Advanced Parallel Processing of Lyapunov ExponentsVerified by Practical Circuit

Tomas GOTTHANS, Jirı PETRZELA, Zdenek HRUBOS

Dept. of Radio Electronics, Brno University of Technology, Purkynova 118, 612 00 Brno, Czech Republic

xgotth00@stud.feec.vutbr.cz, petrzelj@feec.vutbr.cz, xhrubo00@stud.feec.vutbr.cz

Abstract. An advanced method for parallel computationof Lyapunov exponents in nonlinear parameter dependentcontinuous dynamical systems is presented. For computingexponents MATLAB with Parallel Toolbox was used. A sim-plest physical chaotic flow published in [2] was chosen andmathematically analysed. Obtained equation’s parameterswere calculated and synthesized into a circuit. Realizationwas implemented into a analog circuit and compared withthe theory.

KeywordsChaos, Lyapunov exponents, parallel processing, cir-

cuit synthesis, simple chaotic flow.

1. IntrodutionThis article discuss parametric non-linear system anal-

ysis, especially quantifying and detecting chaotic behaviour.

Almost every article dealing with chaotic non-linear be-

haviour is using Lyapunov exponents for system analysis.

Calculating exponents is well known process [1]. It is very

important when analysing time stability.

It is well known fact due ageing system (harmonic os-

cillator) can became erratic. To discover parameters and

system stability, it is good to examine all permutations of

parameters on bounded set.

2. Parallel AnalysisWhen resolving systems, especially it’s exponents,

one algorithm with multiple data was used. We present a

parallel algorithm using decomposition and distribution of

constants and parallel toolbox from Matlab 2009b. Pro-

posed algorithm has been tested on a novel chaotic circuit

and has also been computed with different numbers of CPU

cores compared with standard non distributed method. This

technique has proved to be very useful on new multi-core

central processing units.

Fig. 1. Decompositing values between workers.

When converting sequential algorithm into a dis-

tributed, it needs to be carefully chosen which parameter

will be distributed between workers like in Figure 1.

Distributing more than one variable lead to diagonal filled

cube results like yellow and red workers in Figure 1.

This is also valid in more than two dimensional state

space. In two dimensional data set it turned out, that

distributing only one variable between workers is correct.

3. Mathematical AnalysisSpecial chaotic system with centre equilibria points

was chosen

x = −y, y = x+ z, z = bxz + ay2 (1)

where dots over state variables denote time derivatives and

a and b are the real numbers. It is computed for which

real numbers (meaning a and b) the system behave chaot-

ically. Setting a, b > 0, the equilibria point is located at

c0 = [0, 0, 0]T . The position of equilibria point is parame-

ter independent. Investigation of vicinity around point c0 is

given by

det(J− λI) = 0 (2)

where J represents the Jacobian matrix made of all first-

order partial derivatives of a vector or scalar-valued function

with respect to another vector. I is the identity matrix (unit

978-1-61284-324-7/11/$26.00 ©2011 IEEE

Fig. 2. Numerically integrated system (1) for a = 3 and b = 1.

matrix) of size n is the n x n square matrix with ones on the

main diagonal and zeros elsewhere.

∣∣∣∣∣∣∣

−λ 1 0

−1 −λ −10 0 −λ

∣∣∣∣∣∣∣

= −λ3 − λ = 0. (3)

Characteristic polynomial root given by Equation 3.

are λ = [0,+i,−i]T . When examinating stability around

equilibria point, we get centre or elliptical isocline. That

means, tending to unstable behaviour.

The flow compression around equilibria leads to ex-

treme sensitivity to initial conditions. Figure 4. shows

time behaviour for initial conditions ic1 = [0.1, 0, 0]T

,

ic2 = [0.11, 0, 0]T

and ic3 = [0.2, 0, 0]T

.

Speciality of this system is the stable point in zero.

Without any initial condition, the circuit stay in zero even af-

ter power on. But because scatter of components and many

more imperfections, the circuit escape stable point and start

to oscillate. Analysing the largest Lyapunov exponent in

Figure 3, can be seen the robustness of the flow. This prove,

that chaotic behaviour is not caused only by transient per-

formance.

4. Circuitry RealizationThere exist several ways how to practically realize

chaotic oscillators [3]. Most of these techniques are

straightforward and have been already published. To syn-

thesize circuit from differential equations system, integrator

synthesis was chosen. Only three basic building blocks

are necessary: inverting integrator, summing amplifier and

analog multipliers. The block schematics can be seen at

Figure 6.

Fig. 3. The largest Lyapunov exponent for data set.

Fig. 4. Sensitivity to initial conditions.

Fig. 5. Spectrum of Lyapunov exponents (λ1, λ2, λ3) for a = 3

and b = 1.

Fig. 6. Block diagram of circuit realization.

Scaling the values of the passive elements are

C = 100 nF , R3 = 10 kΩ, R4 = 1 kΩ. Re-

sistors R1 ≈ 33 Ω and R2 ≈ 100 Ω are sup-

posed to be variable in order to trace the evolution

of chaos. There is a direct proportion between a, band R1, R2. Thus frequency re-normalization is an

easy and straight-forward process covering identical

change of all integration constants simultaneously.

5. Experimental ResultsThe parallel method has proved to be a good improve-

ment to contemporary approach. In Table 1. we can see the

evident speed up of the process. Using parallel computing

can use the full potential af computing units. Comparing the

speed, it was used data set of size 10 by 10 field. The circuit

implementation was realized and measured by Agilent

Infinium 54820a.

No. of CPUs 1 CPU 2 CPUs 4 CPUs 8 CPUs

Elapsed time[s] 353.6 180.3 116.2 88.4

Tab. 1. Compairing consumed time of computings.

Comparing experimentally obtained (Fig. 7. and

Fig. 8.) and mathematical results (Fig. 9. and Fig. 10.) it

confirmed a very good relationship between theoretical ex-

pectations and the experimental measurements.

6. Conclusion and ImprovementsIn this article the novel method for parallel comput-

ing of Lyapunov exponents for large data sets has been

Fig. 7. Measured data from ralized circuit R1 = 36Ω and

R2 = 108Ω. Projection X-Y.

Fig. 8. Measured data from ralized circuit R1 = 36Ω and

R2 = 108Ω.Projection Z-Y.

Fig. 9. Mathematical results obtained by numerical integration.

Fig. 10. Mathematical results obtained by numerical integra-

tion.

presented. The method was tested on various systems and

showed to be very efficient.

Finally was chosen a special system of differential

equations (1) with centre equilibria. The system was math-

ematically analysed and a circuit realization was imple-

mented and measured to prove method validity. This sim-

ple chaotic flow proved to be very robust and can be easily

implemented into communication systems.

AcknowledgementResearch described in the paper was financially

supported by the Czech Grant Agency under grant

102/08/H027. The second author would also like to thank

Grant Agency of the Czech Republic for their support

through project number 102/09/P217.

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