Story

We’ve seen how we can understand some of the main features of the bifurcation diagram. How can we quantify some of these features?

5

GEM2505M

Thus far we considered the stability of a single fixed point. How about the stability of a period-k orbit?

Stability

What is, e.g., the stability of a period 2 orbit?

The most straightforward answers is that it is determined by the slope of the kth composition as we have seen before.

But, in mathematical rather than graphical terms, what do we do?

6

GEM2505M

The slope is given by the derivative which is (using the chain rule):

Stability

But what do we see? The term in brackets is just x1! And therefore, the slope of the second composition is given by:

The second composition of the logistic map is given by:

7

GEM2505M

Stability

Again the absolute value must be smaller than one in order for the orbit to be attracting.

This same procedure works of course also for higher iterates and we can conclude that the stability of a period-n orbit is given by:

Since we know from the bifurcation diagram that x0 … xn-1

change for different nonlinearities, we can wonder whether there is a ‘most stable’ orbit.

If it’s funny once, it’s funny twice!

8

GEM2505M

As is a product and unequal to zero for all periodic orbits with a period larger than 1, we can immediately infer that a super-stable orbit contains the point x = 0.

Stability

The smallest absolute value is of course 0. Hence an orbit which has a fixed point with a slope of 0 (a horizontal line) is the most stable orbit and therefore called super-stable.

Indeed there is:

Super-stable orbits

A super-stable orbit contains the point x = 0

Slope = 0

Second composition

9

GEM2505M

Stability

And as before, we can find these graphically right away by identifying where the periodic orbits in the bifurcation diagram intersect the x-axis.

Super-stable orbits

Points on super-stable orbits

How many super-stable orbits are there?

10

GEM2505M

Lyapunov ExponentsOf course, if we talk about stability, we would like to have some kind of a number, a quantifier, that can tell us in a relative sense how stable an orbit is.

As such one could think that the product of derivatives would provide such a quantifier. This is not really the case, however, since the number of terms in the product depends on the periodicity and in the case of a chaotic orbit would be infinite.

One option would be to divide by the number of terms.

11

GEM2505M

Lyapunov ExponentsHowever …

When considering sensitive dependence on initial conditions one can see that errors grow exponentially fast.

Both axes linear Y-axis log, x-axis linear

12

GEM2505M

Lyapunov ExponentsPut differently ….

Both axes linear Both axes linear

Take the log of the data Take the log of the data, anddivide by the x-value.

Now we have a straight line.

13

GEM2505M

Lyapunov ExponentsPut differently …we see that if something grows exponentially in time, then the log of that something divided by the time remains constant.

Therefore we could argue that a reasonable quantifier for the stability of an orbit is:

14

GEM2505M

Lyapunov Exponents

We obtain:

And if the orbit is not periodic, we should take the limit

is called the Lyapunov exponent of the orbit.

or more generally

Since:

15

GEM2505M

Lyapunov ExponentsThe preceding few slides are plausible enough but do not really stress the fundamental connection between the Lyapunov exponent and the derivative or the growth of an error.In order to do so, let us choose a somewhat different approach.

The difference between two initially nearby orbits can be expressed as:

Not in exam

16

GEM2505M

Lyapunov ExponentsDividing both sides by we obtain:

forNote: This is the first derivative of the function f n

In other words:

Not in exam

17

GEM2505M

Lyapunov ExponentsAccording to the chain rule we have:

And consequently:

Not in exam

18

GEM2505M

Lyapunov ExponentsDropping the “approximate” and taking the log:

Which, after reversing the order, taking the limit, dividing by n and changing the ln of a product to a sum of ln, again becomes:

Not in exam

19

GEM2505M

Lyapunov Exponents

If we have a period-k orbit, the Lyapunov exponent becomes:

Periodic orbits

E.g. for period 2 we have:

with x1 and x2 the two periodic points.

