Research Article
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Year 2022, Volume 4, Issue 4, 197 - 204, 31.12.2022
https://doi.org/10.51537/chaos.1196653

Abstract

References

  • Addabbo, T., M. Alioto, A. Fort, S. Rocchi, and V. Vignoli, 2006 The digital tent map: Performance analysis and optimized design as a low-complexity source of pseudorandom bits. IEEE Transactions on Instrumentation and Measurement 55: 1451–1458.
  • Ajai, R. A., H. N. Balakrishnan, and N. Nagaraj, Sep. 2022 Analysis of logistic map based neurons in neuorchaos learning architectures for data classification. Meeting for the Dissemination and Research in the Study of Complex Systems and their Applications (EDIESCA 2022) .
  • Alligood, K. T., T. Sauer, and J. A. Yorke, 2000 Chaos: an introduction to dynamical systems. Springer New York.
  • Alvarez, G. and S. Li, 2006 Some basic cryptographic requirements for chaos-based cryptosystems. International journal of bifurcation and chaos 16: 2129–2151.
  • Balakrishnan, H. N., A. Kathpalia, S. Saha, and N. Nagaraj, 2019 Chaosnet: A chaos based artificial neural network architecture for classification. Chaos: An Interdisciplinary Journal of Nonlinear Science 29: 113125.
  • Banerjee, S., J. A. Yorke, and C. Grebogi, 1998 Robust chaos. Physical Review Letters 80: 3049.
  • Barrera, R. A. and G. G. Robert, 2022 Chaotic sets and hausdorff dimension for lüroth expansions. Journal of Mathematical Analysis and Applications p. 126324.
  • Boyarski, A. and P. Gora, 1998 Laws of chaos. invariant measures and dynamical systems in one dimension. APPLICATIONS OF MATHEMATICS-PRAHA- 43: 480–480.
  • Campos-Cantón, I., E. Campos-Cantón, J. Murguía, and H. Rosu, 2009 A simple electronic circuit realization of the tent map. Chaos, Solitons & Fractals 42: 12–16.
  • Campos-Cantón, I., L. M. Torres-Treviño, E. Campos-Cantón, and R. Femat, 2013 Generation of a reconfigurable logical cell using evolutionary computation. Discrete Dynamics in Nature and Society 2013.
  • Dajani, K. and C. Kraaikamp, 2002 Ergodic theory of numbers, volume 29 of carus mathematical monographs. Mathematical Association of America, Washington, DC p. 1.
  • Devaney, R. L., 2018 An introduction to chaotic dynamical systems. CRC press.
  • Ditto, W. L., A. Miliotis, K. Murali, S. Sinha, and M. L. Spano, 2010 Chaogates: Morphing logic gates that exploit dynamical patterns. Chaos: An Interdisciplinary Journal of Nonlinear Science 20: 037107.
  • Glendinning, P., 2017 Robust chaos revisited. The European Physical Journal Special Topics 226: 1721–1738.
  • Harikrishnan, N., A. Kathpalia, and N. Nagaraj, 2022a Causeeffect preservation and classification using neurochaos learning. NeurIPS 2022 p. accepted.
  • Harikrishnan, N. and N. Nagaraj, 2021 When noise meets chaos: Stochastic resonance in neurochaos learning. Neural Networks 143: 425–435.
  • Harikrishnan, N., S. Pranay, and N. Nagaraj, 2022b Classification of sars-cov-2 viral genome sequences using neurochaos learning. Medical & Biological Engineering & Computing pp. 1–11.
  • Hasler, M. and T. Schimming, 2000 Chaos communication over noisy channels. International Journal of Bifurcation and Chaos 10: 719–735.
  • Hernandez, E. D. M., G. Lee, and N. H. Farhat, 2003 Analog realization of arbitrary one-dimensional maps. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 50: 1538–1547.
  • Jaimes-Reátegui, R., S. Afanador-Delgado, R. Sevilla-Escoboza, G. Huerta-Cuellar, J. H. García-López, et al., 2014 Optoelectronic flexible logic gate based on a fiber laser. The European Physical Journal Special Topics 223: 2837–2846.
  • Korn, H. and P. Faure, 2003 Is there chaos in the brain? ii. experimental evidence and related models. Comptes rendus biologies 326: 787–840.
  • Kumar, D., K. Nabi, P. K. Misra, and M. Goswami, 2018 Modified tent map based design for true random number generator. In 2018 IEEE International Symposium on Smart Electronic Systems (iSES)(Formerly iNiS), pp. 27–30, IEEE.
  • Li, S., G. Chen, and X. Mou, 2005 On the dynamical degradation of digital piecewise linear chaotic maps. International journal of Bifurcation and Chaos 15: 3119–3151.
  • Miliotis, A., S. Sinha, and W. L. Ditto, 2008 Exploiting nonlinear dynamics to store and process information. International Journal of Bifurcation and Chaos 18: 1551–1559.
  • Murali, K., S. Sinha, and W. L. Ditto, 2005 Construction of a reconfigurable dynamic logic cell. Pramana 64: 433–441.
  • Nagaraj, N., 2008 Novel applications of chaos theory to coding and cryptography. Ph.D. thesis, NIAS.
  • Nagaraj, N., 2009 A dynamical systems proof of kraft–mcmillan inequality and its converse for prefix-free codes. Chaos: An Interdisciplinary Journal of Nonlinear Science 19: 013136.
  • Nagaraj, N., 2011 Huffman coding as a nonlinear dynamical system. International Journal of Bifurcation and Chaos 21: 1727– 1736.
  • Nagaraj, N., 2012 One-time pad as a nonlinear dynamical system. Communications in Nonlinear Science and Numerical Simulation 17: 4029–4036.
  • Nagaraj, N., 2019 Using cantor sets for error detection. PeerJ Computer Science 5: e171.
  • Nagaraj, N., M. C. Shastry, and P. G. Vaidya, 2008 Increasing average period lengths by switching of robust chaos maps in finite precision. The European Physical Journal Special Topics 165: 73–83.
  • Nagaraj, N. and P. G. Vaidya, 2009 Multiplexing of discrete chaotic signals in presence of noise. Chaos: An Interdisciplinary Journal of Nonlinear Science 19: 033102.
  • Nagaraj, N., P. G. Vaidya, and K. G. Bhat, 2009 Arithmetic coding as a non-linear dynamical system. Communications in Nonlinear Science and Numerical Simulation 14: 1013–1020.
  • Palacios-Luengas, L., J. Pichardo-Méndez, J. Díaz-Méndez, F. Rodríguez-Santos, and R. Vázquez-Medina, 2019 Prng based on skew tent map. Arabian Journal for Science and Engineering 44: 3817–3830.
  • Rissanen, J. and G. G. Langdon, 1979 Arithmetic coding. IBM Journal of research and development 23: 149–162.
  • Sethi, D., N. Nagaraj, and H. N. Balakrishnan, 2022 Neurochaos feature transformation for machine learning. in EDIESCA 2022 .
  • Sinha, S. andW. L. Ditto, 1998 Dynamics based computation. physical review Letters 81: 2156.
  • Strogatz, S. H., 2018 Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. CRC press.
  • Valtierra, J. L., E. Tlelo-Cuautle, and Á. Rodríguez-Vázquez, 2017 A switched-capacitor skew-tent map implementation for random number generation. International Journal of Circuit Theory and Applications 45: 305–315.
  • Wong, K.-W., Q. Lin, and J. Chen, 2010 Simultaneous arithmetic coding and encryption using chaotic maps. IEEE Transactions on Circuits and Systems II: Express Briefs 57: 146–150.

