ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Kuznetsov S. P. Self-oscillating system generating rough hyperbolic chaos. Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, vol. 27, iss. 6, pp. 39-62. DOI: 10.18500/0869-6632-2019-27-6-39-62

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Language: 
Russian
Heading: 
Bifurcation in Dynamical Systems. Deterministic Chaos. Quantum Chaos.
Article type: 
Article
UDC: 
517.9:534.1
DOI: 
10.18500/0869-6632-2019-27-6-39-62

Self-oscillating system generating rough hyperbolic chaos

Autors: 
Kuznetsov Sergey Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Abstract: 

Topic and aim. The aim of the work is design of rough chaos generator, whose attractor implements dynamics close to Anosov flow on a manifold of negative curvature, as well as constructing and analyzing mathematical model, and
conducting circuit simulation of the dynamics using the Multisim software.

Investigated models. A mathematical model is considered that is a set of ordinary differential equations of the ninth order with algebraic nonlinearity, and a circuit representing the chaos generator is designed.

Results. A numerical study of the dynamics of the mathematical model was carried out, which confirmed existence of the attractor composed of trajectories close to the geodesic flow on the surface of negative curvature (Schwarz P-surface). A circuit simulation of the electronic generator, in which the dynamics corresponds to the proposed mathematical model, is carried out. The illustrations of the system dynamics are presented in the form of
oscilloscope traces, power spectra, pictures of the trajectory flow on the attractor. For the mathematical model, the Lyapunov exponents were calculated and the hyperbolic nature of the attractor was verified by analyzing histograms of the intersection angles of stable and unstable manifolds.

Discussion. The proposed electronic generator demonstrates chaos with intrinsic structural stability due to hyperbolic nature of the attractor, which implies insensitivity of the dynamics with respect to small variations in the system parameters, manufacturing imperfections, and interferences. The hyperbolic attractor is characterized by approximate uniformity of stretching and compression for phase volume elements in continuous time, which determines
rather good spectral properties of the signal. Although the consideration has been carried out for a low-frequency device, it seems possible to develop and modify the circuit also to create generators of rough chaos in high and ultra-high frequency bands.
 

