# Chaos ....

## Introduction

Chaos is a phenonenon with many meanings depending on whom you ask. For the purpose of this tutorial, I will say that a system is chaotic if there is sensitive dependence on initial conditions. That is, if I start with two initial conditions that are very close to each other, then their difference will diverge exponentially in time.

For discrete dynamical systems, it is possible to get chaotic behavior with a one-dimensional map, as you saw with the standard map. For continuous systems, the usual model for chaos is the Lorenz attractor but the mechanism through which chaos occurs for this model is somewhat difficult to explain.

## The "simplest" chaotic differential equation

When the linearization of a system about an equilibrium has a zero eigenvalue for some set of parameters, this is a sign that perhaps new solutions may exist for nearby sets of parameters. If the zero eigenvalue is simple, then normal form analysis can be applied to reduce the system to a one-dimensional dynamical system. The dynamics of this one-dimensional system describes the behavior of the full system. More complicated equations arise if the zero eigenvalue is not simple. In the remainder of this tutorial, we examine an equation that arises when the linearized system has a triple zero eigenvalue that is geometrically simple. This means that the characteristic polynomial starts with cubic terms. The Jordan canonical form for the three-dimensional subspace with this degeneracy is:

Since the characteristic polynomial has a triple degeneracy, we need three free parameters to "unfold" this triple zero eigenvalue. Thus, we will study the differential equation:

Notice that there is only one nonlinear term, a quadratic. Also note that if the three parameters `mu,nu,gamma` are set to zero, the linearized system is J.

## Period doubling cascades

A detailed analysis of this equation is beyond the scope of the tutorial. However, we can explore the dynamics letting one parameter, say , `gamma` vary. Thus, we fix `q=2,nu=2,mu=1` and let `gamma` be our principle parameter.

Clearly (x,y,z)=(0,0,0) is always a solution, so we will study stability and bifurcation of this. The easiest way to do this is to use AUTO. (The stability of the fixed point could be done analytically using the Routh-Hurwitz criterion, but any other analysis is best doen numerically, even for this simple system.)

Get the AUTO window up by clicking (File) (Auto). Within AUTO, do the following:

- Click (Axes) (Hi Lo) and set
  - Main Parm gamma
  - Xmin 0
  - Ymin -3
  - Xmax 4
  - Ymax 3
  and click (Ok)
- Click (Numerics) and set
  - Dsmin .0001
  - Dsmax .1
  - Par Min 0
  - Par Max 3.75
  - Norm Min 0.0
  - Norm Max 50
  and click (Ok)
- Click (Run) (Steady State) You will see a line travel across the screen indicating the zero steady state. It loses stability at gamma=2 Click (Grab) and move the cross until Label 2 is reached. (Just click (Tab) to go there quickly.) You will see that the type of point is HB meaning a Hopf bifurcation has occurred.
- Click (Enter) to accept this point and then click (Run) (Periodic) to compute the periodic solution that comes from the point. Note the solution is stable (filled circles) but loses stability.
- Click (Grab) and (Tab) 3 times until label 4 appears in the info window in the AUTO window. Note it appears at gamma=3.126 and that its type is PD This indicates a period doubling bifurcation which means that the basic limit cycle loses stability and a new cycle arises that has twice the period. Click (Enter) to accept this point.
- Click (Run) and (Doubling) to compute the branch of period two solutions. This extends for a bit and itself becomes unstable through yet another period doubling bifurcation. When AUTO has finished the calculation, click on (Grab) and hit (Tab) 5 times to get the next period doubling point. It is Label 6 at gamma=3.357
- Click (Enter) to accept this and then click (Run) (Doubling) once again. This branch also becomes unstable at a period doubling point for gamma=3.412.

We have found three period doublings. This is a classic scenario for the onset of chaotic behavior: the period doubling cascade. These calculations can be continued, but it would take a while, so we will stop the continuation here.

## Some three-dimensional views

Before continuing on, lets look at some of these solutions. We will look at the limit cycles at the three points of instability.

- Click (Grab) and click (Tab) until label 4 appears and click (Enter). Click (File) (Import Orbit). This loads this orbit into the Data Browser so you can look at it.
- In the XPP main window, click (Viewaxes) (3d). Set the following parameters:
  - X-axis X
  - Y-axis Y
  - Z-axis Z
  - Xlabel X
  - Ylabel Y
  - Zlabel Z
  Ignore all the other stuff!! Click (Ok). Click on (Window) (Fit) to let XPP figure out the window dimensions. The orbit will be plotted in three-dimensions. Freeze this curve with some color, e.g.color 7. (If you don't remember how to freeze a curve, see the first tutorial.

  Go back to AUTO and get the next period doubling point, Label 6 using (Grab) and (Tab). Click (File) (Import Orbit) again. Then in the XPP main window click (Window) (Fit) and freeze this curve with a different color. Note how it loops around the phasespace twice. Its period is roughly twice the other orbit's.

  Repeat this for the third period doubling point (Label 8) and freeze it with still another color. You will see that it loops around the phase-space 4 times! You should have a picture that looks like:

  Just for fun, click on (3d-params) in the main XPP window and change Phi, Theta to rotate the three-d plot. You can actually make a movie by rotating and then using the Kinescope command. As this sometimes causes trouble on the X-server, I won't officially do it here, but you can fool around at your discretion.

## Chaos at last...

In the previous section, we saw that there was an apparent period doubling cascade. This is a classic scenario and sometimes "at the end" of the cascade, there is a regime in parameter space where chaotic behavior can occur. Before continuing go into the AUTO window and click on (File) (Reset Diagram) and answer (Yes) so we don't forget and leave AUTO junk all over the disk.

