1. Synchronization and Encryption with a Pair of Simple Chaotic Circuits * Ken Kiers Taylor University, Upland, IN * Some of our results may be found in: Am. J. Phys. 72 (2004) 503. Special thanks to J.C. Sprott and to the many TU students and faculty who have participated in this project over the years

2. 1.Introduction 2.Theory 3.Experimental results with a single chaotic circuit 4.Synchronization and encryption 5.Concluding remarks Outline:

3. 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 …and it must contain a nonlinearity

4. 2. Theory: Consider the following differential equation: …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: …it turns out that Eq. (1) can be modeled by a simple electronic circuit, where x represents the voltage at a node. → and the functions D(x) are modeled using diodes (1)

5. Theory (continued) V1V1 V2V2 V out (inverting) summing amplifier V in V out (inverting) integrator …first: consider the “building blocks” of our circuit…. alternatively:

6. Theory (continued) The circuit:...the sub-circuit models the “one-sided absolute value” function…. experimental data for D(x)=-6min(x,0) → → R v acts as a control parameter to bring the circuit in and out of chaos

7. 3.Experimental Results: A few experimental details*: circuit ran at approximately 3 Hz digital pots provided 2000-step resolution in R v 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 data sent back to the PC via the serial port analog chaotic circuit digital potentiometers PIC microcontroller with A/D personal computer * Am. J. Phys. 72 (2004) 503.

8. Bifurcation Plot → successive maxima of x as a f’n of R v period one period four period two chaos (signal never repeats) Exp. (k  ) Theory (k  ) Diff. (k  ) Diff. (%) a53.252.90.30.6 b65.0 0.0 c78.878.70.1 d101.7 0.0 e125.2125.5-0.3-0.2 Comparison of bifurcation points:

9. Experimental phase space plots: experiment and theory superimposed(!)

10. Power spectrum as a function of frequency “fundamental” at approximately 3 Hz “harmonics” at integer multiples of fundamental “period doubling” is also “frequency halving”…. Chaos gives a “noisy” power spectrum…. period one chaos period four period two

11. Experimental first- and second-return maps for return maps show fractal structure …sure enough…! intersections with diagonal give evidence for unstable period-one and –two orbits successive maxima of a chaotic attractor

12. Demonstration of chaos….

13. two nearly identical copies of the same circuit coupled together in a 4:1 ratio second circuit synchronizes to first (x 2 matches x 1 ) changes in the first circuit can be detected in the second through its inability to synchronize use this to encrypt/decrypt data Encryption of a digital signal: changes in R V correspond to zeros and ones one bit

14. Encryption of an analog signal addition of a small analog signal to x 1 leads to a failure of x 2 to synchronize subtraction of x 2 from x 1 +σ yields a (noisy) approximation to σ

15. 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 Concluding Remarks

16. Extra Slides

17. An Example: The Logistic Map r = 2r = 3.2r = 4 nxnxn xnxn xnxn 00.40000 10.480000.768000.96000 20.499200.570160.15360 30.500000.784250.52003 40.500000.541450.99840 50.500000.794500.00641 60.500000.522460.02547 70.500000.798390.09927 80.500000.515090.35767 90.500000.799270.91897 100.500000.513400.29786 110.500000.799430.83656 120.500000.513100.54692 130.500000.799450.99120 140.500000.513050.03491 150.500000.799450.13476 Reference: “Exploring Chaos,” Ed. Nina Hall r = 2r = 3.2r = 4 nxnxn xnxn xnxn 0 0.35000 0.40010 1 0.455000.728000.96008 2 0.495950.633650.15331 3 0.499970.742840.51921 4 0.500000.611290.99852 5 0.500000.760360.00590 6 0.500000.583070.02345 7 0.500000.777920.09160 8 0.500000.552840.33283 9 0.500000.791070.88822 10 0.500000.528900.39715 11 0.500000.797330.95769 12 0.500000.517110.16208 13 0.500000.799060.54324 14 0.500000.513800.99252 15 0.500000.799390.02969 period one period two chaos …the chaotic case is very sensitive to initial conditions…!

18. Bifurcation Diagram for the Logistic Map r = 2r = 3.2r = 4 nxnxn xnxn xnxn 00.40000 10.480000.768000.96000 20.499200.570160.15360 30.500000.784250.52003 40.500000.541450.99840 50.500000.794500.00641 60.500000.522460.02547 70.500000.798390.09927 80.500000.515090.35767 90.500000.799270.91897 100.500000.513400.29786 110.500000.799430.83656 120.500000.513100.54692 130.500000.799450.99120 140.500000.513050.03491 150.500000.799450.13476 Reference: http://en.wikipedia.org/wiki/Image:LogisticMap_BifurcationDiagram.png

19. 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 experiments that are more quantitative in nature… but not much in between…? …enter the chaotic circuit low-cost excellent agreement between theory and experiment differential equations straightforward to solve