2. Motivation As we’ve seen, chaos in nonlinear oscillator systems, such as the driven damped pendulum discussed last time is very complicated! –The nonlinear oscillator problem (& its differential equation) is complex!! As we’ve seen, chaos may happen, but its not easily understood, because a computer solution to the problem is always needed.  To try to understand chaos further, instead of a nonlinear oscillator, we now investigate a system with simpler math, but which contains some of the same qualitative behavior (Chaos) as the nonlinear oscillator.  We investigate a discrete system which obeys a “nonlinear mapping”.

3. Mapping Notation & terminology: x = A physical observable of the system. n = The time sequence of the system  For example, the time progression of nonlinear oscillator system can be found by investigating how (n+1) th state depends on n th state. –Instead of the nonlinear oscillator, we investigate a system with simpler math, but which contains Chaos!  Consider a discrete system which has a nonlinear mapping & is called the “Logistic Map”.

4. A simple example of a discrete nonlinear system (“map”) is the difference eqtn: x n+1 = (2x n +3) 2 More generally, a nonlinear mapping is written: x n+1 = f(x n ), where f(x n ) is a specified function. Poincaré Sections (discussed last time) are 2 dimensional nonlinear maps for the driven, damped pendulum. Here, we illustrate nonlinear mapping using a simple nonlinear difference equation which contains Chaos

5. A very general nonlinear map is the equation x n+1 = f(α,x n ) where 0  x n  1. The specified function f(α,x n ) generates x n+1 from x n in a specified manner. α is a parameter characteristic of the system. –The collection of points generated is called a “Map” of f(α,x n ). Generating these points is called “Mapping” f(α,x n ). –If f(α,x n ) is nonlinear, we often need to solve x n+1 = f(α,x n ) numerically by iteration. –We consider only one dimen. here. Generalization to higher dimensions is straightforward, but tedious.

6. To be specific, consider the discrete, nonlinear map: f(α,x) = αx(1-x). This results in the iterative difference equation: x n+1 = αx n (1-x n ) (1) Obviously, if x were continuous, (1) is a trivial quadratic equation for x! When x is discrete, (1) is called the “Logistic Equation” –Applications to physics???? –Application to biology: Studying the population growth of fish in an isolated pond. x 1 = # fish in the pond at beginning of the 1 st year (normalized to 0  x 1  1). x n = (relative) # fish in the pond at beginning of n th year (normalized to 0  x n  1) –If x 1 is small, the population may grow rapidly for small n, but as n increases, x n may decrease because of overpopulation!

7. The “Logistic Equation”: x n+1 = αx n (1-x n ) –The x n are scaled so that 0  x n  1. α is a model-dependent parameter. In some sense, it represents the average effects of environmental factors on the fish population. –Experience shows that 0  α  4. This prevents negative or infinite populations!

8. The “Logistic Equation”: f(α,x n ) = x n+1 = αx n (1-x n ) A way to illustrate the numerical iteration solution schematically is by a graph, called the Logistic Map. It plots x n+1 vs. x n. Shown here for α = 2.0. Procedure: x n+1 = 2x n (1-x n ) Start with an initial value x 1 on the horizontal axis. Move up vertically until the curve is intersected. Then move to the left to find x 2 on the vertical axis.Then, start with this value of x 2 on horizontal axis & Repeat the procedure. Do this for several iterations. This converges to x = 0.5. The fish population stabilizes at half its maximum.This result is independent of the initial choice of x 1 as long as its not 0 or 1. An obvious & not surprising result!

9. An Easier Procedure! x n+1 = 2x n (1-x n ) –Add the 45º line x n+1 = x n to the graph, as in the figure. –After intersecting the curve vertically from x 1, move horizontally to intersect the 45º line to find x 2, & move up vertically to find x 3, etc. –This gets the same result as before, but it does so faster. Convergence to x n = 0.5 is faster!

10. The “Logistic Equation”: f(α,x n ) = x n+1 = αx n (1-x n ) –In practice, biologists study the behavior of this system as the parameter α is varied. –Naively (thinking “linearly”) one might think that the solutions would vary smoothly & continuously with changing α. –In fact, this has been found to be true for all α < 3.0. This means that for α < 3.0, stable fish populations result. Shown is the schematic iteration procedure for α = 2.9. The numerical iterative solutions follow the square, spiral path to a converged result.