Key words: 
dynamical system
chaos
Attractor
geodesic flow
Anosov dynamics
hyperbolicity
Lyapunov exponent
Reference: 
  1. Dmitriev A.S., Panas A.I., Starkov S.O. Experiments on speech and music signals transmission using chaos // International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 1995. Vol. 5. P. 1249–1254.
  2. Kennedy M., Setti G., Rovatti R. Chaotic electronics in telecommunications. CRC press, 2000.
  3. Bollt E.M. Review of chaos communication by feedback control of symbolic dynamics // International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 2003. Vol. 13. P. 269–285.
  4. Koronovskii A.A., Moskalenko O.I., Hramov A.E. On the use of chaotic synchronization for secure communication. Physics–Uspekhi, 2009, vol. 52, no. 12, pp. 1213–1238. 
  5. Kaddoum G. Wireless chaos-based communication systems: A comprehensive survey // IEEE Access. 2016. Vol. 4. P. 2621–2648. 
  6. Isaeva O.B., Jalnine A.Yu., Kuznetsov S.P. Chaotic communication with robust hyperbolic transmitter and receiver // IEEE Xplore. Progress In Electromagnetics Research Symposium. Proceedings: St Petersburg, Russia, 22–25 May 2017. P. 3129–3136.
  7. Baptista M.S. Cryptography with chaos // Physics Letters A. 1998. Vol. 240, no. 1-2. P. 50–54.
  8. Kocarev L. Chaos-based cryptography: a brief overview // IEEE Circuits and Systems Magazine. 2001. Vol. 1, no. 3. P. 6–21.
  9. Dachselt F., Schwarz W. Chaos and cryptography // IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. 2001. Vol. 48, no. 12. P. 1498–1509.
  10. Antonik P., Gulina M., Pauwels J., Massar S. Using a reservoir computer to learn chaotic attractors, with applications to chaos synchronization and cryptography // Physical Review E. 2018. Vol. 98, no. 1. 012215.
  11. Verschaffelt G., Khoder M., Van der Sande G. Random number generator based on an integrated laser with on-chip optical feedback // Chaos: An Interdisciplinary Journal of Nonlinear Science. 2017. Vol. 27. 114310.
  12. Bakiri M., Guyeux C., Couchot J.F., Oudjida A.K. Survey on hardware implementation of random number generators on FPGA: Theory and experimental analyses // Computer Science Review. 2018. Vol. 27. P. 135–153.
  13. Yeoh W.Z., Teh J.S., Chern H.R. A parallelizable chaos-based true random number generator based on mobile device cameras for the Android platform // Multimedia Tools and Applications. 2019. Vol. 78, no. 12. P. 15929–15949.
  14. Karakaya B., Gulten A., Frasca M. ¨ A true random bit generator based on a memristive chaotic circuit: Analysis, design and FPGA implementation // Chaos, Solitons & Fractals. 2019. Vol. 119. P. 143–149.
  15. Liu Z., Zhu X., Hu W., Jiang F. Principles of chaotic signal radar // International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 2007. Vol. 17. P. 1735–1739.
  16. Pappu C.S., Flores B.C., Debroux P.S., Boehm J.E. An electronic implementation of Lorenz chaotic oscillator synchronization for bistatic radar applications // IEEE Transactions on Aerospace and Electronic Systems. 2017. Vol. 53, no. 4. P. 2001–2013.
  17. Jiang T., Long J., Wang Z., Qiao S., Cui W., Ma W., Ran L. Experimental investigation of a direct chaotic signal radar with Colpitts oscillator // Journal of Electromagnetic Waves and Applications. 2010. Vol. 24, no. 8-9. P. 1229–1239.
  18. Xiong G., Xi C., He J., Yu W. Radar target detection method based on cross-correlation singularity power spectrum // IET Radar, Sonar & Navigation. 2019. Vol. 13, no. 5. P. 730–739.
  19. Xu H., Li L., Li Y., Zhang J., Han H., Liu L., Li J. Chaos-based through-wall life-detection radar // International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 2019. Vol. 29. 1930020.
  20. Banerjee S., Yorke J.A., Grebogi C. Robust chaos // Physical Review Letters. 1998. Vol. 80, no. 14. P. 3049–3052.
  21. Potapov A., Ali M.K. Robust chaos in neural networks // Physics Letters A. 2000. Vol. 277, no. 6. P. 310–322.
  22. Elhadj Z., Sprott J.C. On the robustness of chaos in dynamical systems: Theories and applications // Frontiers of Physics in China. 2008. Vol. 3, no. 2. P. 195–204.
  23. Elhadj Z. and Sprott J.C. Robust Chaos and Its Applications. World Scientific, Singapore, 2011. 472 p.  
  24. Glendinning P.A., Simpson D.J.W. Robust Chaos and the Continuity of Attractors // arXiv preprint arXiv: 1906.11974. 2019.
