Existence and Control of Hidden Oscillations in a Memristive Autonomous Duffing Oscillator

Part of the Studies in Systems, Decision and Control book series (SSDC, volume 133)


Studying the memristor based chaotic circuit and their dynamical analysis has been an increasing interest in recent years because of its nonvolatile memory. It is very important in dynamic memory elements and neural synapses. In this chapter, the recent and emerging phenomenon such as hidden oscillation is studied by the new implemented memristor based autonomous Duffing oscillator. The stability of the proposed system is studied thoroughly using basin plots and eigenvalues. We have observed a different type of hidden attractors in a wide range of the system parameters. We have shown that hidden oscillations can exist not only in piecewise linear but also in smooth nonlinear circuits and systems. In addition, to control the hidden oscillation, the linear augmentation technique is used by stabilizing a steady state of augmented system.


Memristor system Hidden oscillations Linear augmentation Memristor stability 



Vaibhav Varshney, Awadhesh Prasad and acknowledge DST, Government of India for the financial support. S. Sabarathinam acknowledges the DST-SERB for the financial assistance through National Post Doctoral Fellow (NPDF) scheme. K. Thamilmaran acknowledge DST PURSE scheme for financial assisitance and Awadhesh Prasad and M. D. Shrimali also acknowledge joint project DST-RFBR for funding.


