# Chaos in a Simple Circuit

Synchronization and Encryption with a Pair of Simple Chaotic Circuits* Ken Kiers Taylor University, Upland, IN Special thanks to J.C. Sprott and to the many TU students and faculty who have participated in this project over the years * Some of our results may be found in: Am. J. Phys. 72 (2004) 503. Outline: 1. Introduction 2. Theory 3. Experimental results with a single chaotic circuit 4. Synchronization and encryption 5. Concluding remarks 1. Introduction: What is chaos? A chaotic system exhibits extreme sensitivity to initial conditions(uncertainties grow exponentially with time). Examples: the weather (butterfly effect), driven pendulum What are the minimal requirements for chaos? For a discrete system system of equations must contain a nonlinearity For a continuous system differential equation must be at least third order 2. Theory: Consider the following differential equation: x Ax x D(x) (1) where the dots are time derivatives, A and are constants and D(x) is a nonlinear function of

x. For certain nonlinear functions, the solutions are chaotic, for example: D( x) | x |, D( x) 6 min( x, 0) it turns out that Eq. (1) can be modeled by a simple electronic circuit, where x represents the voltage atand a node. the functions D(x) are modeled using diodes Theory (continued) first: consider the building blocks of our circuit. V1 V2 (inverting) summing amplifier Vout Vout (V1 V2 ) (inverting) integrator Vout Vin Vout alternativel y: 1 Vin dt RC Vin RC dVout dt

Theory (continued) The circuit: R x x x D( x) Rv ...the sub-circuit models the onesided absolute value function. R V0 R0 Rv acts as a control parameter to bring the circuit in experimental data for D(x)=-6min(x,0) 3. Experimental Results: digital potentiomete rs analog chaotic circuit A few experimental details*: circuit ran at approximately 3 Hz digital pots provided 2000-step resolution in Rv microcontroller controled digital pots and measured x and its time derivatives from the circuit A/D at 167 Hz; 12-bit resolution over 0-5 V * Am. dataJ. sent to the PC via the serial Phys.back 72 (2004) 503.

port PIC microcontroller with A/D personal computer Bifurcation Plot successive maxima of x as a fn of Rv chaos (signal never repeats) period one Comparison of bifurcation points: period two period four Exp. (k) Theory (k) Diff. (k) Diff. (%) a 53.2 52.9 0.3 0.6 b

65.0 65.0 0.0 0.0 c 78.8 78.7 0.1 0.1 d 101.7 101.7 0.0 0.0 e 125.2 125.5 -0.3 -0.2 Experimental phase space plots: experiment and theory superimposed(!) x x

x, x and x are measured directly from the circuit Power spectrum as a function of frequency fundamental at approximately 3 Hz period doubling is also frequency halving. Chaos gives a noisy power spectrum. harmonics at integer multiples of fundamental period one period two period four chaos Experimental first- and second-return successive maxima maps of fora chaotic attractor return maps show fractal structure sure enough! intersections with diagonal give evidence for unstable period-one and two orbits

Demonstration of chaos. one bit ; '1'(#2* <. () \$('. & ) () "* ; '1'(#2*<. () \$('. & ) () "* 9": - '\$. * ! "#\$% && '(('(() "* '\$* ! "#\$% ) "* +'", - '(* +'", - '(* . - (* !/ * two nearly identical copies of the same circuit 0- &0-&& '\$1* & '\$1* +'", - +'", '(* - '(* coupled together in a 4:1 ratio 567! / *8*564!4 second circuit synchronizes to first (x2 matches x1) changes in the first circuit can be detected in the second through its inability to synchronize 02#3) * =)-,'(* ) '3) "* +'", +'", - '(* '\$*

. - (* ! 4* Encryption of a digital signal: changes in RV correspond to zeros and Encryption of an analog signal addition of a small analog signal to x1 leads to a failure of x2 to synchronize ! "#\$%&' C&", #; )\$\$%&' ( )&*+)\$' ( )&*+)\$' 6' ), ' 7 ! 8' - +\$' subtraction of x2 from x1+ yields a (noisy) approximation to . )/, "0'\$- '1%' 2, *&34\$%5' ! !)9),)9), /' /' (*+)\$' )&*+)\$' ( )& !8': '7 . +; . ;+;), ;/'), /' ( )&*+)\$' ( )&*+)\$'

>[email protected]!8': '7B': '>?=! = . 0"<%' D%*%)<%&' ( )&*+)\$' ( )&*+)\$' ), ' - +\$' !=' Concluding Remarks Chaos provides a fascinating and accessible area of study for undergraduates The one-sided absolute value circuit is easy to construct and provides both qualitative demonstrations and possibilities for careful comparisons with theory Agreement with theory is better than one percent for bifurcation points and peaks of power spectra for this circuit Chaos can also be used as a means of encryption Extra Slides An Example: The Logistic Map r=2 r = 3.2 r=4 n xn xn xn 0 0.40000 0.40000

1 0.48000 2 xn 1 r xn (1 xn ) r=2 r = 3.2 r=4 n xn xn xn 0.40000 0 0.35000 0.35000 0.40010 0.76800 0.96000 1 0.45500 0.72800 0.96008 0.49920

0.57016 0.15360 2 0.49595 0.63365 0.15331 3 0.50000 0.78425 0.52003 3 0.49997 0.74284 0.51921 4 0.50000 0.54145 0.99840 4 0.50000 0.61129 0.99852 5

0.50000 0.79450 0.00641 5 0.50000 0.76036 0.00590 6 0.50000 0.52246 0.02547 6 0.50000 0.58307 0.02345 7 0.50000 0.79839 0.09927 7 0.50000 0.77792 0.09160

8 0.50000 0.51509 0.35767 8 0.50000 0.55284 0.33283 9 0.50000 0.79927 0.91897 9 0.50000 0.79107 0.88822 10 0.50000 0.51340 0.29786 10 0.50000 0.52890

0.39715 11 0.50000 0.79943 0.83656 11 0.50000 0.79733 0.95769 12 0.50000 0.51310 0.54692 12 0.50000 0.51711 0.16208 13 0.50000 0.79945 0.99120 13 0.50000

0.79906 0.54324 14 0.50000 0.51305 0.03491 14 0.50000 0.51380 0.99252 15 0.50000 0.79945 0.13476 15 0.50000 0.79939 0.02969 period one period two chaos the chaotic case is very sensitive to initial conditions! Reference: Exploring Chaos, Ed.

Bifurcation Diagram for the Logistic Map xn 1 r xn (1 xn ) r=2 r = 3.2 r=4 n xn xn xn 0 0.40000 0.40000 0.40000 1 0.48000 0.76800 0.96000 2 0.49920 0.57016 0.15360 3

0.50000 0.78425 0.52003 4 0.50000 0.54145 0.99840 5 0.50000 0.79450 0.00641 6 0.50000 0.52246 0.02547 7 0.50000 0.79839 0.09927 8 0.50000 0.51509 0.35767

9 0.50000 0.79927 0.91897 10 0.50000 0.51340 0.29786 11 0.50000 0.79943 0.83656 12 0.50000 0.51310 0.54692 13 0.50000 0.79945 0.99120 14 0.50000 0.51305

0.03491 15 0.50000 0.79945 0.13476 Reference: http://en.wikipedia.org/wiki/Image:LogisticMap_BifurcationDiagram.png A chaotic circuit. some personal history with chaos. looking for a low-cost, high-precision chaos experiment there seem to be many qualitative low-cost experiments as well as some very expensive but not much experiments that in are more quantitative in between? nature enter the chaotic circuit low-cost excellent agreement between theory and experiment differential equations straightforward to solve

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