Motivation

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 nth 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: xn+1 = (2xn+3)2 • More generally, a nonlinear mappingis written: xn+1 = f(xn),where f(xn) is aspecified 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 xn+1 = f(α,xn) where 0  xn  1. The specified function f(α,xn) generates xn+1 from xn in a specified manner. αis a parameter characteristic of the system. • The collection of points generated is called a “Map” of f(α,xn). Generating these points is called “Mapping” f(α,xn). • If f(α,xn) is nonlinear, we often need to solve xn+1 = f(α,xn) 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 iterativedifference equation: xn+1 = αxn(1-xn) (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. x1 = # fish in the pond at beginning of the 1styear (normalized to 0  x1  1). xn = (relative) # fish in the pond at beginning of nthyear (normalized to 0  xn  1) • If x1is small, the population may grow rapidly for small n, but as n increases, xnmay decrease because of overpopulation!

7. The “Logistic Equation”: xn+1 = αxn(1-xn) • The xnare scaled so that 0  xn  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,xn) = xn+1 = αxn(1-xn) A way to illustrate the numerical iteration solution schematically is by a graph, called the Logistic Map. It plots xn+1 vs.xn. Shown here for α = 2.0. • Procedure: xn+1 = 2xn(1-xn) Start with an initial value x1on the horizontal axis. Move up vertically until the curve is intersected. Then move to the left to find x2on the vertical axis.Then, start with this value of x2on 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 x1as long as its not 0 or 1. An obvious & not surprising result!

9. An Easier Procedure!xn+1 = 2xn(1-xn) • Add the 45º line xn+1 = xn to the graph, as in the figure. • After intersecting the curve vertically from x1, move horizontally to intersect the 45º line to find x2, & move up vertically to find x3, etc. • This gets the same result as before, but it does so faster. Convergence to xn = 0.5 is faster!

10. The “Logistic Equation”: f(α,xn) = xn+1 = αxn(1-xn) • In practice, biologists study the behavior of this system as the parameter αis 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. Solution 1  • “Logistic Equation”: f(α,xn) = xn+1 = αxn(1-xn) • However, surprisingly (if you think “linearly”!), it has been found that, for αjust > 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 2

12. Bifurcation • “Logistic Equation”: f(α,xn) = xn+1 = αxn(1-xn) • 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 aBIFURCATION.

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

14. “Logistic Equation”: f(α,xn) = xn+1 = αxn(1-xn) • 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! Forα = 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 xN+i = xi • 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”. 2 cycle  1 cycle   4 cycle

15. α = 3.45  α = 3.0  • “Logistic Equation”: f(α,xn) = xn+1 = αxn(1-xn) • The bifurcation occurring at α = 3.0is called aPitchfork 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 x2+i = xi. 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 αnear 3.57.  α = 3.57

16. “Logistic Equation”: f(α,xn) = xn+1 = αxn(1-xn) • 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,xn) = xn+1 = αxn(1-xn) Let Δαn αn - αn+1be defined as the width (in α) between successive period doubling bifurcations. From the figure, let α1 = 3.0 = the α value where the first bifurcation occurs & α2 = 3.449490 = the value where the second one occurs. Alsodefine: δn (Δαn)/(Δαn+1) Letδn δ as n   Find (numerically) δnfor the first few bifurcations & also find the limit δ

18. Solution is in the Table: As αn thelimit of 3.5699456, the number of doublings   and δn δ = 4.669202. This value ofδ 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”