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Hidden Attractors in Fundamental Problems and Engineering Models: A Short Survey

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AETA 2015: Recent Advances in Electrical Engineering and Related Sciences

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 371))

Abstract

Recently a concept of self-excited and hidden attractors was suggested: an attractor is called a self-excited attractor if its basin of attraction overlaps with neighborhood of an equilibrium, otherwise it is called a hidden attractor. For example, hidden attractors are attractors in systems with no equilibria or with only one stable equilibrium (a special case of multistability and coexistence of attractors). While coexisting self-excited attractors can be found using the standard computational procedure, there is no standard way of predicting the existence or coexistence of hidden attractors in a system.In this plenary lecture the concept of self-excited and hidden attractors is discussed, and various corresponding examples of self-excited and hidden attractors are considered. The material is mostly based on surveys [1–4].

International Conference on Advanced Engineering - Theory and Applications, 2015 (Ho Chi Minh City, Vietnam), plenary lecture.

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Notes

  1. 1.

    http://www.youtube.com/watch?v=M6sy-fxIhF0.

  2. 2.

    See discussion of rigorous definitions in [48, 49].

References

  1. Leonov GA, Kuznetsov NV (2011) Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems. IFAC Proc Vol (IFAC-PapersOnline) 18(1):2494–2505

    Google Scholar 

  2. Leonov GA, Kuznetsov NV (2013) Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractors in Chua circuits. Int J Bifurcat Chaos 23(1):1330002

    Google Scholar 

  3. Kuznetsov N, Leonov G (2014) Hidden attractors in dynamical systems: systems with no equilibria, multistability and coexisting attractors. IFAC Proc (IFAC-PapersOnline) 19:5445–5454

    Google Scholar 

  4. Leonov G, Kuznetsov N, Mokaev T (2015) Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion. Eur Phys J Special Topics 224(8):1421–1458

    Article  Google Scholar 

  5. Rayleigh JWS (1877) The theory of sound. Macmillan, London

    MATH  Google Scholar 

  6. Duffing G (1918) Erzwungene Schwingungen bei Veranderlicher Eigenfrequenz. F. Vieweg u. Sohn, Braunschweig

    MATH  Google Scholar 

  7. van der Pol B (1926) On relaxation-oscillations. Philos Mag J Sci 7(2):978–992

    Google Scholar 

  8. Tricomi F (1933) Integrazione di unequazione differenziale presentatasi in elettrotechnica. Annali della R. Shcuola Normale Superiore di Pisa 2(2):1–20

    Google Scholar 

  9. Belousov BP (1958) A periodic reaction and its mechanism. In: Collection of short papers on radiation medicine for 1958. Medical Publications, Moscow (in Russian)

    Google Scholar 

  10. Lorenz EN (1963) Deterministic nonperiodic flow. J Atmos Sci 20(2):130–141

    Article  Google Scholar 

  11. Leonov GA, Kuznetsov NV, Vagaitsev VI (2011) Localization of hidden Chua’s attractors. Phys Lett A 375(23):2230–2233

    Article  MathSciNet  MATH  Google Scholar 

  12. Leonov GA, Kuznetsov NV, Vagaitsev VI (2012) Hidden attractor in smooth Chua systems. Physica D 241(18):1482–1486

    Article  MathSciNet  MATH  Google Scholar 

  13. Leonov G, Kuznetsov N, Mokaev T (2015) Homoclinic orbit and hidden attractor in the Lorenz-like system describing the fluid convection motion in the rotating cavity. Commun Nonlinear Sci Numer Simul 28:166—174. doi:10.1016/j.cnsns.2015.04.007

    Google Scholar 

  14. Kuznetsov N, Leonov GA, Mokaev TN (2015) Hidden attractor in the Rabinovich system. arXiv:1504.04723v1

    Google Scholar 

  15. Rabinovich MI (1978) Stochastic autooscillations and turbulence. Uspehi Physicheskih 125(1):123–168

    MathSciNet  Google Scholar 

  16. Pikovski AS, Rabinovich MI, Trakhtengerts VY (1978) Onset of stochasticity in decay confinement of parametric instability. Sov Phys JETP 47:715–719

