Summary
This chapter discusses the problem of “chaos generation”, namely efficient techniques for the design of circuits to be employed as generators of chaotic signals (or sample sequences) in engineering applications.
As intelligent computing techniques exploiting chaotic dynamics take momentum, synthesisers of chaotic waveforms become important design primitives. Consequently, design approaches suitable for cost containment, high robustness, and low susceptibility to external interference need to be formalised. This topic is particularly significant since digital hardware can deliver dynamics matching the characteristics of chaotic models only in the short term, and truly chaotic behaviour needs analog subsystems, for which design efforts are higher.
The chapter opens with a concise history of chaotic circuits. This illustrates how the focus has progressively shifted from mere demonstrators to circuits and systems finally optimised for typical merit indexes of engineering applications. It is shown how the move from initial, mostly speculative designs to a real confrontation with emerging applications marked a key point in the design of “chaos generators”. On one hand it posed precise statistical requirements and on the other hand it let important implementation robustness issues emerge.
In the central part of the chapter, after a quick comparison of continuous-time vs discrete-time chaotic models, the focus goes on the latter. A brief review of techniques for their electronic implementation is presented illustrating how significant advantages can be obtained by concentrating on those that enable an efficient reuse of existing hardware. A discussion on how chaotic circuits can be derived from (ADCs) is presented.
In the last part of the chapter, the effectiveness of the proposed approach is proved by measurements made on real prototypes. This part is also the occasion to underline that for those applications requiring “true analog chaos” the ability to achieve it by building blocks ready available on most electronic systems can lead to a strategic advantage.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
McCauley, J.L.: Chaos, Dynamics and Fractals. An algorithmic approach to deterministic chaos. Cambridge University Press, Cambridge (1992)
Kocarev, L., Szczepanski, J., Amigo, J., Tomovski, I.: Discrete chaos — I: Theory. IEEE Transactions on Circuits and Systems—Part I: Regular Papers 53(6), 1300–1309 (2006)
Addabbo, T., Alioto, M., Fort, A., Rocchi, S., Vignoli, V.: Low-hardware complexity prbgs based on a piecewise-linear chaotic map. IEEE Transactions on Circuits and Systems—Part II: Express Briefs 53(5), 329–333 (2006)
Wolfram, S.: A new kind of science. Wolfram Media (2002)
Maxwell, J.C.: Essay for the Eranus Club on science and free will (1873)
Steiner, F.: Quantum chaos. Invited Contribution to the Festschrift Schlaglichter der Forschung: zum 75. Jahrestag der Universität Hamburg 1994 (1994), http://xxx.lanl.gov/ps/chao-dyn/9402001
Van der Pol, B., Van der Mark, J.: Frequency demultiplication. Nature 120(3019), 363–364 (1927)
Kennedy, M.P., Chua, L.O.: Van der Pol and chaos. IEEE Transactions on Circuits and Systems 33(10), 974–980 (1986)
Chua, L.O.: The genesis of Chua’s circuit, EECS Department, University of California, Berkeley, Tech. Rep. UCB/ERL M92/1 (1992), http://www.eecs.berkeley.edu/Pubs/TechRpts/1992/1924.html
Rabinder, N.M. (ed.): Chua’s Circuit: a Paradigm for Chaos. Nonlinear Science. World Scientific, Singapore (1993)
Chua, L.O., Wu, C.W., Huang, A., Zhong, G.-Q.: A universal circuit for studying and generating chaos — Part I: Routes to chaos. IEEE Transactions on Circuits and Systems—Part I: Fundamental Theory and Applications 40(10), 732–744 (1993)
Rodriguez-Vazquez, A.B., Huertas, J.L., Chua, L.: Chaos in a switched-capacitor circuit. IEEE Transactions on Circuits and Systems 32(10), 1083–1085 (1985)
Delgado-Restituto, M., Medeiro, F., Rodríguez-Vázquez, A.: Nonlinear, switched current CMOS IC for random signal generation. Electronics Letters 25, 2190–2191 (1993)
Bernstein, G.M., Lieberman, M.A.: Secure random number generation using chaotic circuit. IEEE Transactions on Circuits and Systems 37(9), 1157–1164 (1990)
Callegari, S., Rovatti, R., Setti, G.: Robustness of chaos in analog implementations. In: Kennedy, M.P., Rovatti, R., Setti, G. (eds.) Chaotic Electronics in Telecommunications, ch. 12, pp. 397–442. CRC International Press, Boca Raton (2000)
Delgado-Restituto, M., Rodriguez-Vazquez, A.: Integrated chaos generators. Proceedings of the IEEE 90(5), 747–767 (2002)