  25. Gusso A., Dantas W. G., Ujevic S. Prediction of robust chaos in micro and nanoresonators under two-frequency excitation // Chaos: An Interdisciplinary Journal of Nonlinear Science. 2019. Vol. 29, no. 3. 033112.
  26. Deshpande A., Chen Q., Wang Y., Lai Y.C., Do Y. Effect of smoothing on robust chaos // Physical Review E. 2010. Vol. 82, no. 2. 026209.
  27. Shilnikov L. Mathematical problems of nonlinear dynamics: A tutorial // International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 1997. Vol. 7, no. 9. P. 1353–2001.
  28. Gonchenko S.V., Shil’nikov L.P., Turaev D.V. Quasiattractors and homoclinic tangencies // Computers & Mathematics with Applications. 1997. Vol. 34, no. 2-4. P. 195–227.
  29. Botella–Soler V., Castelo J.M., Oteo J.A., Ros J. Bifurcations in the Lozi map // Journal of Physics A: Mathematical and Theoretical. 2011. Vol. 44, no. 30. 305101.
  30. Elhadj Z. Lozi Mappings: Theory and Applications. CRC Press, 2013.
  31. Belykh V.N., Belykh I. Belykh map // Scholarpedia. 2011. Vol. 6, no. 10. P. 5545.
  32. Kuznetsov S.P. Belykh attractor in Zaslavsky map and its transformation under smoothing. Izvestiya VUZ. Applied Nonlinear Dynamics, 2018, vol. 26, no. 1, pp. 64–79 (in Russian).
  33. Smale S. Differentiable dynamical systems. Bull. Amer. Math. Soc. (NS), 1967, vol. 73, pp. 747–817.
  34. Anosov D.V., Aranson S.Kh., Grines V.Z., Plykin R.V., Sataev E.A., Safonov A.V., Solodov V.V., Starkov A.N., Stepin A.M., Shlyachkov S.V. Dynamical Systems IX: Dynamical Systems with Hyperbolic Behaviour (Encyclopaedia of Mathematical Sciences, Vol. 9). Springer, 1995. 236 p.
  35. Katok A. and Hasselblatt B. Introduction to the Modern Theory of Dynamical Systems, (Encyclopedia of Mathematics and its Applications, Vol. 54). Cambridge Univ. Press, 1996. 824 p.
  36. Hasselblatt B., Pesin Ya. Hyperbolic dynamics // Scholarpedia. 2008. Vol. 3, no. 6. C. 2208.
  37. Pugh C., Peixoto M.M. Structural stability. Scholarpedia. 2008. Vol. 3, no. 9. C. 4008.
  38. Anosov D.V. Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov, 1967, vol. 90, pp. 3–210 (in Russian).
  39. Anosov D. V., Sinai Y. G. Some smooth ergodic systems. Russian Mathematical Surveys, 1967, vol. 22, no. 5, pp. 103–167.
  40. Sinai Ya.G. The stochasticity of dynamical systems. Selected Translations. Selecta Math. Soviet., 1981, vol. 1, no. 1, pp. 100–119. 
  41. Kuznetsov S.P. Hyperbolic Chaos: A Physicist’s View. Berlin, Heidelberg: Springer-Verlag, 2012. 336 p.
  42. Kuznetsov S.P. Dynamical Chaos and Hyperbolic Attractors: From Mathematics to Physics. Moscow; Izhevsk: ICS, 2013, 488 p. (in Russian).
  43. Kuznetsov S.P. Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics. Physics-Uspekhi, 2011, vol. 54, no. 2, pp. 119–144.
  44. Kuznetsov S.P., Seleznev E.P. A strange attractor of the Smale–Williams type in the chaotic dynamics of a physical system. JETP, 2006, vol. 102, no. 2, pp. 355–364. 
  45. Kuznetsov S.P., Ponomarenko V.I. Realization of a strange attractor of the Smale–Williams type in a radiotechnical delay-feedback oscillator. Technical Physics Letters, 2008, vol. 34, no. 9, pp. 771–773.
  46. Turukina L.V., Pikovsky A. Hyperbolic chaos in a system of resonantly coupled weakly nonlinear oscillators // Physics Letters A. 2011. Vol. 375, no. 11. P. 1407–1411.
  47. Ponomarenko V.I., Seleznev Е.P., Kuznetsov S.P. Autonomous system generating hyperbolic chaos: Circuit simulation and experiment. Izvestiya VUZ, Applied Nonlinear Dynamics, 2013, vol. 21, no. 5, pp. 17–30 (in Russian). 
  48. Isaeva O.B., Kuznetsov S.P., Sataev I.R., Savin D.V., Seleznev E.P. Hyperbolic chaos and other phenomena of complex dynamics depending on parameters in a nonautonomous system of two alternately activated oscillators // International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 2015. Vol. 25, no. 12. 1530033.
  49. Mangiarotti S., Letellier C. Topological analysis for designing a suspension of the Henon map // Physics Letters A. 2015. Vol. 379, no. 47-48. P. 3069–3074.
  50. Balazs N.L., Voros A. Chaos on the pseudosphere // Physics Reports. 1986. Vol. 143, no. 3. P. 109–240.
  51. Hadamard J. Les surfaces a courbures opposees et leurs lignes geodesique // J. Math. Pures Appl. 1898. Vol. 4. P. 27–73.
  52. Herniter M.E. Schematic Capture with Multisim. Prentice Hall, 2004.