Click on (Graphic stuff) (Freeze) (Remove all) to get rid of all the frozen curves and set the angles Phi,Theta back to 45 using the (3d-params) command. Go into the (nUmerics) menu and set (Total) to 150. (Escape) back to XPP main and set the parameter gamma=3.5. Click (I)(G) to integrate. When done, click (W) (F) to automatically fit the window to the data.

You should see a banded surface that is twisted like a Mobius strip. This is chaotic attractor. To see it is chaotic, click on (Xi vs T) to plot Z as a function of t. Click on (I) (N) and put the following initial conditions in:

- X -.3
- Y 1
- Z 3

Freeze this curve with some nonzero color. Now, click (I) (N) again and use the following initial data:

- X -.3001
- Y 1
- Z 3

Window the graph so that the x-axis goes from 100 to 150 and the y-axis goes from -1.5 to 3.5. Note thjat the curves are quite different; small bumps and large bumps are switched and so on. This shows sensitive dependence on initial conditions the hallmark of a chaotic system. We changed the fourth decimal place of an initial condition and the trajectory changed completely after some time. What this says is that the trajectories diverge exponentially in time. The rate of this divergence is called the maximal Liapunov exponent. Stable fixed points have a negative maximal Liapunov exponent, stable perioidic orbits have a zero maximal Lyapunov exponent and chaotic orbits have a positive maximal Lyapunov exponent.

## Computing the Poincare map

Often in chaotic systems, a great deal can be learned by computing a Poincare section. We will compute the section through X=0 as it is clear that at every cycle, X passes through zero. (To see this plot X vs t .) We will need to compute for a longer period of time.

- Delete any frozen curves by clicking (G) (F) (R). Then edit curve 0 so that it is line-type 0. Use (G) (E) 0 (Enter) and click (Ok) when done.
- Enter the (nUmerics) menu and change (Total) from 150 to 1000.
- Click (Poincare) (Section) and fill in:
  - Variable: X
  - Section: 0
  - Sign: 1
  - Stop on section: N
  and click on (Ok).
- (Esc) back to the main menu and integrate with (I) (L).
- Use the (View axes) to plot `Y and Z` for this map, then use (W) (F) to fit the window, and observe that they are essentially linearly related. Thus, we need only look at `Z` and this relation gives us `Y.` That is, the Poincare map on two dimensions is fundamentally one dimensional.
- We will use the Ruelle plot to find an approximate map. Change the view to have `Z` on both axes and use (W) (F) to fit the window to the data. You should see a diagonal line.
- Click to (nUmerics) and then (Ruelle) to get the Ruelle plot option. Shift the x-axis by 1 and leave the remaining axes unshifted. This will allow you to see `Z(n) vs Z(n-1)` and thus see if there is a nice map. (Escape) to the main XPP window and click (Restore) to redraw. It is apparent that the map is roughly quadratic. In fact, it is very well fitted by

  a quadratic approximation of the real Poincare map.

With this knowledge, you should close the continuous ODE file and fire up the discrete map approximation.

## The quadratic approximation map, unstable periodic points, and Liapunov exponents

If you haven't already, fire up the quadratic approximation to the Poincare map of the dynamical system. Click on (nUmerics) and change Total to 2000 and Method to (Discrete). (Escape) to the main menu and edit the curve so that the Linetype is 0 and we plot dots. Change the view to plot `Z` on both axes as we will be using the Ruelle plots to look at things. Integrate the equations by clicking (I) (G). Click (W) (F) to fit the window to the data. You will see a diagonal line. Then use the (G) (F) (F) to freeze this diagonal. You can keep all the defaults and just click on (Ok).

Now we will look for periodic points of different periods. A point is k- periodic if Z(n)=Z(n-k) and Z(n) != Z(n-l) if l < k thus, we need to shift the x-axis by different amounts and look at the intersections with the diagonal line.

- First we look for period 1 or fixed points of the map. Click (nUmerics) (rUelle) and shift the x-axis by 1. (Esc) to the main menu and click (Restore). Note there is one intersection and it is at about `Z=2.43.` The slope at the intersection is greater than 1 in magnitude so it is unstable.
- Now look for period 2 points, by repeating the above and shifting the x-axis by 2 instead. Note the appearance of two additional intersections as well as the original fixed point. (Fixed points are trivially periodic of all periods.)
- Try to find fixed points of period 3,4,5. Are there any? Be careful that you don't mistake the fixed point for a higher period point.
- Integrate for a total of 4000 and then look for higher period points that are not powers of 2. I think there is are period 10,11 points. The resolution is bad for much higher than 11.

The existence of many unstable periodic points is often a hallmark of chaotic behavior. As we mentioned previously, another hallmark of chaos is sensitivity to initial conditions. For a one-dimensional map, this is easy to quantify. We need to compute the maximal Liapunov exponent:

where f'(xn) is the derivative of the map evaluated at the nth iterate.

Since the log of the product is the sum of the logs we can write this as the limit of an iteration. Close the above XPP file and fire up the XPP file for computing this. The source for this is worth looking at to see how I did it. Basically, I compute the iterate and then divide by the number of iterates (kept in the `t` variable as an auxiliary variable `lexp.`

Click on (View axes) (2d) and set the Y-axis to be `lexp` . Set the dimensions so that the Y-axis runs between -1 and 1 and the X-axis runs between 0 and 2000. Click (Ok) and then go into the (nUmerics) menu, set Total= 2000, Bounds=1000, and Method= (D)iscrete. (Esc) back to the main window and click (I) (G) to integrate the equations. You should see a curve that rises and then flattens out at about .332. This is the Lyapunov exponent and since it is positive, there is good evidence that the system is chaotic.

## Homework 5.1

# References

##

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