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Unconventional Algorithms and Hidden Chaotic Attractors

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Chaotic Systems with Multistability and Hidden Attractors

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 40))

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Abstract

Evolutionary algorithms (EAs) and deterministic chaos, which is a complex behavior produced by complex as well as simple dynamical systems, are tightly joined to create an interdisciplinary fusion of two interesting areas. This chapter discusses the use of EAs for numerical identification of the existence of the so-called hidden attractors (a full report is in [1]), which are part of the chaotic dynamics, as well as their synthesis [2, 3].

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Notes

  1. 1.

    https://www.youtube.com/watch?v=jP-QMmzGL5I.

  2. 2.

    https://www.youtube.com/watch?v=faB5bIdksi8.

  3. 3.

    http://nmp.jpl.nasa.gov/st5/.

  4. 4.

    http://www.math.spbu.ru/user/nk/.

  5. 5.

    navy.cs.vsb.cz.

References

  1. I. Zelinka, Evolutionary identification of hidden chaotic attractors. Eng. Appl. Artif. Intell. 50, 159–167 (2016)

    Google Scholar 

  2. I. Zelinka, D. Davendra, R. Jasek, R. Senkerik, Z. Oplatkova, Analytical Programming – A Novel Approach for Evolutionary Synthesis of Symbolic Structures (INTECH Open Access Publisher, 2011)

    Google Scholar 

  3. J.R. Koza, Human-competitive results produced by genetic programming. Genet. Program. Evolvable Mach. 11(3–4), 251–284 (2010)

    Article  Google Scholar 

  4. N.V. Kuznetsov, G.A. Leonov, V.I. Vagaitsev, Analytical-numerical method for attractor localization of generalized Chua’s system. Periodic Control Syst. 4(11), 29–33 (2010)

    Google Scholar 

  5. T. Kapitaniak, G.A. Leonov, Multistability: uncovering hidden attractors. Eur. Phys. J. Spec. Top. 224(8), 1405–1408 (2015)

    Article  Google Scholar 

  6. H. Jiang, Y. Liu, Z. Wei, L. Zhang, Hidden chaotic attractors in a class of two-dimensional maps. Nonlinear Dyn. 1–9 (2016)

    Google Scholar 

  7. D. Dudkowski, S. Jafari, T. Kapitaniak, N.V. Kuznetsov, G.A. Leonov, A. Prasad, Hidden attractors in dynamical systems. Physics Reports (2016)

    Google Scholar 

  8. R.C. Hilborn, S. Coppersmith, A.J. Mallinckrodt, S. Mckay, Chaos and nonlinear dynamics: an introduction for scientists and engineers. Am. J. Phys. 62(9), 235 (1994)

    Google Scholar 

  9. H. Kantz, T. Schreiber, Nonlinear Time Series Analysis (Cambridge University Press, 2004)

    Google Scholar 

  10. E. Schöll, H.G. Schuster, Handbook of Chaos Control (Wiley, 2008)

    Google Scholar 

  11. G.A. Leonov, N.V. Kuznetsov, M.A. Kiseleva, E.P. Solovyeva, A.M. Zaretskiy, Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor. Nonlinear Dyn. 77(1–2), 277–288 (2014)

    Google Scholar 

  12. F.R. Tahir, S. Jafari, V. Pham, C. Volos, X. Wang, A novel no-equilibrium chaotic system with multiwing butterfly attractors. Int. J. Bifurc. Chaos 25(04) (2015)

    Google Scholar 

  13. P.R. Sharma, M.D. Shrimali, A. Prasad, N.V. Kuznetsov, G.A. Leonov, Controlling dynamics of hidden attractors. Int. J. Bifurc. Chaos 25(4) (2015)

    Google Scholar 

  14. G.A. Leonov, Iwcfta2012 keynote speech i – hidden attractors in dynamical systems: from hidden oscillation in hilbert-kolmogorov, aizerman and kalman problems to hidden chaotic attractor in chua circuits. Int. J. Bifurc. Chaos 23(1), xv–xvii (2012)

    Google Scholar 

  15. M. Molaie, S. Jafari, J.C. Sprott, Simple chaotic flows with one stable equilibrium. Int. J. Bifurc. Chaos 23(11), 151–167 (2013)

    Google Scholar 

  16. J.C. Sprott, S. Jafari, V.T. Pham, Z.S. Hosseini, A chaotic system with a single unstable node. Phys. Lett. A 379(36), 2030–2036 (2015)