    Google Scholar 

  17. Glukhovskii AB, Dolzhanskii FV (1980) Three-component geostrophic model of convection in a rotating fluid. Academy of Sciences, USSR, Izvestiya, Atmospheric and Oceanic Physics 16:311–318

    MathSciNet  Google Scholar 

  18. Leonov GA, Boichenko VA (1992) Lyapunov’s direct method in the estimation of the Hausdorff dimension of attractors. Acta Applicandae Mathematicae 26(1):1–60

    Article  MathSciNet  MATH  Google Scholar 

  19. Andronov AA, Vitt EA, Khaikin SE (1937) Theory of Oscillators (in Russian). ONTI NKTP SSSR (trans: Pergamon Press, 1966)

    Google Scholar 

  20. Pisarchik A, Feudel U (2014) Control of multistability. Phys Rep 540(4):167–218

    Article  MathSciNet  Google Scholar 

  21. Ueda Y, Akamatsu N, Hayashi C (1973) Computer simulations and non-periodic oscillations. Trans IEICE Jpn 56A(4):218–255

    Google Scholar 

  22. Hilbert D (1901–1902) Mathematical problems. Bull Amer Math Soc (8):437–479

    Google Scholar 

  23. Leonov GA, Kuznetsova OA (2010) Lyapunov quantities and limit cycles of two-dimensional dynamical systems. Analytical methods and symbolic computation. Regular and Chaotic Dynamics 15(2–3):354–377

    Article  MathSciNet  MATH  Google Scholar 

  24. Leonov GA, Kuznetsov NV, Kuznetsova OA, Seledzhi SM, Vagaitsev VI (2011) Hidden oscillations in dynamical systems. Trans Syst Control 6(2):54–67

    Google Scholar 

  25. Kuznetsov NV, Kuznetsova OA, Leonov GA (2013) Visualization of four normal size limit cycles in two-dimensional polynomial quadratic system. Diff Equ Dyn Syst 21(1–2):29–34

    Article  MathSciNet  MATH  Google Scholar 

  26. Kalman RE (1957) Physical and mathematical mechanisms of instability in nonlinear automatic control systems. Trans ASME 79(3):553–566

    MathSciNet  Google Scholar 

  27. Aizerman MA (1949) On a problem concerning the stability in the large of dynamical systems. Uspekhi Mat. Nauk (in Russian) 4:187–188

    MathSciNet  Google Scholar 

  28. Pliss VA (1958) Some Problems in the theory of the stability of motion. Izd LGU, Leningrad (in Russian)

    MATH  Google Scholar 

  29. Fitts RE (1966) Two counterexamples to Aizerman’s conjecture. Trans IEEE AC-11(3):553–556

    Google Scholar 

  30. Barabanov NE (1988) On the Kalman problem. Sib Math J 29(3):333–341

    Article  MathSciNet  MATH  Google Scholar 

  31. Bernat J, Llibre J (1996) Counterexample to Kalman and Markus-Yamabe conjectures in dimension larger than 3. Dyn Continuous Discrete Impulsive Syst 2(3):337–379

    MathSciNet  MATH  Google Scholar 

  32. Leonov GA, Bragin VO, Kuznetsov NV (2010) Algorithm for constructing counterexamples to the Kalman problem. Dokl Math 82(1):540–542

    Article  MATH  Google Scholar 

  33. Bragin VO, Vagaitsev VI, Kuznetsov NV, Leonov GA (2011) Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua’s circuits. J Comput Syst Sci Int 50(4):511–543

    Article  MathSciNet  MATH  Google Scholar 

  34. Leonov GA, Kuznetsov NV (2011) Algorithms for searching for hidden oscillations in the Aizerman and Kalman problems. Dokl Math 84(1):475–481

