Synchronization and Encryption with a Pair of Simple Chaotic Circuits *

1. Ken Kiers Taylor University, Upland, IN Synchronization and Encryption with a Pair of Simple Chaotic Circuits* 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.

2. Outline: • Introduction • Theory • Experimental results with a single chaotic circuit • Synchronization and encryption • Concluding remarks

3. What is chaos? Introduction: 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. For certain nonlinear functions, the solutions are chaotic, for example: Consider the following differential equation: 2. Theory: (1) …where the dots are time derivatives, A and  are constants and D(x) is a nonlinear function of x. …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

5. V1 (inverting) summing amplifier Vout V2 (inverting) integrator Vin Vout …first: consider the “building blocks” of our circuit…. Theory (continued) alternatively:

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

7. PIC microcontroller with A/D analog chaotic circuit personal computer digital potentiometers 3. Experimental Results: 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 • data sent back to the PC via the serial port * Am. J. Phys. 72 (2004) 503.

8. chaos (signal never repeats) period one Bifurcation Plot→ successive maxima of x as a f’n of Rv Comparison of bifurcation points: period two period four

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

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

11. 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 Experimental first- and second-return mapsfor

12. Demonstration of chaos….

13. one bit 4. Synchronization and Encryption • two nearly identical copies of the same circuit • coupled together in a 4:1 ratio • second circuit synchronizes to first (x2 matches x1) • 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 RV correspond to zeros and ones

14. addition of a small analog signal to x1 leads to a failure of x2 to synchronize • subtraction of x2 from x1+σ yields a (noisy) approximation to σ Encryption of an analog signal

15. 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

16. Extra Slides

17. period one period two chaos An Example: The Logistic Map …the chaotic case is very sensitive to initial conditions…! Reference:“Exploring Chaos,” Ed. Nina Hall

18. Bifurcation Diagram for the Logistic Map Reference: http://en.wikipedia.org/wiki/Image:LogisticMap_BifurcationDiagram.png

19. 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…? …some personal history with chaos…. A chaotic circuit…. • …enter the chaotic circuit • low-cost • excellent agreement between theory and experiment • differential equations straightforward to solve