1. 1 Class #27 Notes :60 Homework Is due before you leave Problem 10-25 has been upgraded to extra-credit. Probs 10-21 and 10-24 are CORE PROBLEMS. Make sure you understand these. When you return – Will spend another lecture on tops, tensors and Euler’s theorem before the exam.

2. 2 Class #27 of 30 Nonlinear Systems and Chaos Most important concepts  Sensitive Dependence on Initial conditions  Attractors Other concepts  State-space orbits  Non-linear diff. eq.  Driven oscillations  Second Harmonic Generation  Subharmonics  Period-doubling cascade  Bifurcation plot  Poincare diagram  Mappings  Feigenbaum number  Universality :02

3. 3 Chaos on the ski-slope :60 7 “Ideal skiers” follow the fall-line and end up very different places

4. 4 Insensitive dependence on initial conditions :60

5. 5 Sensitive dependence on initial conditions :60

6. 6 Resampled pendulum data :60 Gamma=0.3 Gamma=1.0826Gamma=1.105 Gamma=1.077

7. 7 Bifurcation plot :60      

8. 8 Bifurcation plot and universality :60 For ANY chaotic system, the period doubling route to chaos takes a similar form The intervals of “critical parameter” required to create a new bifurcation get ever shorter by a ratio called the Feigenbaum #.

9. 9 In and out of chaos :60

10. 10 Poincare plot :60

11. 11 Poincare plot :60 Poincare plot is set of allowed states at any time t.  States far from these points converge on these points after transients die out  Because it has fractal dimension, the Poincare plot is called a “strange attractor”

12. 12 State-space of flows :60

13. 13 Cooking with state-space :60 Dissipative system  The net volume of possible states in phase space ->0 Bounded behavior  The range of possible states is bounded The evolution of the dynamic system “stirs” phase space. The set of possible states gets infinitely long and with zero area. It becomes fractal A cut through it is a “Cantor Set”

14. 14 Mapping vs. Flow Gamma=1.0826 Gamma=1.105 A Flow is a continuous system A flow moves from one state to another by a differential equation Our DDP is a flow A mapping is a discrete system. State n-> State n+1 according to a difference equation Evaluating a flow at discrete times turns it into a mapping Mappings are much easier to analyze.

15. 15 Logistic map :02 “Interesting” values of r are. R=2.8 (interesting because it’s boring) R=3.2 (Well into period doubling_ R= 3.4 (Period quadrupling) R=3.7 (Chaos) R=3.84 (Period tripling)