    Article  MathSciNet  MATH  Google Scholar 

  35. Kuznetsov NV, Leonov GA, Seledzhi SM (2011) Hidden oscillations in nonlinear control systems. IFAC Proc Vol (IFAC-PapersOnline) 18(1):2506–2510

    Google Scholar 

  36. Alli-Oke R, Carrasco J, Heath W, Lanzon A (2012) A robust Kalman conjecture for first-order plants. In: Proceedings of IEEE Control and Decision Conference (2012)

    Google Scholar 

  37. Heath WP, Carrasco J, de la Sen M (2015) Second-order counterexamples to the discrete-time Kalman conjecture. Automatica 60:140–144

    Article  MathSciNet  MATH  Google Scholar 

  38. Andrievsky BR, Kuznetsov NV, Leonov GA, Pogromsky AY (2013) Hidden oscillations in aircraft flight control system with input saturation. IFAC Proc Vol (IFAC-PapersOnline) 5(1):75–79

    Google Scholar 

  39. Andrievsky BR, Kuznetsov NV, Leonov GA, Seledzhi SM (2013) Hidden oscillations in stabilization system of flexible launcher with saturating actuators. IFAC Proc Vol (IFAC-PapersOnline) 19(1):37–41

    Google Scholar 

  40. Lauvdal T, Murray R, Fossen T (1997) Stabilization of integrator chains in the presence of magnitude and rate saturations: a gain scheduling approach. In: Proceedings of IEEE Control and Decision Conference, vol 4, pp 4404–4005

    Google Scholar 

  41. Kuznetsov N, Leonov G, Yuldashev M, Yuldashev R (2014) Nonlinear analysis of classical phase-locked loops in signal’s phase space. In: IFAC Proc Vol (IFAC-PapersOnline) 19:8253–8258

    Google Scholar 

  42. Kuznetsov N, Kuznetsova O, Leonov G, Neittaanmaki P, Yuldashev M, Yuldashev R (2014) Simulation of nonlinear models of QPSK Costas loop in Matlab Simulink. In: 2014 6th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), 2015 January, IEEE, pp 66–71

    Google Scholar 

  43. Kuznetsov N, Kuznetsova O, Leonov G, Seledzhi S, Yuldashev M, Yuldashev R (2014) BPSK Costas loop: simulation of nonlinear models in Matlab Simulink. In: 2014 6th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT).2015 January, IEEE, pp 83–87

    Google Scholar 

  44. Kudryasoha EV, Kuznetsova OA, Kuznetsov NV, Leonov GA, Seledzhi SM, Yuldashev MV, Yuldashev RV (2014) Nonlinear models of BPSK Costas loop. In: ICINCO 2014 - Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics, vol 1, pp 704–710

    Google Scholar 

  45. Kuznetsov N, Kuznetsova O, Leonov G, Neittaanmaki P, Yuldashev M, Yuldashev R (2015) Limitations of the classical phase-locked loop analysis. In: International Symposium on Circuits and Systems (ISCAS), IEEE (2015), pp 533–536 http://arxiv.org/pdf/1507.03468v1.pdf

  46. Best R, Kuznetsov N, Kuznetsova O, Leonov G, Yuldashev M, Yuldashev R (2015) A short survey on nonlinear models of the classic Costas loop: rigorous derivation and limitations of the classic analysis. In: American Control Conference (ACC), IEEE, pp 1296–1302 http://arxiv.org/pdf/1505.04288v1.pdf

  47. Bianchi G, Kuznetsov N, Leonov G, Yuldashev M, Yuldashev R (2015) Limitations of PLL simulation: hidden oscillations in SPICE analysis. In: arXiv:1506.02484 (2015) http://arxiv.org/pdf/1506.02484.pdf, http://www.mathworks.com/matlabcentral/fileexchange/52419-hidden-oscillations-in-pll (accepted to IEEE 7th International Congress on Ultra Modern Telecommunications and Control Systems)

