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Proceedings of the 5th International Conference on Applications in Nonlinear Dynamics pp 54–60Cite as

Linear Chaos in a Tape Recorder

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Part of the book series: Understanding Complex Systems ((UCS))

Abstract

A mathematical model of an analog tape recorder is developed and shown to exhibit linear chaos. The playback dynamics act as a wave and are modeled by a linear partial differential equation with a simple analytic solution. This linear dynamical system is shown to exhibit three properties commonly used to define chaotic dynamics: the solution set is dense with periodic orbits, contains transitive orbits, and exhibits extreme sensitivity to initial conditions. Thus, a tape recorder provides a common physical example of linear chaos.

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References

  1. J.M.T. Thompson, H.B. Stewart, Nonlinear Dynamics and Chaos (Wiley, New York, 1986)

    Google Scholar 

  2. C.R. MacCluer, Chaos in linear distributed systems. J. Dyn. Syst. Meas. Control 114, 322 (1992)

    Article  Google Scholar 

  3. A. Gulisashvili, C.R. MacCluer, Linear chaos in the unforced quantum harmonic oscillator. J. Dyn. Syst. Meas. Control 118, 337 (1996)

    Article  Google Scholar 

  4. R. deLaubenfels, H. Emamirad, V. Protopopescu, Linear, chaos and approximation. J. Approx. Theory 105, 176 (2000)

    Article  MathSciNet  Google Scholar 

  5. K.-G. Grosse-Erdmann, A.P. Manguillot, Linear Chaos (Springer, Berlin, 2011)

    Book  Google Scholar 

  6. D.F. Drake, Information’s Role in the Estimation of Chaotic Signals, Ph.D. dissertation (Georgia Institute of Technology, 1998)

    Google Scholar 

  7. S.T. Hayes, Chaos from linear systems: implications for communicating with chaos, and the nature of determinism and randomness. J. Phys. Conf. Ser. 23, 215 (2005)

    Google Scholar 

  8. Y. Hirata, K. Judd, Constructing dynamical systems with specified symbolic dynamics. Chaos 15, 033102 (2005)

    Article  MathSciNet  Google Scholar 

  9. N.J. Corron, S.T. Hayes, S.D. Pethel, J.N. Blakely, Chaos without nonlinear dynamics. Phys. Rev. Lett. 97, 024101 (2006)

    Google Scholar 

  10. D.F. Drake, D.B. Williams, Linear, random representations of chaos. IEEE Trans. Signal Process. 55, 1379 (2007)

    Article  MathSciNet  Google Scholar 

  11. N.J. Corron, S.T. Hayes, S.D. Pethel, J.N. Blakely, Synthesizing folded band chaos. Phys. Rev. E 75, 045201R (2007)

    Google Scholar 

  12. D.W. Hahs, N.J. Corron, J.N. Blakely, Synthesizing antipodal chaotic waveforms. J. Franklin Inst. 351, 2562 (2014)

    Article  MathSciNet  Google Scholar 

  13. E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 1993)

    Google Scholar 

  14. N.J. Corron, An exactly solvable chaotic differential equation. Dyn. Contin. Discret. Impuls. Syst. A 16, 777 (2009)

    Google Scholar 

  15. N.J. Corron, J.N. Blakely, M.T. Stahl, A matched filter for chaos. Chaos 20, 023123 (2010)

    Google Scholar 

  16. N.J. Corron, J.N. Blakely, Exact folded-band chaotic oscillator. Chaos 22, 023113 (2012)

    Article  MathSciNet  Google Scholar 

  17. N.J. Corron, J.N. Blakely, Chaos in optimal communication waveforms. P. Roy. Soc. Lond. A 471, 20150222 (2015)

    Article  MathSciNet  Google Scholar 

  18. A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, P. Davis, Fast physical random bit generation with chaotic semiconductor lasers. Nat. Photon. 2, 728 (2008)

    Article  Google Scholar 

  19. M.S. Willsey, K.M. Cuomo, A.V. Oppenheim, Selecting the Lorenz parameters for wideband radar waveform generation. Int. J. Bifurc. Chaos 21, 2539 (2011)

    Article  Google Scholar 

  20. H. Leung (ed.), Chaotic Signal Processing (SIAM, 2014)

    Google Scholar 

  21. M. Eisencraft, R. Attux, R. Suyama (eds.), Chaotic Signals in Digital Communications (CRC Press, Boca Raton, 2014)

    Google Scholar 

  22. R.L. Devaney, Introduction to Chaotic Dynamical Systems (Addison-Wesley, Boston, 1989)

    Google Scholar 

  23. J. Banks, J. Brooks, G. Cairns, G. Davis, P. Stacy, On Devaney’s definition of chaos. Am. Math. Mon. 99, 332–334 (1992)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author recognizes Dr. Daniel Hahs, Dr. Shawn Pethel, and Dr. Shangbing Ai for helpful discussions regarding the interpretation and presentation of the research results.

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Authors and Affiliations

  1. Charles M. Bowden Laboratory, U.S. Army AMRDEC, Redstone Arsenal, Huntsville, AL, USA

    Ned J. Corron

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  1. Ned J. Corron
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Correspondence to Ned J. Corron .

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Editors and Affiliations

  1. Space and Naval Warfare Systems Center, San Diego, CA, USA

    Visarath In

  2. Space and Naval Warfare Systems Center, San Diego, CA, USA

    Patrick Longhini

  3. Department of Mathematics and Statistics, San Diego State University, San Diego, CA, USA

    Antonio Palacios

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© 2019 This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply

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Corron, N.J. (2019). Linear Chaos in a Tape Recorder. In: In, V., Longhini, P., Palacios, A. (eds) Proceedings of the 5th International Conference on Applications in Nonlinear Dynamics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-10892-2_7

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  • DOI: https://doi.org/10.1007/978-3-030-10892-2_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10891-5

  • Online ISBN: 978-3-030-10892-2

  • eBook Packages: EngineeringEngineering (R0)

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