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Electronic Chaos

2009 Fall

Steven Wright and Amanda Baldwin

Special Thanks to Mr. Dan Brunski

Outline● Motivation and History

● The Pasco Setup

● Feedback and Mapping

● Feigenbaum

● Lyapunov Exponents

● MatLab files

● Conclusions

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www.nathanselikoff.com/strangeattractors/

Motivation● Contained in the field on nonlinear dynamics-

evolves 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)

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•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

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

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

CxDxxxx )(...

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:

)0,min(6)(

,0)min(VR

R– = )D(V = V in

1

2inout

xxD

•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

D(x)

Plot of experimental measurement of D(x)

Chaotic Attractor

● 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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Phase Space Plots {F[x]; F = dx/dt}

Concepts of Chaos Theory Feigenbaum‘s number

let Dan = an - an-1 be the width between successive period doublings

Dan+1

n an Da dn

1 3.0 2 3.449490 0.449490 4.7515 3 3.544090 0.094600 4.6562 4 3.564407 0.020317 4.6684 5 3.568759 0.004352

3.5699456 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

– is the separation of the trajectories

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)(tZ

Concepts of Chaos Theory Lyapunov Exponents

1

2

Alexander Brunner Chaos and Stability

Application to the Logistic Map

• a > 0 means chaos and a < 0 indicates nonchaotic behaviour

Alexander Brunner Chaos and Stability

a

a < 0

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●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

Matlab

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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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●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

References