    Article  MathSciNet  Google Scholar 

  17. E. Ott, C. Grebogi, J.A. York, Controlling chaos. Phys. Rev. Lett. 64(1), 2837 (1990)

    Google Scholar 

  18. I. Zelinka, R. Senkerik, E. Navratil, Investigation on evolutionary optimization of chaos control. Chaos, Solitons Fractals 40(1), 111–129 (2009)

    Article  Google Scholar 

  19. R. Senkerik, I. Zelinka, E. Navratil, Optimization of feedback control of chaos by evolutionary algorithms. IFAC Proc. 39(8), 77–82 (2006)

    Article  Google Scholar 

  20. I. Zelinka, R. Senkerik, E. Navratil, Investigation on realtime deterministic chaos control by means of evolutionary algorithms. IFAC Proc. 39(8), 190–196 (2006)

    Article  Google Scholar 

  21. K. Pyragas, Control of chaos via extended delay feedback. Phys. Lett. A 206(5), 323–330 (1995)

    Article  MathSciNet  Google Scholar 

  22. H. Richter, K.J. Reinschke, Optimization of local control of chaos by an evolutionary algorithm. Physica D: Nonlinear Phenomena 144(3), 309–334 (2000)

    Article  MathSciNet  Google Scholar 

  23. H. Richter, An evolutionary algorithm for controlling chaos: the use of multi-objective fitness functions, in International Conference on Parallel Problem Solving from Nature (Springer, 2002), pp. 308–317

    Google Scholar 

  24. Z. Oplatkova, R. Senkerik, I. Zelinka, J. Holoska, Synthesis of control law for chaotic henon system preliminary study, in ECMS (2010), pp. 277–282

    Google Scholar 

  25. Z. Oplatková, R. Šenkeřík, S. Bělašková, I. Zelinka, Synthesis of control rule for synthesized chaotic system by means of evolutionary techniques, MENDEL 2010 (2010)

    Google Scholar 

  26. Z. Oplatková, R. Senkerik, I. Zelinka, J. Holoska, Synthesis of control law for chaotic logistic equation-preliminary study, in 2010 Fourth Asia International Conference on Mathematical/Analytical Modelling and Computer Simulation, IEEE (2010), pp. 65–70

    Google Scholar 

  27. W. Just, Principles of time delayed feedback control, Handbook of Chaos Control (Wiley, 1999), pp. 21–41

    Google Scholar 

  28. K. Pyragas, Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170(6), 421–428 (1992)

    Article  Google Scholar 

  29. G. Hu, F. Xie, J. Xiao, J. Yang, Z. Qu, Control of patterns and spatiotemporal chaos and its applications, Handbook of Chaos Control (Wiley-VCH, 1999), pp. 43–86

    Google Scholar 

  30. I. Zelinka, G. Chen, S. Celikovsky, Chaos synthesis by means of evolutionary algorithms. Int. J. Bifurc. Chaos 18(04), 911–942 (2008)

    Google Scholar 

  31. I. Zelinka, S. Celikovskỳ, H. Richter, G. Chen, Evolutionary Algorithms and Chaotic Systems, vol. 267 (Springer, 2010)

    Google Scholar 

  32. A.H. Wright, A. Agapie, Cyclic and chaotic behavior in genetic algorithms, in Proceedings of the 3rd Annual Conference on Genetic and Evolutionary Computation (2001), pp. 718–724

    Google Scholar 

  33. K.J. Persohn, R.J. Povinelli, K.J. Persohn, R.J. Povinelli, Analyzing logistic map pseudorandom number generators for periodicity induced by finite precision floating-point representation. Chaos, Solitons Fractals 45(3), 238–245 (2012)

    Article  Google Scholar 

  34. M. Drutarovsky, P. Galajda, A robust chaos-based true random number generator embedded in reconfigurable switched-capacitor hardware, in 2007 17th International Conference on Radioelektronika, IEEE (2007), pp. 1–6

    Google Scholar 

  35. X.-Y. Wang, X. Qin, A new pseudo-random number generator based on cml and chaotic iteration. Nonlinear Dyn. 70(2), 1589–1592 (2012)

    Article  MathSciNet  Google Scholar 

  36. N.K. Pareek, V. Patidar, K.K. Sud, A random bit generator using chaotic maps. Int. J. Netw. Secur. 10(1), 32–38 (2010)

    Google Scholar 

  37. E. Araujo, L.D.S. Coelho, Particle swarm approaches using lozi map chaotic sequences to fuzzy modelling of an experimental thermal-vacuum system. Appl. Soft Comput. 8(4), 1354–1364 (2008)

    Google Scholar 

  38. B. Alatas, E. Akin, A.B. Ozer, Chaos embedded particle swarm optimization algorithms. Chaos, Solitons Fractals 40(4), 1715–1734 (2009)