  1. Ahamed AI, Lakshmanan M (2013) Nonsmooth bifurcations, transient hyperchaos and hyperchaotic beats in a memristive muralilakshmananchua circuit. Int J Bifurc Chaos 23:1350098Google Scholar
  2. Bao BC, Xu JP, Liu Z (2010) Initial state dependent dynamical behaviors in a memris-tor based chaotic circuit. Chin Phys Lett 27:070504Google Scholar
  3. Bar-Eli K (1985) On the stability of coupled chemical oscillators. Physica D: Nonlinear Phenom 14:242–252Google Scholar
  4. Brezetskyi S, Dudkowski D, Kapitaniak T (2015) Rare and hidden attractors in Van der Pol-Duffing oscillators. Eur Phys J Spec Top 224:1459–1467Google Scholar
  5. Chaudhuri U, Prasad A (2014) Complicated basins and the phenomenon of amplitude death in coupled hidden attractors. Phys Lett A 378:713–718Google Scholar
  6. Cheng BB, Ping XJ, Hua ZG, Hua MZ, Ling Z (2011) Chaotic memristive cir-cuit: equivalent circuit realization and dynamical analysis. Chin Phys B 20:120502Google Scholar
  7. 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 Chuas circuit. Nonlinear Dyn 81:215–226Google Scholar
  8. Chen M, Yu J, Bao B-C (2015) Finding hidden attractors in improved memristor-based Chua’s circuit. Electron Lett 51:462–464Google Scholar
  9. Chen M, Yu J, Yu Q, Li C, Bao B (2014) A memristive diode bridge-based canonical chuas circuit. Entropy 16:6464–6476Google Scholar
  10. Chua LO (1971) Memristor-the missing circuit element. IEEE Circuit Theory 18:507–519CrossRefGoogle Scholar
  11. Dudkowski D, Jafari S, Kapitaniak T, Kuznetsov NV, Leonov GA, Prasad A (2016) Hidden attractors in dynamical systems. Phys Rep 637:1–50Google Scholar
  12. Dudkowski D, Prasad A, Kapitaniak T (2015) Perpetual points and hidden attractors in dynamical systems. Phys Lett A 379:2591–2596Google Scholar
  13. Fujisaka H, Yamada T (1983) Stability theory of synchronized motion in coupled-oscillator systems. Prog Theor Phys 69:32–47Google Scholar
  14. Ira B, Schwartz et al (1997) Tracking controlled chaos: theoretical foundations and applications. Chaos: An Interdiscip J Nonlinear Sci 7:664–679Google Scholar
  15. Itoh M, Chua LO (2008) Memristor oscillators. Int J Bifurc Chaos 18:3183–3206Google Scholar
  16. Jafari S, Sprott JC, Nazarimehr F (2015) Recent new examples of hidden attractors. Eur Phys J Spec Top 224:1469–1476Google Scholar
  17. Javaloyes J, Perrin M, Politi A (2008) Collective atomic recoil laser as a synchronization transition. Phys Rev E 78:011108Google Scholar
  18. Kapitaniak T, Leonov GA (2015) Multistability: uncovering hidden attractors. Eur Phys J Spec Top 224:1405–1408Google Scholar
  19. Kim KH, Gaba S, Wheeler D, Cruz-Albrecht JM, Hussain T, Srinivasa N, Lu W (2011) A functional hybrid memristor crossbar-array/cmos system for data storage and neuromorphic applications. Nano Lett 12:389–395Google Scholar
  20. Kim MY, Roy R, Aron JL, Carr TW, Schwartz IB (2005) Scaling behavior of laser population dynamics with time-delayed coupling: theory and experiment. Phys Rev Lett 94:088101Google Scholar
  21. Kiseleva MA, Kuznetsov NV, Leonov GA (2016) Hidden attractors in electromechanical systems with and without equilibria. IFAC-PapersOnLine 49:51–55Google Scholar
  22. Kruse D, von Cube C, Zimmermann C, Courteille PW (2003) Observation of lasing mediated by collective atomic recoil. Phys Rev Lett 91:183601Google Scholar
  23. Kumar P, Prasad A, Ghosh R (2008) Stable phase-locking of an external-cavity diode laser subjected to external optical injection. J Phys B: At Mol Opt Phys 41:135402Google Scholar
  24. Kumar P, Prasad A, Ghosh R (2009) Strange bifurcation and phase-locked dynamics in mutually coupled diode laser systems. J Phys B: At Mol Opt Phys 42:145401Google Scholar
  25. Kuznetsov AP, Kuznetsov SP, Mosekilde E, Stankevich NV (2015) Co-existing hidden attractors in a radio-physical oscillator system. J Phys A: Math Theor 48:125101Google Scholar
  26. Kuznetsov NV, Leonov GA (2014) Hidden attractors in dynamical systems: systems with no equilibria, multistability and coexisting attractors. IFAC Proceedings Volumes 47:5445–5454Google Scholar
  27. Lakshmanan M, Rajaseekar S (2012) Nonlinear dynamics: integrability, chaos and patterns. Springer Science & Business MediaGoogle Scholar
  28. Leonov GA, Kuznetsov NV, Kuznetsova OA, Seledzhi SM, Vagaitsev VI (2011) Hidden oscillations in dynamical systems. Trans Syst Contr 6:54–67Google Scholar
  29. Leonov GA, Kuznetsov NV, Mokaev TN (2015) Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion. Eur Phys J Spec Top 224:1421–1458Google Scholar
  30. Leonov GA, Kuznetsov NV (2013) Hidden attractors in dynamical systems. From hidden oscillations in HilbertKolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. Int J Bifurc Chaos 23:1330002Google Scholar
  31. Leonov GA, Kuznetsov NV, Vagaitsev VI (2011) Localization of hidden Chuas attractors. Phys Lett A 375:2230–2233Google Scholar
  32. Li C, Sprott JC (2014) Coexisting hidden attractors in a 4-D simplified Lorenz system. Int J Bifurc Chaos 24:1450034Google Scholar
  33. Li Q, Zeng H, Yang XS (2014) On hidden twin attractors and bifurcation in the Chuas circuit. Nonlinear Dyn 77:255–266Google Scholar
  34. Mohanty SP (2013) Memristor: from basics to deployment. IEEE Potentials 32:34–39Google Scholar
  35. Mouttet B (2008) Proposal for memristors in signal processing. In: International Conference on Nano-Networks. Springer, Berlin, Heidelberg, pp 11–13Google Scholar
  36. Muthuswamy B, Kokate PP (2009) Memristor-based chaotic circuits. IETE Tech Rev 26:417–429Google Scholar
  37. Ott E, Grebogi C, Yorke JA (1990) Controlling chaos. Phys Rev Lett 64:1196Google Scholar
  38. Pershin YV, Fontaine SL, Ventra MD (2009) Memristive model of amoeba learning. Phys Rev E 80:021926Google Scholar