The Unreasonable Effectiveness of the Chaotic Tent Map in Engineering Applications

Year 2022, Volume 4, Issue 4, 197 - 204, 31.12.2022
https://doi.org/10.51537/chaos.1196653

Abstract

From decimal expansion of real numbers to complex behaviour in physical, biological and human-made systems, deterministic chaos is ubiquitous. One of the simplest examples of a nonlinear dynamical system that exhibits chaos is the well known 1-dimensional piecewise linear Tent map. The Tent map (and their skewed cousins) are instances of a larger family of maps namely Generalized Luröth Series (GLS) which are studied for their rich number theoretic and ergodic properties. In this work, we discuss the unreasonable effectiveness of the Tent map and their generalizations (GLS maps) in a number of applications in electronics, communication and computer engineering. To list a few of these applications: (a) GLS-coding: a lossless data compression algorithm for i.i.d sources is Shannon optimal and is in fact a generalization of the popular Arithmetic Coding algorithm used in the image compression standard JPEG2000; (b) GLS maps are used as neurons in the recently proposed Neurochaos Learning architecture which delivers state-of-the-art performance in classification tasks; (c) GLS maps are ideal candidates for chaos-based computing since they can simulate XOR, NAND and other gates and for dense storage of information for efficient search and retrieval; (d) Noise-resistant versions of GLS maps are useful for signal multiplexing in the presence of noise and error detection; (e) GLS maps are shown to be useful in a number of cryptographic protocols - for joint compression and encryption and also for generating pseudo-random numbers. The unique properties and rich features of the Tent Map (GLS maps) that enable these wide variety of engineering applications will be investigated. A list of open problems are indicated as well.

