Synchronization and Encryption with a Pair of Simple Synchronization and Encryption with a Pair of Simple

Chaotic CircuitsChaotic 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

1. Introduction

2. Theory

3. Experimental results with a single chaotic circuit

4. Synchronization and encryption

5. Concluding remarks

Outline:

1.1. Introduction: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

2. 2. Theory:Theory:

Consider the following differential equation:

)(xDxxAx

…where the dots are time derivatives, A and are constants and D(x) is a nonlinear function of x.

( ) | |,

( ) min( , )6 0

D x x

D x 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)

Theory (continued)Theory (continued)

V1

V2

Vout

(inverting) summing amplifier

)( 21 VVVout

dtVRC

V inout 1

Vin Vout

(inverting) integrator

…first: consider the “building blocks” of our circuit….

dt

dVRCV out

in

alternatively:

Theory (continued)Theory (continued)

The circuit:

00

)( VR

RxDxx

R

Rx

v

...the sub-circuit models the “one-

sided absolute value” function….

experimental data for D(x)=-6min(x,0)

→→ Rv acts as a control parameter to bring the circuit in and out of chaos

3.3. Experimental Results: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

analog chaotic circuit

digital potentiomete

rs

PIC microcontroller

with A/D

personal computer

* Am. J. Phys. 72 (2004) 503.

Bifurcation PlotBifurcation Plot →→ successive maxima of successive maxima of x x as a fas a f’’n n ofof R Rvv

period one

period fourperiod two

chaos (signal never repeats)

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

Comparison of bifurcation points:

Experimental phase space plots:Experimental phase space plots:

x

xcircuit thefromdirectly measured are and , xxx

experiment and theory superimposed(!)

Power spectrumPower spectrum as a function of as a function of frequencyfrequency

“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

Experimental Experimental first- and second-return first- and second-return mapsmaps forfor

return maps show fractal

structure

…sure enough…!

intersections with diagonal give evidence for

unstable period-one and –two orbits

successive maxima of a successive maxima of a chaoticchaotic attractorattractor

Demonstration of chaos….Demonstration of chaos….

• 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

one bit

Encryption of an analog signalEncryption of an analog signal

• 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

• 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 RemarksConcluding Remarks

Extra Slides

An Example: The Logistic MapAn Example: The Logistic Map )( nnn xxrx 11

  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:Reference: ““Exploring Chaos,Exploring Chaos,”” Ed. Ed. Nina HallNina Hall

  r = 2 r = 3.2 r = 4

n xn xn xn

0 0.35000 0.35000 0.40010

1 0.45500 0.72800 0.96008

2 0.49595 0.63365 0.15331

3 0.49997 0.74284 0.51921

4 0.50000 0.61129 0.99852

5 0.50000 0.76036 0.00590

6 0.50000 0.58307 0.02345

7 0.50000 0.77792 0.09160

8 0.50000 0.55284 0.33283

9 0.50000 0.79107 0.88822

10 0.50000 0.52890 0.39715

11 0.50000 0.79733 0.95769

12 0.50000 0.51711 0.16208

13 0.50000 0.79906 0.54324

14 0.50000 0.51380 0.99252

15 0.50000 0.79939 0.02969

period oneperiod two

chaos

……the chaotic case is very the chaotic case is very sensitive to initial sensitive to initial conditions…!conditions…!

Bifurcation Diagram for the Logistic Bifurcation Diagram for the Logistic MapMap

)( nnn xxrx 11

  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:Reference: http://en.wikipedia.org/wiki/Image:LogisticMap_BifurcationDiagram.png http://en.wikipedia.org/wiki/Image:LogisticMap_BifurcationDiagram.png

A chaotic circuit….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