Transcript Demonstration PowerPoint

Electronic Chaos
2009 Fall
Steven Wright and Amanda Baldwin
Special Thanks to Mr. Dan Brunski
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Outline
● Motivation and History
● The Pasco Setup
● Feedback and Mapping
● Feigenbaum
● Lyapunov Exponents
● MatLab files
● Conclusions
www.nathanselikoff.com/strangeattractors/
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Motivation
● Contained in the field on nonlinear dynamicsevolves in time
● Chaos theory offers ordered models for seemingly
disorderly systems, such as:
– Weather patterns
– Turbulent Flow
– Population dynamics
– Stock Market Behavior
– Traffic Flow
– Nonlinear circuits
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Chaos Circuit
●Chaotic behavior in a circuit
─Time continuous system
─Simplest circuits that exhibit chaos have a nonlinear component
●Chaotic vs Periodic behavior
─Dependent on the initial conditions
─Makes them unpredictable in the long run
●PASCO system controls initial conditions (control
parameter) with a variable resistor
─Control resistance via Electronic Chaos System
─2000 steps available
─“Tap”0 = 2.5 kΩ
─Each successive “Tap” increase ≈ 40Ω
─Chaotic regions determined for this circuit by the bifurcation plot
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Bifucation Plot
● We took data for tap 603-703 using
10000 data points
● Plot of our data
─ Agrees with the plot of this region in the
manual? (I’m now not so convinced)
•For each ‘R’, every local maxima of the
waveform (x vs. t)—Xmax—is plotted
•Number of points for each ‘R’
corresponds to the period of the
waveform and number of loops on phase
portrait
–The white areas are periodic
–Chaotic regions theoretically have
an infinite number of points and are
the darker regions
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Differential Equation
●Dynamical Equation of an Electronic Circuit
─Dependent on time t
● N-First order equations
● Autonomous ODE
─How large must N be for chaos to exist for this circuit
● N ≥ 3 is sufficient
● Chaos and periodic solutions are available
● General form of the equation for the circuit

..
.
x   x   x  x  D( x)  C
Where α,β,γ, and C are real constants; D(x) is the nonlinear function of the
nonlinear component in the circuit
● This is known as the jerk equation because it is a 3rd order differential
dependent on time
PASCO Chaos Circuit
●D(x) is the nonlinear
component
●Node analysis equations
come from Kirchoff’s
current law
●Chaos controlled by
potentiometer
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The nonlinear element D(x)
●The nonlinear element is described by the equation:
Plot of experimental
measurement of D(x)
R2
Vout = D(Vin ) = –
min(Vin ,0)
R1
D( x)  6 min(x,0)
D(x)

•For all positive values of x this
returns D = 0
•For all negative values of x this
returns D = -6 * x
•2 Diodes & Op Amp
Chaotic Attractor
Phase Space Plots {F[x]; F = dx/dt}
● The number of loops around the basin of attraction corresponds to the period of the
waveform described by the differential equation
● We can conclude:
─ For R = 76.35 kΩ, Xmax = -0.1.1932 the circuit is periodic
─ For R = 78.62 kΩ, Xmax = -0.099818 the circuit is chaotic
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Feigenbaum‘s number
Concepts of Chaos Theory
let Dan = an - an-1 be the width between successive
period doublings
Dan+1
n
1
2
3
4
5
an
3.0
3.449490
3.544090
3.564407
3.568759
3.5699456
Da
dn
0.449490 4.7515
0.094600 4.6562
0.020317 4.6684
0.004352
4.6692
limn®¥ dn » 4.669202 is called the Feigenbaum´s number d
Alexander Brunner
Chaos and Stability
Feigenbaum Numbers
●The limit δ is a universal property when the function f
(α,x) has a quadratic maximum
●It is also true for two-dimensional maps
●The result has been confirmed for several cases
(First found by Mitchell Feigenbaum in the 1970s)
●Chaotic systems with a single quadratic maximum undergo
bifurcations, or period doubling similar to the logistic map
●Every bifurcation level can be related to the next by the
limiting constant δ
●Since such chaotic systems bifurcate at the same rate,
Feigenbaum's constant can be used to predict if and when
chaos will arise.
Lyapunov Exponents
• Gives a measure for the predictability of a dynamic system
– characterizes the rate of separation of infinitesimally close
trajectories
– Describes the avg rate which predictability is lost
• Calculated by similar means as eigenvalues of the Jacobian
matrix J 0(x0)
• Usually Calculate the Maximal Lyapunov Exponent
– Gives the best indication of predictability
– Positive value usually taken as an indication that the system is
chaotic
– Z (t )is the separation of the trajectories

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Lyapunov Exponents
Concepts of Chaos Theory
1
2
Alexander Brunner
Chaos and Stability
Application to the Logistic Map
• a > 0 means chaos and a < 0 indicates nonchaotic behaviour
a
a<0
Alexander Brunner
Chaos and Stability
Matlab
●Script Files
─2008S-1DPlots displays a series of interesting plots wrt x and the
derivatives
─2008S-ElectronicBifurcation looks for local maxima of data series
and plots vs resistance in kΩ
─2008S-ElectronicSingleTap has 4 plots
● Moving wave
● Phase plot
● Poincare section
● Phase plot
●Movies
─In script files, set “savemovie” option to true
─Saves each frame sequentially in 125 dpi tif file
─Use mencoder to turn tif files into a movie
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Conclusion
Chaotic circuits can be created using simple op-amps to create nonlinear
components
Very small changes in the resistance can easily shift the circuit from stable to
chaotic
Bifurcation diagrams can show us the regions of stability and chaos for a given
resistance and Xmax
Phase space plots of the circuit display an attractor, and give us regions of stability
or chaos for x and dx/dt
Lyapunov exponents help us determine the rate at which predictability of the
circuit is lost, i.e. where chaos begins
Feigenbaum number gives us an idea if and when chaos will arise, as a result of
the period doubling on the bifurcation diagram
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References
●Electronic Chaos System 1.0 User’s Guide, by Ken Kiers
●Precision Measurements of a Simple Chaotic Circuit, by Ken
Kiers, Dory Schmidt, and J,C, Sprott
●Chaos amnd Time-Series Analysis, by J.C. Sprott, Oxford
Press 2006
●Matlab Notes and Commentary, by Dan Brunski
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