References

  • Addabbo, T., M. Alioto, A. Fort, S. Rocchi, and V. Vignoli, 2006 The digital tent map: Performance analysis and optimized design as a low-complexity source of pseudorandom bits. IEEE Transactions on Instrumentation and Measurement 55: 1451–1458.
  • Ajai, R. A., H. N. Balakrishnan, and N. Nagaraj, Sep. 2022 Analysis of logistic map based neurons in neuorchaos learning architectures for data classification. Meeting for the Dissemination and Research in the Study of Complex Systems and their Applications (EDIESCA 2022) .
  • Alligood, K. T., T. Sauer, and J. A. Yorke, 2000 Chaos: an introduction to dynamical systems. Springer New York.
  • Alvarez, G. and S. Li, 2006 Some basic cryptographic requirements for chaos-based cryptosystems. International journal of bifurcation and chaos 16: 2129–2151.
  • Balakrishnan, H. N., A. Kathpalia, S. Saha, and N. Nagaraj, 2019 Chaosnet: A chaos based artificial neural network architecture for classification. Chaos: An Interdisciplinary Journal of Nonlinear Science 29: 113125.
  • Banerjee, S., J. A. Yorke, and C. Grebogi, 1998 Robust chaos. Physical Review Letters 80: 3049.
  • Barrera, R. A. and G. G. Robert, 2022 Chaotic sets and hausdorff dimension for lüroth expansions. Journal of Mathematical Analysis and Applications p. 126324.
  • Boyarski, A. and P. Gora, 1998 Laws of chaos. invariant measures and dynamical systems in one dimension. APPLICATIONS OF MATHEMATICS-PRAHA- 43: 480–480.
  • Campos-Cantón, I., E. Campos-Cantón, J. Murguía, and H. Rosu, 2009 A simple electronic circuit realization of the tent map. Chaos, Solitons & Fractals 42: 12–16.
  • Campos-Cantón, I., L. M. Torres-Treviño, E. Campos-Cantón, and R. Femat, 2013 Generation of a reconfigurable logical cell using evolutionary computation. Discrete Dynamics in Nature and Society 2013.
  • Dajani, K. and C. Kraaikamp, 2002 Ergodic theory of numbers, volume 29 of carus mathematical monographs. Mathematical Association of America, Washington, DC p. 1.
  • Devaney, R. L., 2018 An introduction to chaotic dynamical systems. CRC press.
  • Ditto, W. L., A. Miliotis, K. Murali, S. Sinha, and M. L. Spano, 2010 Chaogates: Morphing logic gates that exploit dynamical patterns. Chaos: An Interdisciplinary Journal of Nonlinear Science 20: 037107.
  • Glendinning, P., 2017 Robust chaos revisited. The European Physical Journal Special Topics 226: 1721–1738.
  • Harikrishnan, N., A. Kathpalia, and N. Nagaraj, 2022a Causeeffect preservation and classification using neurochaos learning. NeurIPS 2022 p. accepted.
  • Harikrishnan, N. and N. Nagaraj, 2021 When noise meets chaos: Stochastic resonance in neurochaos learning. Neural Networks 143: 425–435.
  • Harikrishnan, N., S. Pranay, and N. Nagaraj, 2022b Classification of sars-cov-2 viral genome sequences using neurochaos learning. Medical & Biological Engineering & Computing pp. 1–11.
  • Hasler, M. and T. Schimming, 2000 Chaos communication over noisy channels. International Journal of Bifurcation and Chaos 10: 719–735.
  • Hernandez, E. D. M., G. Lee, and N. H. Farhat, 2003 Analog realization of arbitrary one-dimensional maps. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 50: 1538–1547.
  • Jaimes-Reátegui, R., S. Afanador-Delgado, R. Sevilla-Escoboza, G. Huerta-Cuellar, J. H. García-López, et al., 2014 Optoelectronic flexible logic gate based on a fiber laser. The European Physical Journal Special Topics 223: 2837–2846.
  • Korn, H. and P. Faure, 2003 Is there chaos in the brain? ii. experimental evidence and related models. Comptes rendus biologies 326: 787–840.
  • Kumar, D., K. Nabi, P. K. Misra, and M. Goswami, 2018 Modified tent map based design for true random number generator. In 2018 IEEE International Symposium on Smart Electronic Systems (iSES)(Formerly iNiS), pp. 27–30, IEEE.