11. “Logistic Equation”: f(α,x n ) = x n+1 = αx n (1-x n ) –However, surprisingly (if you think “linearly”!), it has been found that, for α just > 3.0, more than one solution for the fish population exists! Shown is α = 3.1. Schematically, the numerical iterative solutions follow the square, spiral path, but they never converge to one point! Instead, the iteration alternates back & forth between 2 solutions! Solution 1   Solution 2

12. Bifurcation “Logistic Equation”: f(α,x n ) = x n+1 = αx n (1-x n ) –For α just < 3.0, One solution! –For α just > 3.0, Two solutions! Again, this is very “weird” for linear systems. However, its not unusual at all for nonlinear systems! Generally, a sudden change in the number of solutions to a nonlinear equation when a single parameter (such as α) is changed only slightly is called a BIFURCATION.

13. “Logistic Equation”: f(α,x n ) = x n+1 = αx n (1-x n ) We can obtain a general view of this eqtn & its solutions by plotting a BIFURCATION DIAGRAM This is a plot of the converged x n, after many iterations, as a function of the parameter α. This is shown here for 2.8 < α < 4.0 α  “Periodicity”       Chaos

14. “Logistic Equation”: f(α,x n ) = x n+1 = αx n (1-x n ) The bifurcation diagram shows many new & interesting effects (which are totally weird if you think linearly!) –There are regions & “windows” of stability. –There are regions of Chaos! For α = 2.9, after a few iterations, the iterations converge to a stable solution x = 0.655. Definition: N Cycle  A solution that returns to its initial value after N iterations. That is x N+i = x i –For α = 2.9, there is 1 solution.  The “period” = 1 “cycle”. –For α = 3.1, after a few iterations, get 2 solutions which are alternately (oscillating between) x = 0.558 and x = 0.765.  For α = 3.1, the “period” = 2 “cycle”. 1 cycle  2 cycle   4 cycle

15. “Logistic Equation”: f(α,x n ) = x n+1 = αx n (1-x n ) The bifurcation occurring at α = 3.0 is called a Pitchfork Bifurcation because of the shape of bifurcation diagram there. At α = 3.1, the period is doubled (2 cycle), so the solutions have the form x 2+i = x i. At α = 3.45, this 2 cycle bifurcation bifurcates again, to a 4 cycle (4 solutions to the eqtn!)!!! This period doubling continues over & over again as α is increased & the intervals between the doublings decrease. This continues up to an  # of cycles (CHAOS!) at α near 3.57. α = 3.0  α = 3.45   α = 3.57

16. “Logistic Equation”: f(α,x n ) = x n+1 = αx n (1-x n ) Chaos occurs for many α values between 3.57 & 4.0. There are still windows of periodicity. A wide window of this occurs around α = 3.84. Interesting behavior occurs for α = 3.82831. A 3 cycle occurs for several periods & appears stable. Then, it suddenly & violently changes for a few cycles & then returns to the 3 year cycle. This intermittent behavior (stability & instability; chaos & back again) obviously could be devastating to a biological system (the fish population)! α = 3.82831   α = 3.84

17. Example “Logistic Equation”: f(α,x n ) = x n+1 = αx n (1-x n ) Let Δα n  α n - α n+1 be defined as the width (in α) between successive period doubling bifurcations. From the figure, let α 1 = 3.0 = the α value where the first bifurcation occurs & α 2 = 3.449490 = the value where the second one occurs. Also define: δ n  (Δα n )/(Δα n+1 ) Let δ n  δ as n   Find (numerically) δ n for the first few bifurcations & also find the limit δ

18. Solution is in the Table: As α n  the limit of 3.5699456, the number of doublings   and δ n  δ = 4.669202. This value of δ has been found to be a universal property of the period doubling route to chaos when the function being considered has a quadratic maximum. Not confined to 1 dimension! Also true for 2 dimensions ! 4.669202  “Feigenbaum’s number”