= 0.75

Recall:

20

GEM2505M

We have seen that for increasing , the orbits bifurcate. What would the Lyapunov exponent be exactly at a bifurcation point? (e.g. = 0.75)

?What would the Lyapunov exp. be?

1. Depends on (not all bif. points have the same )2. 1 or minus 13. 04. 1/2k with k the periodicity just before the bif.

Lyapunov Exponents

21

GEM2505M

Lyapunov Exponents

Super-stable orbits go through 0. Consequently, the Lyapunov exponent is given by:

Super-stable orbits

Why? Since the ln of 0 is minus infinity (and all the other terms are finite).

Points on super-stable orbits

E.g.

22

GEM2505M

Lyapunov Exponents

Similarly to the bifurcation diagram, we can plot versus .

Versus 1 2 3 4

1) Second bifurcation2) Period 4 super-stable orbit3) Third bifurcation4) Period 3 super-stable orbit

23

GEM2505M

We just saw that the Lyapunov exponent of a super-stable orbit is minus infinity. Yet in the graph of the Lyapunov exponent versus the smallest exponent is around minus 2.5.

?Why would that be?

1. Our calculation is wrong2. The graph is always wrong3. The resolution of the graph is limited4. There is no ‘infinity’ in the real world

Lyapunov Exponents

24

GEM2505M

Homoclinic PointsHomoclinic points were discovered by Henry Poincaré in his studies of the solar system. In a similar form they also exist in the logistic map.

= 1.75

Third Iterate

Plot exactly touches diagonal

From the left, zigzags in to fixed point (cannot pass it)

From the right, zigzags away from fixed point.

25

GEM2505M

Homoclinic PointsThe point where the plot touches the diagonal is a so-called saddle point which is both attracting and repelling, depending on the side from which it is approached.

Here, homoclinic points are all those points on the repelling side (i.e. right hand side) of the saddle that when iterated will eventually end up on the saddle via the attracting side.

Note: Here we do not have stable and unstable manifolds since these require two or more dimensions.

26

GEM2505M

IntermittencyWhen the plot is very close to touching but does not actually touch the diagonal yet, a small channel is left.

= 1.7498

= 1.7498

While passing through this channel, the x-values of the orbit do not change much leading to ‘laminar’ looking sections in the time series.

= 1.7496

Every third time step is plotted.

27

GEM2505M

Intermittency

Starting form the opening point of the period three window ( = 1.75), when decreasing the non-linearity , the length of the laminar regions decreases from infinitely long to very short.

Route to chaos

Hence this is an alternative route to chaos as compared to the period-doubling route to chaos discussed previously.

Histogram

= 1.7496

28

GEM2505M

Fractal

Fractals in the logistic map

The orbit of the logistic map at = 2.0 is not fractal as can readily be seen from the histogram to the right.

Histogram for = 2.0

However, there are fractal structures in the bifurcation diagram. For example the set of super-stable points.

29

GEM2505M

Fractal

Fractal dimension

Another fractal may be at the accumulation point where the orbit is neither periodic nor chaotic. Some estimates are that D) = 0.538.

accumulation point

1.401155

Conceptually, how can one understand this? If one approaches the accumulation point from the chaotic side (starting at say = 1.6), one can see that first there are two bands, then four, eight, etc. this is similar to the construction of the Cantor set.

1.60

1.37

Remove

30

GEM2505M

Fractal

or enlarged …

31

GEM2505M

FractalRelationship to theMandelbrot setThe Logistic map can be written as:

Which is exactly the real part of the iterative map used for the Mandelbrot set.

x = -2.0 x = 0.25

Period three window

32

GEM2505M

Stability

Lyapunov Exponents

Intermittency

Key Points of the Day

33

GEM2505M

Is nature based on stability or instability?

Think about it!

Stable,House,Cards,Unstable!

34

GEM2505M

References

http://www.cmp.caltech.edu/~mcc/Chaos_Course/Lesson4/Demo1.html

http://www.expm.t.u-tokyo.ac.jp/~kanamaru/Chaos/e/