  48. Leonov G, Kuznetsov N, Yuldashev M, Yuldashev R (2015) Hold-in, pull-in, and lock-in ranges of PLL circuits: rigorous mathematical definitions and limitations of classical theory. In: IEEE Transactions on Circuits and Systems—I: Regular Papers (http://arxiv.org/pdf/1505.04262v2.pdf)

    Google Scholar 

  49. Kuznetsov N, Leonov G, Yuldashev M, Yuldashev R (2015) Rigorous mathematical definitions of the hold-in and pull-in ranges for phase-locked loops. In: 1st IFAC Conference on Modelling, Identification and Control of Nonlinear Systems, IFAC Proceedings Volumes (IFAC-PapersOnline), pp 720–723

    Google Scholar 

  50. Shakhtarin B (1969) Study of a piecewise-linear system of phase-locked frequency control. Radiotechnica and electronika (in Russian) 8:1415–1424

    Google Scholar 

  51. Belyustina L, Brykov V, Kiveleva K, Shalfeev V (1970) On the magnitude of the locking band of a phase-shift automatic frequency control system with a proportionally integrating filter. Radiophys Quantum Electron 13(4):437–440

    Article  Google Scholar 

  52. Sommerfeld A (1902) Beitrage zum dynamischen ausbau der festigkeitslehre. Z Ver Dtsch Ing 46:391–394

    MATH  Google Scholar 

  53. Blekhman I, Indeitsev D, Fradkov A (2007) Slow motions in systems with inertially excited vibrations. IFAC Proc Vol (IFAC-PapersOnline) 3(1):126–131

    Google Scholar 

  54. Eckert M (2013) Arnold sommerfeld: science, life and turbulent times 1868–1951. Springer

    Google Scholar 

  55. Nose S (1984) A molecular dynamics method for simulations in the canonical ensemble. Mol Phys 52(2):255–268

    Article  MathSciNet  Google Scholar 

  56. Hoover W (1985) Canonical dynamics: equilibrium phase-space distributions. Phys Rev A 31:1695–1697

    Article  Google Scholar 

  57. Sprott J (1994) Some simple chaotic flows. Phys Rev E 50(2):R647–R650

    Article  MathSciNet  Google Scholar 

  58. Venkatasubramanian V (2001) Stable operation of a simple power system with no equilibrium points. In: Proceedings of the 40th IEEE Conference on Decision and Control. vol 3, pp 2201–2203

    Google Scholar 

  59. Wei Z, Wang R, Liu A (2014) A new finding of the existence of hidden hyperchaotic attractors with no equilibria. Math Comput Simul 100:13–23

    Article  MathSciNet  Google Scholar 

  60. Pham VT, Jafari S, Volos C, Wang X, Golpayegani S (2014) Is that really hidden? The presence of complex fixed-points in chaotic flows with no equilibria. Int J Bifurcat Chaos 24(11):1450146

    Google Scholar 

  61. Pham VT, Rahma F, Frasca M, Fortuna L (2014) Dynamics and synchronization of a novel hyperchaotic system without equilibrium. Int J Bifurcat Chaos 24(06):1450087

    Google Scholar 

  62. Pham VT, Volos C, Gambuzza L (2014) A memristive hyperchaotic system without equilibrium. Sci World J 04:368986

    Google Scholar 

  63. Pham VT, Volos C, Jafari S, Wei Z, Wang X (2014) Constructing a novel no-equilibrium chaotic system. Int J Bifurcat Chaos 24(05):1450073

    Article  MathSciNet  MATH  Google Scholar 

  64. Tahir F, Jafari S, Pham VT, Volos C, Wang X (2015) A novel no-equilibrium chaotic system with multiwing butterfly attractors. Int J Bifurcat Chaos 25(04):1550056

    Article  MathSciNet  Google Scholar 

  65. Vaidyanathan S, Volos CK, Pham V (2015) Analysis, control, synchronization and SPICE implementation of a novel 4-D hyperchaotic Rikitake dynamo system without equilibrium. J Eng Sci Technol Rev 8(2):232–244