  39. Pershin YV, Ventra MD (2008) Spin memristive systems: spin memory effects in semiconductor spintronics. Phys Rev B 78:113309Google Scholar
  40. Pershin YV, Ventra MD (2009) Frequency doubling and memory effects in the spin Hall effect. Phys Rev B 79:153307Google Scholar
  41. Pham VT, Volos CK, Vaidyanathan S, Le TP, Vu VY (2015) A Memristor-based hyperchaotic system with hidden attractors: dynamics, synchronization and circuital emulating. J Eng Sci Technol Rev 8:2Google Scholar
  42. 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 8:1157–1163Google Scholar
  43. Pikovsky AS, Rosenblum MG, Kurths J (2001) Synchronization: a universal concept in nonlinear sciences. Cambridge University Press, CambridgeGoogle Scholar
  44. Prasad A et al (2003) Complicated basins in external-cavity semiconductor lasers. Phys Lett A 314:44–50Google Scholar
  45. Prodromakis T, Peh BP, Papavassiliou C, Toumazou C (2011) A versatile memristor model with nonlinear dopant kinetics. IEEE Trans Electron Devices 58:3099–3105Google Scholar
  46. Radwan AG, Zidan MA, Salama KN (2010) Hp memristor mathematical model for periodic signals and dc. In: Circuits and Systems (MWSCAS), 2010, 53rd IEEE International Midwest Symposium, pp 861–864Google Scholar
  47. Rajendran J, Manem H, Karri R, Rose GS (2010) Memristor based programmable threshold logic array. In: Proceedings of the 2010 IEEE/ACM International Symposium on Nanoscale Architectures. IEEE Press, pp 5–10Google Scholar
  48. Resmi V, Ambika G, Amritkar RE (2010) Synchronized states in chaotic systems coupled indirectly through a dynamic environment. Phys Rev E 81:046216Google Scholar
  49. Resmi V, Ambika G, Amritkar RE, Rangarajan G (2012) Amplitude death in complex networks induced by environment. Phys Rev E 85:046211Google Scholar
  50. Rosa ER, Ott E, Hess MH (1998) Transition to phase synchronization of chaos. Phys Rev Lett 80:1642Google Scholar
  51. Sabarathinam S, Thamilmaran K (2017) Effect of variable memristor emulator in a Duffing nonlinear oscillator. AIP Conference Proceedings-AIP Publishing, 1832Google Scholar
  52. Sabarathinam S, Volos CK, Thamilmaran K (2017) Implementation and study of the nonlinear dynamics of a memristor-based Duffing oscillator. Nonlinear Dyn 87:37–49Google Scholar
  53. Saha P, Saha DC, Ray A, Chowdhury AR (2015) Memristive non-linear system and hidden attractor. Eur Phys J Spec Top 224:1563–1574Google Scholar
  54. Sharma PR, Sharma A, Shrimali MD, Prasad A (2011) Targeting fixed-point solutions in nonlinear oscillators through linear augmentation. Phys Rev E 83:067201Google Scholar
  55. Sharma PR, Shrimali MD, Prasad A, Feudel U (2013) Controlling bistability by linear augmentation. Phys Lett A 377:2329–2332Google Scholar
  56. Sharma PR, Shrimali MD, Prasad A, Kuznetsov NV, Leonov GA (2015a) Control of multistability in hidden attractors. Eur Phys J Spec Top 224:1485–1491Google Scholar
  57. Sharma PR, Shrimali M, Prasad A, Kuznetsov NV, Lenov GA (2015b) Controlling dynamics of hidden attractors. Int J Bifurc Chaos 25:1550061Google Scholar
  58. Sharma PR, Singh A, Prasad A, Shrimali MD (2014) Controlling dynamical behavior of drive-response system through linear augmentation. Eur Phys J Spec Top 223:1531–1539Google Scholar
  59. Sharma A, Shrimali MD, Dana SK (2012) Phase-flip transition in nonlinear oscillators coupled by dynamic environment. Chaos: An Interdiscip J Nonlinear Sci 22:023147Google Scholar
  60. Sinha S et al (1990) Adaptive control in nonlinear dynamics. Physica D: Nonlinear Phenom 43:118–128Google Scholar
  61. Strukov DB, Snider GS, Stewart DR, Williams RS (2008) The missing memristor found. Nature 453:80–83Google Scholar
  62. Talukdar AH (2011) Nonlinear dynamics of memristor based 2nd and 3rd order oscillators. PhD thesisGoogle Scholar
  63. Thomas A (2013) Memristor-based neural networks. J Phys D: Appl Phys 46:093001Google Scholar
  64. Toth R, Taylor AF, Tinsley MR (2006) Collective behavior of a population of chemically coupled oscillators. J Phys Chem B 110:10170–10176Google Scholar
  65. Tour JM, He T (2008) Electronics: the fourth element. Nature 453:42–43Google Scholar
  66. Triandaf I, Schwartz IB (2000) Tracking sustained chaos: a segmentation method. Phys Rev E 62:3529Google Scholar
  67. Varshney V, Sabarathinam S, Thamilmaran K, Prasad A (Submitted-2017) Hidden oscillations in a memristive autonomous Duffing oscillator-A case study. Int J Bifurc ChaosGoogle Scholar
  68. Wang D, Hu Z, Yu X, Yu J (2009) A pwl model of memristor and its application example. In: International Conference on Communications, Circuits and Systems, (2009) ICCCAS 2009, pp 932–934Google Scholar
  69. Wei Z, Zhang W (2014) Hidden hyperchaotic attractors in a modified LorenzStenflo system with only one stable equilibrium. Int J Bifurc Chaos 24:1450127Google Scholar
  70. Xu C, Dong X, Jouppi NP, Xie Y (2011) Design implications of memristor-based RRAM cross-point structures. In: Design, Automation & Test in Europe Conference & Exhibition (DATE), 2011. IEEE (2011)Google Scholar
  71. Zhang G, Hu J, Shen Y (2015) New results on synchronization control of delayed memristive neural networks. Nonlinear Dyn 81:1167–1178Google Scholar
  72. Zhang W, Zou X (2012) Synchronization ability of coupled cell-cycle oscillators in changing environments. BMC Syst Biol 6:S13Google Scholar
  73. Zhao H, Lin Y, Dai Y (2014) Hidden attractors and dynamics of a general autonomous van der PolDuffing oscillator. Int J Bifurc Chaos 24:1450080Google Scholar

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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Physics and AstrophysicsUniversity of DelhiDelhiIndia
  2. 2.Department of PhysicsCentral University of RajasthanAjmerIndia
  3. 3.School of PhysicsCentre for Nonlinear Dynamics, Bharathidasan UniversityTiruchirappalliIndia

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