Gerosa, A., Bernardini, R., Pietri, S.: A fully integrated chaotic system for the generation of truly random numbers. IEEE Transactions on Circuits and Systems—Part I: Fundamental Theory and Applications 49(7), 993–1000 (2002)
Callegari, S., Rovatti, R., Setti, G.: Embeddable ADC-based true random number generator for cryptographic applications exploiting nonlinear signal processing and chaos. IEEE Transactions on Signal Processing 53(2), 793–805 (2005)
Ott, E.: Chaos in dynamical systems. Cambridge University Press, Cambridge (1993)
Setti, G., Mazzini, G., Rovatti, R., Callegari, S.: Statistical modeling of discrete time chaotic processes: Basic finite dimensional tools and applications. Proceedings of the IEEE 90(5), 662–690 (2002)
Lasota, A., Mackey, M.C.: Chaos, Fractals and Noise. Stochastic Aspects of Dynamics, 2nd edn. Springer, Heidelberg (1995)
Chua, L.O., Wu, C.W., Huang, A., Zhong, G.-Q.: A universal circuit for studying and generating chaos — Part II: Strange attractors. IEEE Transactions on Circuits and Systems—Part I: Fundamental Theory and Applications 40(10), 745–761 (1993)
Yalcin, M.E., Suykens, J.A.K., Vandewalle, J.: True random bit generation from a double-scroll attractor. IEEE Transactions on Circuits and Systems—Part I: Regular Papers 51(7), 1395–1404 (2004)
Tamasevicius, A., Bumeliene, S., Lindberg, E.: Improved chaotic Colpitts oscillator for ultrahigh frequencies. Electronics Letters 40(25), 1569–1570 (2004)
Schwarz, W., Götz, M., Kelber, K., Abel, A., Falk, T., Dachselt, F.: Statistical analysis and design of chaotic systems. In: Kennedy, M.P., Rovatti, R., Setti, G. (eds.) Chaotic Eletronics in Telecommunications, ch. 9. CRC International Press, Boca Raton (2000)
Kuta, S.: Current mode circuit implementations of PWL functions. Analog Integrated Circuits and Signal Processing (16), 285–297 (1998)
Toumazou, C., Lidgey, F.J., Haigh, D.G. (eds.): Analog IC Design: the Current Mode Approach. Peter Peregrinus Ltd., London (1990)
Callegari, S., Rovatti, R., Setti, G.: Generatore di numeri autenticamente casuali con possibilitá di operazione riconfigurabile tra generatore di numeri autenticamente casuali e convertitore analogico-digitale, Italian Patent application for industrial invention BO2007A000 364 (May 2007)
Callegari, S.: Some more robustness conditions for the invariant density of a class of 1D maps under additive noise. In: Proc. of ECCTD, pp. 1038–1041 (September 2007)
Stojanovski, T., Kocarev, L.: Chaos-based random number generators — Part I: Analysis. IEEE Transactions on Circuits and Systems, Part I 48(3), 281–287 (2001)
Knuth, D.E.: The Art Of Computer Programming, 2nd edn. Seminumerical Algorithms, vol. 2. Addison-Wesley, Reading (1982); random numbers
Pareschi, F., Rovatti, R., Setti, G.: Simple and effective post-processing stage for random stream generated by a chaos-based RNG. In: Proceedings of NOLTA 2006, Bologna - IT: IEICE, pp. 383–386 (September 2006)
Razavi, B.: Principles of Data Conversion System Design. Wiley — IEEE Press (1994)
Callegari, S., Rovatti, R., Setti, G.: Circuito elettronico riconfigurabile come convertitore analogico/digitale e generatore di sequenze binarie autenticamente casuali, Italian Patent application for industrial invention BO2005A000 060 (Feburary 2005)
Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes in C: The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge (1992)
Barker, E., Kelsey, J.: Recommendation for Random Number Generation Using Deterministic Random Bit Generators, National Institute for Standards and Technology Special publication 800-90 (June 2006), http://csrc.nist.gov/publications/nistpubs/800-90/SP800-90DRBG_June2006.pdf
Secure Hash Standard, National Institute for Standards and Technology Std. Federal Information Processing Standards FIPS 180-2 (August 2002), http://www.itl.nist.gov/fipspubs/by-num.htm
Kennedy, M.P., Rovatti, R., Setti, G. (eds.): Chaotic Electronics in Telecommunications. CRC International Press, Boca Raton (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Pareschi, F., Callegari, S., Setti, G., Rovatti, R. (2009). Circuits and Systems for the Synthesis of Chaotic Signals in Engineering Applications. In: Kocarev, L., Galias, Z., Lian, S. (eds) Intelligent Computing Based on Chaos. Studies in Computational Intelligence, vol 184. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-95972-4_8
Download citation
DOI: https://doi.org/10.1007/978-3-540-95972-4_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-95971-7
Online ISBN: 978-3-540-95972-4
eBook Packages: EngineeringEngineering (R0)