    Google Scholar 

  66. Cafagna D, Grassi G (2015) Fractional-order systems without equilibria: the first example of hyperchaos and its application to synchronization. Chin Phys B 24(8):080502

    Article  Google Scholar 

  67. Chen G (2015) Chaotic systems with any number of equilibria and their hidden attractors (plenary lecture). In: 4th IFAC Conference on Analysis and Control of Chaotic Systems. http://www.ee.cityu.edu.hk/~gchen/pdf/CHEN_IFAC2015.pdf

  68. Leonov GA, Kuznetsov NV (2009) Localization of hidden oscillations in dynamical systems (plenary lecture). In: 4th International Scientific Conference on Physics and Control. http://www.math.spbu.ru/user/leonov/publications/2009-PhysCon-Leonov-plenary-hidden-oscillations.pdf

  69. Kuznetsov NV, Leonov GA, Vagaitsev VI (2010) Analytical-numerical method for attractor localization of generalized Chua’s system. IFAC Proc Vol (IFAC-PapersOnline) 4(1):29–33

    Google Scholar 

  70. Kuznetsov N, Vagaitsev V, Leonov G, Seledzhi S (2011) Localization of hidden attractors in smooth Chua’s systems. In: International Conference on Applied and Computational Mathematics, pp 26–33

    Google Scholar 

  71. Kuznetsov NV, Kuznetsova OA, Leonov GA, Vagaytsev VI (2011) Hidden attractor in Chua’s circuits. In: ICINCO 2011 - Proceedings of the 8th International Conference on Informatics in Control, Automation and Robotics, vol 1, pp 279–283

    Google Scholar 

  72. Leonov GA, Kuznetsov NV (2012) 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. In: IEEE 2012 Fifth International Workshop on Chaos-Fractals Theories and Applications (IWCFTA), pp 15–17

    Google Scholar 

  73. Kuznetsov N, Kuznetsova O, Leonov G, Vagaitsev V (2013) Analytical-numerical localization of hidden attractor in electrical Chua’s circuit. Inform Control Autom Robot Lect Notes Electr Eng 174(4):149–158

    Google Scholar 

  74. Leonov G, Kiseleva M, Kuznetsov N, Kuznetsova O (2015) Discontinuous differential equations: comparison of solution definitions and localization of hidden Chua attractors. In: 1st IFAC Conference on Modelling, Identification and Control of Nonlinear Systems, IFAC Proceedings Volumes (IFAC-PapersOnline), pp 418–423

    Google Scholar 

  75. Kiseleva M, Kuznetsov N, Leonov G, Neittaanmaki P (2013) Hidden oscillations in drilling system actuated by induction motor. IFAC Proc Vol (IFAC-PapersOnline) 5:86–89

    Google Scholar 

  76. Kiseleva M, Kondratyeva N, Kuznetsov N, Leonov G, Solovyeva E (2014) Hidden periodic oscillations in drilling system driven by induction motor. IFAC Proc Vol (IFAC-PapersOnline) 19:5872–5877

    Google Scholar 

  77. Leonov GA, Kuznetsov NV, Kiseleva MA, Solovyeva EP, Zaretskiy AM (2014) Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor. Nonlinear Dyn 77(1–2):277–288

    Article  Google Scholar 

  78. Leonov GA, Kuznetsov NV (2014) Hidden oscillations in dynamical systems. 16 Hilbert’s problem, Aizerman’s and Kalman’s conjectures, hidden attractors in Chua’s circuits. J Math Sci 201(5):645–662

    Article  MathSciNet  MATH  Google Scholar 

  79. Leonov GA, Kuznetsov NV (2013) Prediction of hidden oscillations existence in nonlinear dynamical systems: analytics and simulation. Adv Intell Syst Comput AISC 210:5–13

    Google Scholar 

  80. Leonov GA, Kiseleva MA, Kuznetsov NV, Neittaanmäki P (2013) Hidden oscillations in drilling systems: torsional vibrations. J Appl Nonlinear Dyn 2(1):83–94

