Liapunov Exponent. September 19, 2011

Transcription

1 Liapunov Exponent September 19, Introduction At times, it is difficult to see whether a system is chaotic or not. We can use the Liapunov Exponent to check if an orbit is stable, which will give us a better idea of how the dynamics of the system works. The Liapunov Exponent is calculated as: { } 1 n 1 λ = lim ln f (x i ) n n We will first derive this equation and then look at an example of how it can be used. 2 General Idea i=1 If we choose two very close initial points, x 0 and x 0 +δ 0, then in a function f(x), δ n would be the difference between the points after n iterations. That is δ n = f n (x 0 +δ 0 ) f n (x 0 ). Figure 1 We can also think of the separation δ n as an exponential function of the original difference δ 0. Here we introduce the Liapunov exponent λ in the separation equation δ n δ 0 e nλ. 1

2 In this equation, if λ < 0, then the orbits converge. If λ > 0, the orbits diverge. If λ = 0, e 0 = 1 so δ n δ 0. Remember, this is not exact, we are just looking at the difference at the ending iteration and using this simple equation to describe it. The following are graphs of e nλ to give a better visual understanding of how λ effects this equation s convergence. (a) (b) Figure 2: Exponential functions of λ < 0 and λ > 0 3 Solving for the Liapunov Exponent We start by solving for λ in the equation δ n δ 0 e nλ, using what we know from the previous section, where we obtain: λ 1 n ln δ n δ 0 = 1 n ln f n (x 0 + δ 0 ) f n (x 0 ) δ 0 = 1 n ln (f n ) (x 0 ) Now (f n ) (x 0 ) will be expanded with the chain rule since f(x k ) = f(f(x k 1 )) so (f n ) (x 0 ) = f (x n 1 )f (x n 2 )... f (x 0 ) = n 1 i=0 f (x i ) Now we plug this into the Liapunov Exponent equation and use the product rule of logarithms to get λ 1 n 1 n ln f (x i ) = 1 n n 1 i=0 i=0 ln f (x i ) 2

3 4 Logistic Map Lets look at the iterated map x i+1 = µx(1 x i ) for different values of µ. (a) (b) (c) (d) Figure 3 Looking at the graphs for 50 iterations of this map in Figure 3, we can see that the map converges to one point for µ = 2.9. If we change µ to 3.3, we have the same two points repeating. At µ = 3.5 it becomes a 4 repeating points, and it looks like it becomes chaotic at µ = 3.75, where we can t see it repeating any points in the graph. The program for graphing Logistic Map is included in Appendix A. 3

4 4.1 Fixed Points A fixed point a is defined by f(a ) = a. That is, when you iterate the function at that point, it always gives the same output as was input. We can see this happening in Figure 3(a), when the map converges to one point for µ = 2.9. This is the fixed point for that value of the parameter µ; all initial conditions eventually converge to it. At µ = 3.3, the logistic map has 2 fixed points. This is a called 2-cycle and is defined as f(a 1 ) = a 2 and f(a 2 ) = a 1, so f 2 (a ) = a for each fixed point. In the same way, µ = 3.5 is a 4-cycle since each fixed point repeats itself every 4 iterations (f 4 (a ) = a ). Now that we have the idea, we can generalize further. The m -cycle fixed point would be defined as f m (a i ) = a i, i = 1, 2,..., m Here m is the number of iterations it takes to get back to that fixed point again. As you can see, there are m many fixed points in a m -cycle. 4.2 Bifurcation Diagram for Logistic Map To get a better understanding of the behavior of the system, we can plot the fixed points with respect to what is chosen as the parameter µ. The program for graphing fixed points with matlab is in appendix B, and the graph in Figure 4 is the output. Looking at this bifurcation diagram, we can see that it period doubles. Figure 4(b) shows a section between 3.55 and 3.6 where the fixed points start increasing their period doubling very rapidly. This is actually showing the onset of chaos. (a) (b) Figure 4 4

5 4.3 Liapunov Exponent for a Logistic Map Using these two methods of investigating the Logistic Map, we may not be certain if or where it is chaotic. This is where The Liapunov Exponent λ comes in handy. If it is positive, then the orbit is unstable, and most likely leads to chaotic behavior. The derviative for the logistic map is µ 2µx. Lets substitute in for a value of µ we think is chaotic by looking at the previous two methods. Setting µ = 3.75 we get f (x i ) = x i. Now we have all we need to calculate the Liapunov Exponent: { } 1 n 1 λ = lim ln f (x i ) n n i=1 = lim n { 1 n 1 n i=1 ln x i } The results for calculating the Liapunov Exponent for different µ values using the program in Appendix A are shown below. µ = 3.25; λ = µ = 3.50; λ = µ = 3.75; λ = µ = 4.00; λ = From these calculations, we can assume that the iterated map becomes chaotic with µ values somewhere between 3.5 and

6 4.4 Comparison of Lambda vs Mu How would we track down the critical value of µ, where λ becomes positive (without slaving away trying each value manually)? Does it go back to a negative value at one point, having stable orbits after it becomes chaotic? An easy way to find this point where it becomes chaotic is by using a loop to run the Liapunov Exponent calculation at each µ value with a certain dµ, then graphing λ vs µ, giving the graph in Figure 5. (a) (b) Figure 5 Zooming in to where the graph first shows a positive value (Figure 5(b)), we can see that the critical value is a little less than 3.57, which is in the range of where we assumed it to be in the previous methods. Also, there are certain µ values after which the Liapunov Exponent becomes negative again. 4.5 What can we learn from these graphs? Graphing the Liapunov Exponents in this way actually gives us a lot of information. At µ = 3, λ touches 0, at the point where the logistic map period doubles as shown in the bifurcation diagram in Figure 4(a). This happens again 3.544, where it period doubles again to 4-cycle. Once it passes 3.57, the cycles are unstable and quickly period double unpredictably. Lets match up the Liapunov Exponent graph and bifurcation diagram to see if we can see a pattern on the next page (Figure 6). 6

