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

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

7 “Ideal skiers” follow the fall-line and end up very different places Chaos on the ski-slope Same as pinball after several rows of pins. 10 cm and 1 mm apart. 7 “Ideal skiers” follow the fall-line and end up very different places :60

Insensitive dependence on initial conditions :60

Sensitive dependence on initial conditions :60

Dependence on initial conditions Insensitive: Large differences in IC’s become exponentially smaller Takes 4 seconds to grow by 2 orders of magnitude. So if could improve accuracy of experiment by 2 orders of magnitude, can get 4 seconds more predictability. Sensitive: Small differences in IC’s become exponentially larger :60

Lyapunov exponent :60

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

Bifurcation plot 0.3 +100 +50 -50 -100 :60 -50 -100 Show representative plots here … period doubling / tripling etc. :60

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 #. Show representative plots here … period doubling / tripling etc. :60

In and out of chaos :60

Poincare plot or Poincare “section” Instead of showing entire phase-space orbit, put single point in phase space once/cycle of pendulum. Show representative plots here … period doubling / tripling etc. Poincare section for 1000 seconds. First figure for 100 seconds. 1.105, reduce damping by half. :60

Poincare plot Poincare plot is set of allowed states at any time t. More illustrations … converting flows to mappings. Logistic map. 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

State-space of flows Show representative plots here … period doubling / tripling etc. :60

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” More illustrations … converting flows to mappings. Logistic map. :60

Mapping vs. Flow A Flow is a continuous system 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

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