    Article  Google Scholar 

  81. Kiseleva M, Kondratyeva N, Kuznetsov N, Leonov G (2015) Hidden oscillations in drilling systems with salient pole synchronous motor. In: 1st IFAC Conference on Modelling, Identification and Control of Nonlinear Systems, IFAC Proceedings Volumes (IFAC-PapersOnline), pp 418–423

    Google Scholar 

  82. Sprott J, Wang X, Chen G (2013) Coexistence of point, periodic and strange attractors. Int J Bifurcat Chaos 23(5):1350093

    Google Scholar 

  83. Wang X, Chen G (2013) Constructing a chaotic system with any number of equilibria. Nonlinear Dyn 71:429–436

    Article  MathSciNet  Google Scholar 

  84. Zhusubaliyev Z, Mosekilde E (2015) Multistability and hidden attractors in a multilevel DC/DC converter. Math Comput Simul 109:32–45

    Article  MathSciNet  Google Scholar 

  85. Wang Z, Sun W, Wei Z, Zhang S (2015) Dynamics and delayed feedback control for a 3d jerk system with hidden attractor. Nonlinear Dyn

    Google Scholar 

  86. Sharma P, Shrimali M, Prasad A, Kuznetsov NV, Leonov GA (2015) Controlling dynamics of hidden attractors. Int J Bifurcat Chaos 25(04):1550061

    Google Scholar 

  87. Dang XY, Li CB, Bao BC, Wu HG (2015) Complex transient dynamics of hidden attractors in a simple 4d system. Chin Phys B 24(5):050503

    Google Scholar 

  88. Kuznetsov A, Kuznetsov S, Mosekilde E, Stankevich N (2015) Co-existing hidden attractors in a radio-physical oscillator system. J Phys A Math Theor 48:125101

    Article  MathSciNet  MATH  Google Scholar 

  89. Pham VT, Volos C, Jafari S, Wang X, Vaidyanathan S (2014) Hidden hyperchaotic attractor in a novel simple memristive neural network. Optoelectron Adv Mater Rapid Commun 8(11–12):1157–1163

    Google Scholar 

  90. Li C, Sprott JC (2014) Coexisting hidden attractors in a 4-D simplified Lorenz system. Int J Bifurcat Chaos 24(03):1450034

    Google Scholar 

  91. Wei Z, Moroz I, Liu A (2014) Degenerate Hopf bifurcations, hidden attractors and control in the extended Sprott E system with only one stable equilibrium. Turk J Math 38(4):672–687

    Article  MathSciNet  MATH  Google Scholar 

  92. Pham VT, Volos C, Vaidyanathan S, Le T, Vu V (2015) A memristor-based hyperchaotic system with hidden attractors: Dynamics, synchronization and circuital emulating. J Eng Sci Technol Rev 2:205–214

    Google Scholar 

  93. Chen M, Yu J, Bao BC (2015) Finding hidden attractors in improved memristor-based Chua’s circuit. Electron Lett 51:462–464

    Article  Google Scholar 

  94. Chen M, Li M, Yu Q, Bao B, Xu Q, Wang J (2015) Dynamics of self-excited attractors and hidden attractors in generalized memristor-based Chua’s circuit. Nonlinear Dyn. doi:10.1007/s11071-015-1983-7

    MathSciNet  Google Scholar 

  95. Wei Z, Zhang W, Wang Z, Yao M (2015) Hidden attractors and dynamical behaviors in an extended Rikitake system. Int J Bifurcat Chaos 25(02):1550028

    Google Scholar 

  96. Burkin I, Khien N (2014) Analytical-numerical methods of finding hidden oscillations in multidimensional dynamical systems. Differ Equ 50(13):1695–1717

    Article  MathSciNet  MATH  Google Scholar 

  97. Wei Z, Zhang W (2014) Hidden hyperchaotic attractors in a modified Lorenz-Stenflo system with only one stable equilibrium. Int J Bifurcat Chaos 24(10):1450127

