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
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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
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3. 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
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4. Insensitive dependence on initial conditions
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5. Sensitive dependence on initial conditions
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6. 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
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7. Lyapunov exponent
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8. :60

9. Resampled pendulum data
Gamma=0.3
Gamma=1.077
Gamma=1.0826
Gamma=1.105
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10. Bifurcation plot 0.3 +100 +50 -50 -100 :60
-50
-100
Show representative plots here … period doubling / tripling etc.
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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 #.
Show representative plots here … period doubling / tripling etc.
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12. In and out of chaos
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13. 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 , reduce damping by half.
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14. 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”
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15. State-space of flows
Show representative plots here … period doubling / tripling etc.
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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”
More illustrations … converting flows to mappings.
Logistic map.
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17. 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

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)
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