    Article  MathSciNet  Google Scholar 

  39. M. Pluhacek, R. Senkerik, D. Davendra, Z.K. Oplatkova, I. Zelinka, On the behavior and performance of chaos driven pso algorithm with inertia weight. Comput. Math. Appl. 66(2), 122–134 (2013)

    Article  MathSciNet  Google Scholar 

  40. M. Pluhacek, R. Senkerik, I. Zelinka, “Impact of Various Chaotic Maps on the Performance of Chaos Enhanced PSO Algorithm with Inertia Weight-an Initial Study, in Nostradamus: Modern Methods of Prediction, Modeling and Analysis of Nonlinear Systems (Springer, 2013), pp. 153–166

    Google Scholar 

  41. N. Kuznetsov, O. Kuznetsova, G. Leonov, V. Vagaitsev, Analytical-Numerical Localization of Hidden Attractor in Electrical Chua’s Circuit (Springer, Berlin-Heidelberg, 2013)

    Google Scholar 

  42. V.O. Bragin, V.I. Vagaitsev, N.V. Kuznetsov, G.A. Leonov, Algorithms for finding hidden oscillations in nonlinear systems: the aizerman and kalman problems and chua’s circuits. Int. J. Comput. Syst. Sci. 50(4), 511–543 (2011)

    Article  MathSciNet  Google Scholar 

  43. I. Zelinka, L. Nolle, Plasma reactor optimizing using differential evolution, Differential Evolution: A Practical Approach to Global Optimization (2005), pp. 499–512

    Google Scholar 

  44. L. Nolle, I. Zelinka, A.A. Hopgood, A. Goodyear, Comparison of an self-organizing migration algorithm with simulated annealing and differential evolution for automated waveform tuning. Adv. Eng. Softw. 36(10), 645–653 (2005)

    Article  Google Scholar 

  45. E.L. Houghton, P.W. Carpenter, Aerodynamics for Engineering Students (Butterworth-Heinemann, 2003)

    Google Scholar 

  46. C.L. Karr, R. Bowersox, V. Singh, Minimization of sonic boom on supersonic aircraft using an evolutionary algorithm, in Genetic and Evolutionary Computation Conference (Springer, 2003), pp. 2157–2167

    Google Scholar 

  47. M. Dorigo, M. Birattari, T. Stutzle, Ant colony optimization. IEEE Comput. Intell. Mag. 1(4), 28–39 (2006)

    Article  Google Scholar 

  48. Q.T. Pham, Dynamic optimization of chemical engineering processes by an evolutionary method. Comput. Chem. Eng. 22(7), 1089–1097 (1998)

    Article  Google Scholar 

  49. Q. Pham, S. Coulter, Modelling the chilling of pig carcasses using an evolutionary method,” in Proceedings of the International Congress of Refrigeration, vol. 3 (1995), pp. 676–683

    Google Scholar 

  50. Y. Li, A. Häu\(\beta \)ler, Artificial evolution of neural networks and its application to feedback control. Artif. Intell. Eng. 10(2), 143–152 (1996)

    Google Scholar 

  51. O. Levenspiel, Chemical Reaction Engineering: An Introduction to the Design of Chemical Reactors (Wiley, 1962)

    Google Scholar 

  52. M. Judy, K. Ravichandran, K. Murugesan, A multi-objective evolutionary algorithm for protein structure prediction with immune operators. Comput. Methods Biomech. Biomed. Eng. 12(4), 407–413 (2009)

    Article  Google Scholar 

  53. O. Ebenhöh, R. Heinrich, Evolutionary optimization of metabolic pathways. theoretical reconstruction of the stoichiometry of atp and nadh producing systems. Bull. Math. Biol. 63(1), 21–55 (2001)

    Article  Google Scholar 

  54. G.B. Fogel, D.W. Corne, Evolutionary Computation in Bioinformatics (Morgan Kaufmann, 2002)

    Google Scholar 

  55. J.H. Holland, Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control and artificial intelligence. Control Artif. Intell. Univ. Mich. Press 6(2), 126–137 (1975)

    Google Scholar 

  56. M. Dorigo, M. Birattari, C. Blum, Ant Colony Optimization and Swarm Intelligence, vol. 5217(8) (Springer, 2004), pp. 767–771

    Google Scholar 

  57. V. Kenneth, Price: An Introduction to Differential Evolution, New Ideas in Optimization (McGraw-Hill, London, 1999)

    Google Scholar 

  58. I. Zelinka, Soma—self-organizing migrating algorithm, in New Optimization Techniques in Engineering (Springer, 2004), pp. 167–217