    Google Scholar 

  98. Li Q, Zeng H, Yang XS (2014) On hidden twin attractors and bifurcation in the Chua’s circuit. Nonlinear Dyn 77(1–2):255–266

    Article  MathSciNet  Google Scholar 

  99. Zhao H, Lin Y, Dai Y (2014) Hidden attractors and dynamics of a general autonomous van der Pol-Duffing oscillator. Int J Bifurcat Chaos 24(06):1450080

    Google Scholar 

  100. Lao SK, Shekofteh Y, Jafari S, Sprott J (2014) Cost function based on Gaussian mixture model for parameter estimation of a chaotic circuit with a hidden attractor. Int J Bifurcat Chaos 24(1):1450010

    Google Scholar 

  101. Chaudhuri U, Prasad A (2014) Complicated basins and the phenomenon of amplitude death in coupled hidden attractors. Phys Lett A 378(9):713–718

    Article  MathSciNet  MATH  Google Scholar 

  102. Pham VT, Volos C, Jafari S, Wang X (2014) Generating a novel hyperchaotic system out of equilibrium. Optoelectron Adv Mater Rapid Commun 8(5–6):535–539

    Google Scholar 

  103. Kingni S, Jafari S, Simo H, Woafo P (2014) Three-dimensional chaotic autonomous system with only one stable equilibrium: analysis, circuit design, parameter estimation, control, synchronization and its fractional-order form. Eur Phys J Plus 129(5)

    Google Scholar 

  104. Li C, Sprott J (2014) Chaotic flows with a single nonquadratic term. Phys Lett A 378(3):178–183

    Article  MathSciNet  Google Scholar 

  105. Molaie M, Jafari S, Sprott J, Golpayegani S (2013) Simple chaotic flows with one stable equilibrium. Int J Bifurcat Chaos 23(11):1350188

    Google Scholar 

  106. Jafari S, Sprott JC, Pham VT, Golpayegani SMRH, Jafari AH (2014) A new cost function for parameter estimation of chaotic systems using return maps as fingerprints. Int J Bifurcat Chaos 24(10):1450134

    Google Scholar 

  107. Jafari S, Sprott J (2013) Simple chaotic flows with a line equilibrium. Chaos Solitons Fractals 57:79–84

    Article  MathSciNet  Google Scholar 

  108. Dudkowski D, Prasad A, Kapitaniak T (2015) Perpetual points and hidden attractors in dynamical systems. Phys Lett A 379(40–41):2591–2596

    Article  MathSciNet  Google Scholar 

  109. Wei Z, Yu P, Zhang W, Yao M (2015) Study of hidden attractors, multiple limit cycles from hopf bifurcation and boundedness of motion in the generalized hyperchaotic rabinovich system. Nonlinear Dyn

    Google Scholar 

  110. Zhusubaliyev ZT, Mosekilde E, Rubanov VG, Nabokov RA (2015) Multistability and hidden attractors in a relay system with hysteresis. Physica D: Nonlinear Phenomena

    Google Scholar 

  111. Bao B, Hu F, Chen M, Xu Q, Yu Y (2015) Self-excited and hidden attractors found simultaneously in a modified Chua’s circuit. Int J Bifurcat Chaos 25(05):1550075

    Article  MATH  Google Scholar 

  112. Danca MF, Feckan M, Kuznetsov N, Chen G (2016) Looking more closely to a Rabinovich-Fabrikant system. Int J Bifurcat Chaos (accepted)

    Google Scholar 

  113. Shahzad M, Pham VT, Ahmad M, Jafari S, Hadaeghi F (2015) Synchronization and circuit design of a chaotic system with coexisting hidden attractors. Eur Phys J Spec Topics 224(8):1637–1652

    Article  Google Scholar 

  114. Brezetskyi S, Dudkowski D, Kapitaniak T (2015) Rare and hidden attractors in Van der Pol-Duffing oscillators. Eur Phys J Spec Topics 224(8):1459–1467