7 (a) (b) Figure 6 7

8 If you look at the places where the exponent is positive, the fixed points are all over the place. When the exponent goes back to negative, there are stable cycles (white space) on the bifurcation diagram. The biggest dip in the Liapunov Exponent post-chaos, is at µ = where λ = This means there is a very stable cycle at this µ value. If you look above at the bifurcation diagram at that point, there are only 3 fixed points pretty far apart from each other, then it becomes 6-cycle, and then chaotic again. Graphing the Logistic map at µ = shows what we predicted, and it would have been a greusome task to try to find this 3-cycle if we did not have these tools at our disposal. Figure 7 8

9 A Matlab Program: Liapunov Calculator for Logistic Map % Liapunov Exponent C a l c u l a t o r f o r I t e r a t e d Map % mu = ; % C o e f f i c i e n t x ( 1 ) =. 1 ; % I n i t i a l Condition n = 1000; % Number o f I t e r a t i o n s f o r Exponent IterGraph = 5 0 ; % Number o f I t e r a t i o n s f o r Graph Sum = 0 ; % I n i t i a l i z e V a r i a b l e s Liap = 0 ; % f o r i =1:n x ( i +1) = mu x ( i ) (1 x ( i ) ) ; % Solve i t e r a t i o n o f f u n c t i o n Sum = Sum + l o g ( abs (mu 2 mu x ( i ) ) ) ; % Sum the l o g o f the d e r i v a t i v e s end % Divide Sum over the amount o f x ( i ) s to get the exponent! LiapunovExponent = Sum/n % Graph o f Function p l o t ( x ( 1 : IterGraph ), ) t i t l e ( [ Graph o f f ( x ) f o r mu =, num2str (mu) ] ) x l a b e l ( I t e r a t i o n ) y l a b e l ( x ( n ) ) 9

10 B Matlab Program: Bifurcation Diagram for Logistic Map % % % B i f u r c a t i o n Diagram f o r L o g i s t i c Map % % % n = 10000; % Number o f I t e r a t i o n s f o r each mu value dmu = ; % Change in mu f o r graph / c a l c u l a t i o n x1 =. 1 ; % I n i t i a l Condition mustart = 1 ; muend = 4 ; % F i r s t mu value % Last mu value f i n a l = [ ] ; % I n i t i a l i z e empty matrix muvalues = mustart : dmu : muend ; % Values o f mu i n graph m = 1 + ( (1/dmu) (muend mustart ) ) ; % How many i t e r a t i o n s needed f o r s p e c i f i f o r j =1:m mu = mustart + ( j 1) dmu % C a l c u l a t e s mu f o r j value x ( 1 ) = x1 ; % Reset I n i t i a l Condition f o r i =1:n x ( i +1)=mu x ( i ) (1 x ( i ) ) ; % Solve I t e r a t i o n end end f i n a l = [ f i n a l ; x ( ( n 150):n ) ] ; % Add v a l u e s from t h i s mu to next row in m p l o t ( muvalues, f i n a l,. ) % Plot a l l v a l u e s o f matrix as dots t i t l e ( B i f u r c a t i o n Diagram f o r L o g i s t i c Map ) x l a b e l ( Mu ) y l a b e l ( Fixed Point ) 10

11 C Matlab Program: Liapunov Exponent Graph for Logistic Map % Liapunov Exponent Graph f o r L o g i s t i c Map % x ( 1 ) =. 1 ; % I n i t i a l Condition n = 10000; % Number o f I t e r a t i o n s f o r each mu value dmu = ; % Change in mu f o r graph / c a l c u l a t i o n mustart = 3 ; muend = 4 ; % F i r s t mu value % Last mu value muvalues = mustart : dmu : muend ; % Domain o f mu i n graph m = 1 + ( (1/dmu) (muend mustart ) ) ; % How many i t e r a t i o n s needed f o r s p e c i f i f o r j =1:m mu = mustart + ( j 1) dmu; % C a l c u l a t e s mu f o r j value Sum = 0 ; % Reset Sum to 0 f o r i =1:n end end x ( i +1) = mu x ( i ) (1 x ( i ) ) ; % Solve i t e r a t i o n o f f u n c t i o n Sum = Sum + l o g ( abs (mu 2 mu x ( i ) ) ) ; % Sum the l o g o f the d e r i v a t i v e s Liap ( j ) = Sum/n ; % Divide Sum over the amount o f x ( i ) s to get the exponent! Liap1 = [ muvalues ; Liap ] ; % Match mu v a l u e s with exponents at each value Liap1 % Transpose t o make purty % Plot Exponents p l o t ( muvalues, Liap ) x l a b e l ( Mu ) 11

12 y l a b e l ( Lambda ) t i t l e ( Liapunov Exponent at D i f f e r e n t Mu Values ) g r i d on % Red Li ne a t Lambda = 0 hold on mzer=z e r o s (m) ; p l o t ( muvalues, mzer, r ) hold o f f 12

13 D Refrence Strogaz Nonlinear Dynamics and Chaos. Cambridge (MA): Westview Press, p