  • Li, S., G. Chen, and X. Mou, 2005 On the dynamical degradation of digital piecewise linear chaotic maps. International journal of Bifurcation and Chaos 15: 3119–3151.
  • Miliotis, A., S. Sinha, and W. L. Ditto, 2008 Exploiting nonlinear dynamics to store and process information. International Journal of Bifurcation and Chaos 18: 1551–1559.
  • Murali, K., S. Sinha, and W. L. Ditto, 2005 Construction of a reconfigurable dynamic logic cell. Pramana 64: 433–441.
  • Nagaraj, N., 2008 Novel applications of chaos theory to coding and cryptography. Ph.D. thesis, NIAS.
  • Nagaraj, N., 2009 A dynamical systems proof of kraft–mcmillan inequality and its converse for prefix-free codes. Chaos: An Interdisciplinary Journal of Nonlinear Science 19: 013136.
  • Nagaraj, N., 2011 Huffman coding as a nonlinear dynamical system. International Journal of Bifurcation and Chaos 21: 1727– 1736.
  • Nagaraj, N., 2012 One-time pad as a nonlinear dynamical system. Communications in Nonlinear Science and Numerical Simulation 17: 4029–4036.
  • Nagaraj, N., 2019 Using cantor sets for error detection. PeerJ Computer Science 5: e171.
  • Nagaraj, N., M. C. Shastry, and P. G. Vaidya, 2008 Increasing average period lengths by switching of robust chaos maps in finite precision. The European Physical Journal Special Topics 165: 73–83.
  • Nagaraj, N. and P. G. Vaidya, 2009 Multiplexing of discrete chaotic signals in presence of noise. Chaos: An Interdisciplinary Journal of Nonlinear Science 19: 033102.
  • Nagaraj, N., P. G. Vaidya, and K. G. Bhat, 2009 Arithmetic coding as a non-linear dynamical system. Communications in Nonlinear Science and Numerical Simulation 14: 1013–1020.
  • Palacios-Luengas, L., J. Pichardo-Méndez, J. Díaz-Méndez, F. Rodríguez-Santos, and R. Vázquez-Medina, 2019 Prng based on skew tent map. Arabian Journal for Science and Engineering 44: 3817–3830.
  • Rissanen, J. and G. G. Langdon, 1979 Arithmetic coding. IBM Journal of research and development 23: 149–162.
  • Sethi, D., N. Nagaraj, and H. N. Balakrishnan, 2022 Neurochaos feature transformation for machine learning. in EDIESCA 2022 .
  • Sinha, S. andW. L. Ditto, 1998 Dynamics based computation. physical review Letters 81: 2156.
  • Strogatz, S. H., 2018 Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. CRC press.
  • Valtierra, J. L., E. Tlelo-Cuautle, and Á. Rodríguez-Vázquez, 2017 A switched-capacitor skew-tent map implementation for random number generation. International Journal of Circuit Theory and Applications 45: 305–315.
  • Wong, K.-W., Q. Lin, and J. Chen, 2010 Simultaneous arithmetic coding and encryption using chaotic maps. IEEE Transactions on Circuits and Systems II: Express Briefs 57: 146–150.

Details

Primary Language English
Subjects Engineering, Multidisciplinary
Journal Section Research Articles
Authors

Nithin NAGARAJ> (Primary Author)
National Institute of Advanced Studies
0000-0003-0097-4131
India

Supporting Institution This article was selected in EDIESCA 2022 for publication in Special Issue on CHTA.
Project Number This article was selected in EDIESCA 2022 for publication in Special Issue on CHTA.
Thanks This article was selected in EDIESCA 2022 for publication in Special Issue on CHTA.
Publication Date December 31, 2022
Published in Issue Year 2022, Volume 4, Issue 4

Cite

APA Nagaraj, N. (2022). The Unreasonable Effectiveness of the Chaotic Tent Map in Engineering Applications . Chaos Theory and Applications , Dissemination and Research in the Study of Complex Systems and Their Applications (EDIESCA 2022) , 197-204 . DOI: 10.51537/chaos.1196653

Chaos Theory and Applications in Applied Sciences and Engineering: An interdisciplinary journal of nonlinear science 23830