2. Chaos Identification For the driven, damped pendulum, we found chaos for some values of the parameters (the driving torque F) & not for others. Similarly, for the Logistic Equation, we found chaos for some values of α & not for others. Questions: –What are the characteristics of chaos? –How can we identify them? Chaos: –We’ve already seen that its not periodic motion! –We’ve also said that it has a sensitive dependence on the initial conditions. –Now, we’ll make these qualitative definitions more quantitative.

3. Another example of chaos in a discrete system governed by a difference equation: Consider another nonlinear map given by x n+1 = f(α,x n ) = αx n [1-(x n ) 2 ] (0  x n  1) Let α = 2.5. Make 2 numerical iterative calculations with 2 different initial values of x 1 : x 0 = 0.700000000 and x 0 = 0.700000001 These 2 initial “guesses” differ by only one part in 10 8 ! Plot the results for x n vs. n & find the iteration #n where the 2 solutions have clearly diverged. Naively, thinking “linearly”, one would guess that such a small difference in initial “guess” would make no difference at all in the solution!

4. The results of the numerical calculation for the 2 initial values on same same graph! Analysis shows no observed difference for the 2 different x 1 until iteration # n  30. For n = 39: Huge differences in the 2 results can clearly be seen in the graph! If you think “linearly”, this is AMAZING, since the initial values differed by only one part in 10 8 ! (This difference is less than the round off error in many computations!) Note: Also, no convergence to a constant x n in either case!

5. Suppose, for this example, the computations are made with zero error ( never possible, of course !) & that a detailed analysis of the computations shows that the difference between the iterated values doubles (on the average) for each iteration. So, after n iterations, the difference between the 2 solutions is expected to be approximately  2 n  e nln(2) –The initial values start off differing by one part in 10 -8.  For 2 solutions at the n th iteration to differ by the order of  1, we must have (2 n )  10 -8  1 or n  27  This implies that after n  27 iterations, the difference |x n - x n+1 | can reach the full range of x! (0  x  1) Another calculation shows that, in order to have the results differ by unity after n = 40 iterations requires that the initial values be the same to one part in 10 12 precision!

6. This example clearly shows a sensitivity to initial conditions which is a characteristic of chaos! This sensitive dependence on initial conditions is sometimes known as the “butterfly effect”. We can numerically determine the 2 results in this case. But, in practical (experimental!) problems, we rarely know the initial conditions to an experimental precision of 10 -8 ! In the example, adding another factor of 10 to the precision of x 0 gains only 4 iterations before the agreement between the 2 solutions again begins to diverge!  Bottom Line: We must accept the REALITY that, for this problem, with this nonlinear map, increasing the precision of the initial conditions gains little in the accuracy of results! This type of statement is also true for the Logistic Map of last time & for more physically realistic problems which display chaos, such as the driven pendulum! For such problems, precise predictions are just not possible!

7. Lyapunov Exponents A method to put the sensitivity of chaotic systems to the initial conditions on a quantitative basis: Calculate the Lyapunov Characteristic Exponent –Defined as the discussion proceeds! Each system variable has its own Lyapunov Exponent. We’ll illustrate what this is using a 1dimensional system with 1 variable & thus 1 exponent. Consider a 1d nonlinear system (map) with 2 initial states differing by only a small amount  ε. One initial value of x is x 0, the other is x 0 + ε. By assumption, |ε/x 0 | << 1. Solve the map numerically (as in the examples) & investigate x n after n iterations for the 2 initial values.

8. Define: the Lyapunov Exponent  λ  the coefficient of the average exponential growth ( or decay !), per iteration, of the difference between the 2 solutions. That is, after n iterations, the difference between the 2 solutions is (approximately): d n   ε e n λ  If λ < 0, d n  0 for large n (   ) & the 2 solutions will converge  No Chaos!  If λ > 0, d n   for large n (   ) & the 2 solutions will diverge  Chaos!

9. Consider a general 1d map given by x n+1 = f(x n ) –A tedious derivation now follows! Define f n (x)  f(x n ) –The initial difference: d 0 = x 0 - x 0  ε –After 1 iteration, the difference in the solutions is: d 1 = f(x 0 + ε ) - f(x 0 ) or d 1 = f 0 (x + ε ) - f 0 (x) Since ε is small ( using the definition of derivative): d 1  ε(df/dx) 0 –After n iterations, the difference is: d n = f n (x + ε)- f n (x 0 )  ε (df/dx) n (1) Definition of the Lyapunov exponent λ: d n  ε e nλ (2) Equate (1) & (2), divide by ε & take the natural log of both sides, giving: nλ  ln[(f n (x + ε ) - f n (x))/(ε)] or, using the definition of the derivative (ε  0): nλ  ln[(df n /dx) 0 ] (3)

10. So, for a general 1d map, x n+1 = f(x n ), the PRESCRIPTION for computing the Lyapunov Exponent is (taking the limit of (3) as n   and ε  0): λ = (1/n)ln[(df n /dx) 0 ] (4) Note: The value of f n (x 0 ) is obtained by iterating f(x 0 ) n times: f n (x 0 ) = f(f(f(f(… (f(x 0 )) …)))) Use the chain rule for the derivative: (df n /dx) 0 = (df n /dx) n (df n-1 /dx) n-1 (df n-2 /dx) n-2 …(df 1 /dx) 1 (df 0 /dx) 0 (5) Combining (4) & (5): ln[product of (df i /dx) i ] = sum[ln (df i /dx) i ]  λ = (1/n) ln[∏ i |df(x i )/dx|] = (1/n)∑ i ln[|df(x i )/dx|] (6) ∏ i,∑ i  i = 0, 1, …..(n-1) (limit of (6) as n   )

11. Summary: For any nonlinear problem, the Lyapunov exponent is computed by λ = (1/n)∑ i ln[|df(x i )/dx|] ∑ i  i = 0, 1, …..(n-1) (take the limit as n   ) If λ < 0, for large n (   ), the solutions with 2 slightly different initial conditions will converge to the same result  No Chaos! If λ > 0, for large n (   ), the solutions with 2 slightly different initial conditions will diverge to different results  Chaos!

12. Back briefly to the Logistic Map: x n+1 = αx n (1-x n ). The author has computed λ vs. α: The sign of λ is important! λ < 0  No Chaos! λ > 0  Chaos! Results for λ vs. α agree with previous results for bifurcation diagram, x n vs. α  Bifurcation occurs if λ = 0  |df/dx| = 1 & the solution is unstable. A “superstable” point is where |df/dx| = 0  λ  -  α = 3.0  α = 3.0  α = 3.45  α = 3.45  chaos!   chaos!     periodicity    periodicity

13. For N dimensional maps (N variables) there will be N Lyapunov exponents! Only one of them needs to be > 0 for the system to have Chaos! For systems with dissipation, as we’ve discussed, the phase space volume decreases as a function of time. It can be shown that this means:  The sum of the Lyapunov exponents is negative for systems with dissipation.

14. Damped, driven pendulum again! Lyapunov Exponent calculation is difficult, but doable numerically. A differential, rather than a difference eqtn. 3d  3 different λ’s. Results were computed using the same parameters as in the discussion of this system. Results for these same parameters. For F = 0.4 (periodic motion). Requires several 100 iterations before transient effects die out. None of the λ are > 0 after 350 iterations. Phase Diagram λ vs. ω

15. For F = 0.6 (chaotic motion). Again, requires several 100 iterations before the transient effects die out. After 350 iterations, one λ is > 0  CHAOS, as was found in the solutions! Phase Diagram λ vs. ω