    Google Scholar 

  59. J.H. Holland, Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence (University of Michigan Press, 1975)

    Google Scholar 

  60. B.S. Kirkpatrick, C. Gelatt, D. Vecchi, Optimization by simulated annealing. Science 220(4598), 671–80 (1983)

    Google Scholar 

  61. V. Černý, Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm. J. Optim. Theory Appl. 45(1), 41–51 (1985)

    Article  MathSciNet  Google Scholar 

  62. R. Rajendra, D.K. Pratihar, Particle swarm optimization, in International Conference on Biomedical Engineering & Informatics (1942), pp. 129–132

    Google Scholar 

  63. H.P. Schwefel, Numerische optimierung von computer – modellen (2010)

    Google Scholar 

  64. B.I. Rechenberg, Evolutionstrategie: Optimierung Technischer Systeme nach Prinzipien dier Biolischen Evolution (Frommann-Holzboog, 2013)

    Google Scholar 

  65. N.V. Kuznetsov, G.A. Leonov, Hidden attractors in dynamical systems: systems with no equilibria, multistability and coexisting attractors. IFAC Proc. Vol. 47(3), 5445–5454 (2014)

    Article  Google Scholar 

  66. M.A. Van Wyk, W.-H. Steeb, Chaos in Electronics, vol. 2 (Springer Science & Business Media, 2013)

    Google Scholar 

  67. J.R. Koza, Genetic Programming: A Paradigm for Genetically Breeding Populations of Computer Programs to Solve Problems (Stanford University, 1990)

    Google Scholar 

  68. C. Ryan, J. Collins, M.O. Neill, Grammatical evolution: Evolving programs for an arbitrary language, in European Conference on Genetic Programming (Springer, 1998), pp. 83–96

    Google Scholar 

  69. C.G. Johnson, Artificial immune system programming for symbolic regression, in European Conference on Genetic Programming (Springer, 2003), pp. 345–353

    Google Scholar 

  70. M. O’Neill, A. Brabazon, Grammatical Differential Evolution, in Proceedings of International Conference on Artificial Intelligence (2006)

    Google Scholar 

  71. J. Lampinen, I. Zelinka, On stagnation of the differential evolution algorithm, in Proceedings of MENDEL (2000), pp. 76–83

    Google Scholar 

  72. I. Zelinka, M. Chadli, D. Davendra, R. Senkerik, M. Pluhacek, J. Lampinen, Do evolutionary algorithms indeed require random numbers? extended study, in Nostradamus 2013: Prediction, Modeling and Analysis of Complex Systems (Springer, 2013), pp. 61–75

    Google Scholar 

  73. K. Aihara, T. Takabe, M. Toyoda, Chaotic neural networks. Phys. Lett. A 144(6–7), 333–340 (1990)

    Google Scholar 

  74. I. Tsuda, Dynamic link of memory—chaotic memory map in nonequilibrium neural networks. Neural Netw. 5(2), 313–326 (1992)

    Google Scholar 

  75. M.T. Thai, P.M. Pardalos, Handbook of Optimization in Complex Networks (Springer, 2012)

    Google Scholar 

  76. I. Zelinka, D. Davendra, S. Roman, J. Roman, Do evolutionary algorithms dynamics create complex network structures? Complex Syst. 20(2), 127 (2011)

    Google Scholar 

  77. I. Zelinka, A survey on evolutionary algorithms dynamics and its complexity-mutual relations, past, present and future. Swarm Evol. Comput. 25, 2–14 (2015)

    Google Scholar 

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Acknowledgements

The following grants are acknowledged for the financial support provided to this research: Grant Agency of the Czech Republic–GACR P103/15/06700S and by Grant of SGS No. SP2016/175, VSB–Technical University of Ostrava.

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Authors and Affiliations

  1. Department of Computer Science, FEI, VSB Technical University of Ostrava, Tr. 17. Listopadu 15, Ostrava, Czech Republic

    Ivan Zelinka

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  1. Ivan Zelinka
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Correspondence to Ivan Zelinka .

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Editors and Affiliations

  1. Institute for Advanced Studies, Shenzhen University, Shenzhen, China

    Xiong Wang

  2. Faculty of Information Technology, The University of Jyväskylä, Jyväskylä, Finland

    Nikolay V. Kuznetsov

  3. Department of Electrical Engineering, City University of Hong Kong, Hong Kong, China

    Guanrong Chen

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Zelinka, I. (2021). Unconventional Algorithms and Hidden Chaotic Attractors. In: Wang, X., Kuznetsov, N.V., Chen, G. (eds) Chaotic Systems with Multistability and Hidden Attractors. Emergence, Complexity and Computation, vol 40. Springer, Cham. https://doi.org/10.1007/978-3-030-75821-9_18

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