    Article  Google Scholar 

  115. Jafari S, Sprott J, Nazarimehr F (2015) Recent new examples of hidden attractors. Eur Phys J Spec Topics 224(8):1469–1476

    Article  Google Scholar 

  116. Zhusubaliyev Z, Mosekilde E, Churilov A, Medvedev A (2015) Multistability and hidden attractors in an impulsive Goodwin oscillator with time delay. Eur Phys J Spec Topics 224(8):1519–1539

    Article  Google Scholar 

  117. Saha P, Saha D, Ray A, Chowdhury A (2015) Memristive non-linear system and hidden attractor. Eur Phys J Spec Topics 224(8):1563–1574

    Article  Google Scholar 

  118. Semenov V, Korneev I, Arinushkin P, Strelkova G, Vadivasova T, Anishchenko V (2015) Numerical and experimental studies of attractors in memristor-based Chua’s oscillator with a line of equilibria. Noise-induced effects. Eur Phys J Spec Topics 224(8):1553–1561

    Google Scholar 

  119. Feng Y, Wei Z (2015) Delayed feedback control and bifurcation analysis of the generalized Sprott B system with hidden attractors. Eur Phys J Spec Topics 224(8):1619–1636

    Article  Google Scholar 

  120. Li C, Hu W, Sprott J, Wang X (2015) Multistability in symmetric chaotic systems. Eur Phys J Spec Topics 224(8):1493–1506

    Article  Google Scholar 

  121. Feng Y, Pu J, Wei Z (2015) Switched generalized function projective synchronization of two hyperchaotic systems with hidden attractors. Eur Phys J Spec Topics 224(8):1593–1604

    Article  Google Scholar 

  122. Sprott J (2015) Strange attractors with various equilibrium types. Eur Phys J Spec Topics 224(8):1409–1419

    Article  Google Scholar 

  123. Pham V, Vaidyanathan S, Volos C, Jafari S (2015) Hidden attractors in a chaotic system with an exponential nonlinear term. Eur Phys J Spec Topics 224(8):1507–1517

    Article  Google Scholar 

  124. Vaidyanathan S, Pham VT, Volos C (2015) A 5-D hyperchaotic Rikitake dynamo system with hidden attractors. Eur Phys J Spec Topics 224(8):1575–1592

    Article  Google Scholar 

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Acknowledgments

This work was supported by Russian Scientific Foundation (project 14-21-00041, sec. 2) and Saint-Petersburg State University (6.38.505.2014, sec. 1).

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Authors and Affiliations

  1. Faculty of Mathematics and Mechanics, Saint-Petersburg University, St. Petersburg, 198504, Russia

    Nikolay V. Kuznetsov

  2. Department of Mathematical Information Technology, University of Jyväskylä, 40014, Jyväskylä, Finland

    Nikolay V. Kuznetsov

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  1. Nikolay V. Kuznetsov
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Correspondence to Nikolay V. Kuznetsov .

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Editors and Affiliations

  1. Faculty of Electrical and Electronics En, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Vo Hoang Duy

  2. MERLIN, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Tran Trong Dao

  3. Electrical Engineering and Computer Scie, VŠB—Technical University, Ostrava-Poruba, Czech Republic

    Ivan Zelinka

  4. Division of Mechanical, Korea Maritime and Ocean University, Busan, Korea (Republic of)

    Hyeung-Sik Choi

  5. Université de Picardie Jules Verne, Laboratoire MIS, Amiens Cedex 1, France

    Mohammed Chadli

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Kuznetsov, N.V. (2016). Hidden Attractors in Fundamental Problems and Engineering Models: A Short Survey. In: Duy, V., Dao, T., Zelinka, I., Choi, HS., Chadli, M. (eds) AETA 2015: Recent Advances in Electrical Engineering and Related Sciences. Lecture Notes in Electrical Engineering, vol 371. Springer, Cham. https://doi.org/10.1007/978-3-319-27247-4_2

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  • DOI: https://doi.org/10.1007/978-3-319-27247-4_2

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