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Understanding Nonlinear Systems and Chaos

Explore the key concepts of nonlinear systems and chaos, including sensitive dependence on initial conditions, attractors, bifurcation plots, and mappings. Learn about dissipative and non-dissipative chaos, as well as the Poincare plot and its connection to strange attractors. Discover the fascinating world of chaos theory and its applications.

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Understanding Nonlinear Systems and Chaos

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  1. Handout #21 • 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 / strange attractors • Mappings • Feigenbaum number • Universality :02

  2. Conditions for chaos • Dissipative Chaos • Requires a differential equation with 3 or more independent variables. • Requires a non-linear coupling between at least two of the variables. • Requires a dissipative term (that will use up energy). • Non-dissipative chaos • Not in this course :60

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

  4. Insensitive dependence on initial conditions :60

  5. Sensitive dependence on initial conditions :60

  6. Dependence on initial conditions Insensitive: Large differences in IC’s become exponentially smaller Sensitive: Small differences in IC’s become exponentially larger :60

  7. Lyapunov exponent :60

  8. :60

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

  10. Bifurcation plot 0.3 +100 +50 0 -50 -100 :60

  11. Bifurcation plot and universality • 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 #. :60

  12. In and out of chaos :60

  13. Poincare plot or Poincare “section” • Instead of showing entire phase-space orbit, put single point in phase space once/cycle of pendulum. :60

  14. Poincare plot • 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” :60

  15. State-space of flows :60

  16. Cooking with state-space • 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” :60

  17. Mapping vs. Flow Gamma=1.0826 • 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. Gamma=1.105

  18. Logistic map “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) :02

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