  53. Thurston W.P. and Weeks J.R., The mathematics of three-dimensional manifolds, Scientific American, 1984, vol. 251, pp. 94–106.
  54. Hunt T.J. and MacKay R.S. Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor // Nonlinearity. 2003. Vol. 16. P. 1499–1510.
  55. Kuznetsov S.P. Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories. Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 649–666.
  56. Meeks W.H., Perez J. A Survey on Classical Minimal Surface Theory. University Lecture Series. Vol. 60. American Mathematical Society, 2012. 182 p.
  57. Prasolov V.V. Intuitive topology. American Mathematical Society, 1995, 93 p. 
  58. Gantmacher F.R. Lectures in Analytical Mechanics, Moscow: Mir, 1975, 264 p.
  59. Goldstein H., Poole Ch.P. Jr., Safko J.L. Classical Mechanics. 3d ed. Boston, Mass.: AddisonWesley, 2001. 680 p.
  60. Likharev K.K. Dynamics of Josephson Junctions and Circuits. CRC Press, 1986. 614 p.
  61. Shakhgildyan V.V. and Lyahovkin A.A. Phase-Locked Loops. Moscow, Svyaz’, 1972. 446 p. (in Russian).
  62. Best R.E. Phase-Locked Loops: Design, Simulation and Applications. McGraw Hill, 2007. 490 p.
  63. Kuznetsov S.P. From geodesic flow on a surface of negative curvature to electronic generator of robust chaos // International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 2016. Vol. 26, no. 14. 1650232.
  64. Kuznetsov S.P. From Anosov’s dynamics on a surface of negative curvature to electronic generator of robust chaos. Izvestiya of Saratov University. New series. Series Physics, 2016, vol. 16, iss. 3, pp. 131–144. 
  65. Pikovski A.S., Rabinovich M.I., Trakhtengerts V.Y. Appearance of chaos at decay saturation of parametric instability. Sov. Phys. JETP, 1978, vol. 47, pp. 715–719.
  66. Horowitz P. Build a Lorenz attractor: http://frank.harvard.edu/∼paulh/misc/lorenz.htm 
  67. Weady S., Agarwal S., Wilen L., Wettlaufer J.S. Circuit bounds on stochastic transport in the Lorenz equations // Physics Letters A. 2018. Vol. 382. P. 1731–1737.
  68. Fitch A.L., Iu H.H., Lu D.D. An analog computer for electronic engineering education // IEEE Transactions on Education. 2010. Vol. 54, no. 4. P. 550–557.
  69. Gonchenko A.S., Gonchenko S.V., Kazakov A.O., Кozlov A.D. Mathematical theory of dynamical chaos and its applications: Review. Part 1. Pseudohyperbolic attractors. Izvestiya VUZ. Applied Nonlinear Dynamics, 2017, vol. 25, no. 2, pp. 4–36 (in Russian).
  70. Bonatti C., Diaz L.J., Viana M. Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probobalistic Perspective. Berlin, Heidelberg, New-York: Springer, 2005. 384 p.
  71. Sveshnikov A.A. Applied Methods of the Theory of Random Functions. Elsevier, 2014. 332 p.
  72. Jenkins G.M., Watts D.G. Spectral Aalysis and its Applications. Holden-Day, 1969. 525 p.
  73. Benettin G., Galgani L., Giorgilli A., Strelcyn J.M. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them // Meccanica. 1980. Vol. 15, no. 1. P. 9–20.
  74. Shimada I., Nagashima T. A numerical approach to ergodic problem of dissipative dynamical systems // Progress of Theoretical Physics. 1979. Vol. 61, no. 6. P. 1605–1616.
  75. Pikovsky A., Politi A. Lyapunov Exponents: A Tool to Explore Complex Dynamics. Cambridge University Press, 2016. 295 p.
  76. Lai Y.-Ch., Grebogi C., Yorke J.A., Kan I. How often are chaotic saddles nonhyperbolic? // Nonlinearity. 1993. Vol. 6. P. 779–798.
  77. Anishchenko V.S., Kopeikin A.S., Kurths J., Vadivasova T.E., Strelkova G.I. Studying hyperbolicity in chaotic systems // Physics Letters A. 2000. Vol. 270. P. 301–307.
  78. Kuptsov P.V. Fast numerical test of hyperbolic chaos // Physical Review E. 2012. Vol. 85. 015203.
  79. Kruglov V. P. Technique and results of numerical test for hyperbolic nature of attractors for reduced models of distributed systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 2014, vol. 22, iss. 6, pp. 79–93 (in Russian).
  80. Kuptsov P.V., Kuznetsov S.P. Numerical test for hyperbolicity of chaotic dynamics in time-delay systems // Physical Review E. 2016. Vol. 94б no. 1. 010201.
  81. Kuptsov P.V., Kuznetsov S.P. Lyapunov analysis of strange pseudohyperbolic attractors: Angles between tangent subspaces, local volume expansion and contraction // Regular and Chaotic Dynamics. 2018. Vol. 23, no. 7-8. P. 908–932.
Received: 
11.08.2019
Accepted: 
10.10.2019
Published: 
02.12.2019
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