1 GEM2505M Frederick H. Willeboordse Taming Chaos

2 Bifurcations and Windows Lectures 8 & 9

GEM2505M 3 Important Notice! This is a double lecture.

GEM2505M 4 Today’s Lecture Bifurcations Windows Destinations Crisis Stability The Story We’ve seen what a bifurcation diagram is. How can we understand some of its features?

GEM2505M 5 Bifurcations Last time we had a first look at the bifurcation diagram. In high resolution … the bewildering structure can clearly be seen.

GEM2505M 6 Depending on the value of , the logistic map can have a periodicity of 1, 2 or more. Bifurcations Why is that so? Period 1 Period 2 In order to find out, let us inspect the cobweb more closely.

GEM2505M 7 As we can see in the bifurcation diagram, something special happens at  = 0.75 so let’s draw some cobwebs on or near this value. Bifurcations What happens here? Spirals in Spirals slowly Spirals out First Iterates  = 0.6  = 0.75  = 0.9

GEM2505M 8 Zooming into the spiral, we can notice something …. Bifurcations  = 0.6  = 0.9 The slope of the intersection of the function plot and the diagonal is changing. Let us look at two lines crossing:      

GEM2505M 9 Bifurcations ? 1.Spirals in 2.Spirals out 3.Period two 4.Chaotic What kind of cobweb do you think we obtain for the graph to the right?  

GEM2505M 10 Bifurcations       The AngleThe CobwebThe Behavior Spirals in Stays Put Spirals out

GEM2505M 11 Next, let us investigate what happens to the second composition near the first bifurcation point. Bifurcations What happens here?  = 0.6  = 0.75 Zigzags in Zigzags in slowly Zigzags in Second Composition  = Fixed Point 2 Fixed Points Only visible when zooming in.

GEM2505M 12 We can see that the slope of the first iterate changes from being smaller than one to larger than one, and that at the same time two new fixed points of the second composition with a slope smaller than one come into existence. Bifurcations What happens here?  = 0.9 Second Composition First Composition same fixed point, slope > 1 new period two fixed point, slope < 1

GEM2505M 13 If it’s fun once, it’s fun twice! Bifurcations What happens here?  = 1.3 Forth Composition Second Composition same fixed points, slope > 1 new period four fixed points, slope < 1  = 1.2 Second Composition

GEM2505M 14 If it’s fun twice, it’s fun thrice! Bifurcations What happens here?  = 1.38 Eighth Composition

GEM2505M 15 Indeed for increasing nonlinearity, period doublings continue up to infinity. However, the distance between successive bifurcation points decreases rapidly (as can be seen from the bifurcation diagram). Bifurcation Diagram In fact, the length ratio between successive branches approaches a constant. kk  k+1 kk =  = for k to infinity

GEM2505M 16 Bifurcation Diagram The constant  is called the Feigenbaum constant. Feigenbaum constant Feigenbaum point The point in the bifurcation diagram where the period doubling reaches infinity is called the Feigenbaum point. M. Feigenbaum

GEM2505M 17 Destinations Let us return to the first iterate and small . ? Does the value of x 0 matter? 1.Yes 2.No 3.Sometimes 4.Depends We start with a certain value of x 0 and see what orbit it leads to. The question now is, do all values of x 0 lead to the same orbits?  = 0.5

GEM2505M 18 Destinations For small  the answer is no. But for large , things are a bit more subtle. Roughly, there are 4 possibilities  = 0.5  = 1.0  = 1.44  = 2.0 All points have exactly the same orbit. All points have the same orbit though it may be shifted. All points have different orbits though there is a gap. All points have completely different orbits.

GEM2505M 19 Destinations What is going on becomes a bit clearer if we look at the second composition. We see that depending on the value of x 0 the orbit goes to a different fixed point.  = 1.0

GEM2505M 20 Destinations Indeed, we can graphically determine where the possible x 0 go. Hence we see that there are basically three regions. Two go to the red point and one to the green point.  = 1.0 Points ongo to Points ongo to

GEM2505M 21 We just saw that there is a period doubling cascade to infinity. Bifurcation Diagram From the bifurcation diagram, it is also clear that there are windows (periodic regions beyond the Feigenbaum point). Window Why would that be?

GEM2505M 22 Thus far we investigated iterates 1,2,4,8 etc. But there is no reason not to consider 3,5,6,7… etc. as well! Windows  = 0.6  = 1.3  = 1.6 Third Composition Clearly, there is only one fixed point. However, local extrema are getting closer to the diagonal.

GEM2505M 23 Indeed at  = 1.75 the third composition touches the diagonal. This creates three attracting fixed points. Windows  = 1.7  = 1.75  = 1.8 Third Composition At  = 1.75, all the fixed points of 2 n compositions are repelling since we’re past the Feigenbaum point.

GEM2505M 24 After the window is created, we back to the previous story. Windows What happens here?  = Sixth CompositionThird Composition same fixed points, slope > 1 new period six fixed points, slope < 1  = Third Composition  = 1.775

GEM2505M 25 If we look at the bifurcation diagram starting form the third iterate closely, we see that at some stage something special happens. Windows It abruptly ends here

GEM2505M 26 This can again be understood by examining the cobweb. Windows  =  = When starting from x 0 = 0.0, the left path stays close while the right path jumps all over the place. Third Composition

GEM2505M 27 Zooming into the central part of the graph we see why. Windows  =  = 1.79 At this point the orbit can escape. Hence when  reaches a value where escape is possible, the window closes. Third Composition

GEM2505M 28 Crisis ? How? 1.The first iterate needs to be on the diagonal 2.The third iterate of the local minimum needs to be on the diagonal 3.The sixth iterate needs to be on the diagonal 4.The sixth iterate of the local minimum needs to be on the diagonal  = Third Composition The event that leads to the closing of the window is called a crisis. The value of  for which is occurs can be determined graphically.

GEM2505M 29 When the sixth iterate is exactly on the diagonal. Crisis  = Why sixth? Since it’s two steps in the graph of the third composition. I.e. we have 2 x 3 steps. Third composition For larger a, the paths will escape this region, for smaller a (until  = 1.75) the paths will remain inside.

GEM2505M 30 The slope of a function is given by it’s derivative. The derivative of Derivative & Slope is: The derivative of this function at this point is given by the slope of this line.

GEM2505M 31 (use ) The fixed point of the first iterate can be obtained by solving the equation: Slope & Fixed Point In other words we need to solve: Use: However only one of these fixed points is in the interval [-1,1] Period 1 fixed point we need.

GEM2505M 32 Now that we know what the value of the fixed point x* is, we can insert this into the derivative to obtain the slope of the function plot at this fixed point. Stability Of course we can draw this and indeed, at  = 0.75, the absolute value of the slope becomes larger than 1.

GEM2505M 33 Thus we see that a period one cycle is stable when the absolute value of the slope is smaller than 1. Stability We can determine this graphically by investigating the angle at which the function plot intersects the diagonal. Or, and this is far more accurate of course, we can evaluate the derivative at the fixed point to find the slope. What is the slope?

GEM2505M 34 Simple Map. Amazing Properties! Key Points of the Day

GEM2505M 35 Can a crisis be a good thing? Think about it! Crisis, Sale, Computer, Simulation!

GEM2505M 36 References