Skip to main content
SpringerLink
Log in

Coexistence of single- and multi-scroll chaotic orbits in a single-link flexible joint robot manipulator with stable spiral and index-4 spiral repellor types of equilibria

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper reports various chaotic phenomena that occur in a single-link flexible joint (SLFJ) robot manipulator. Four different cases along with subcases are considered here to show different types of chaotic behaviour in a flexible manipulator dynamics. In the first three cases, a partial state feedback as joint velocity and motor rotor velocity feedback is considered, and the resultant autonomous dynamics is considered for analyses. In the fourth case, the manipulator dynamics is considered as a non-autonomous system. The system has (1) one stable spiral and one saddle-node foci, (2) two saddle-node foci and (3) one marginally stable nature of equilibrium points. We found single- and multi-scroll chaotic orbits in these cases. However, with the motor rotor velocity feedback, the system has two unstable equilibria. One of them has an index-4 spiral repellor. In the non-autonomous case, the SLFJ robot manipulator system has an inverse crisis route to chaos and exhibits (1) transient chaos with a stable limit cycle and (2) chaotic behaviour. In all the four cases, the SLFJ manipulator dynamics exhibits coexistence of chaotic orbits, i.e. multi-stability. The various dynamical behaviours of the system are analysed using available methods like phase portrait, Lyapunov spectrum, instantaneous phase plot, Poincaré map, parameter space, bifurcation diagram, 0–1 test and frequency spectrum plot. The MATLAB simulation results support various claims made about the system. These claims are further confirmed and validated by circuit implementation using NI Multisim.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32

Similar content being viewed by others

References

  1. Sprott, J.C.: Elegant Chaos, Algebraically Simple Chaotic Flows. World Scientific Publishing Co Pte Ltd, Singapore (2010)

    Book  MATH  Google Scholar 

  2. Sprott, J.C.: Chaos and Time-Series Analysis. Oxford University Press, Oxford (2003)

    MATH  Google Scholar 

  3. Kuznetsov, N.V., Mokaev, T.N., Vasilyev, P.A.: Numerical justification of Leonov conjecture on Lyapunov dimension of Rossler attractor. Commun. Nonlinear Sci. Numer. Simul. 19(4), 1027–1034 (2014)

    Article  MathSciNet  Google Scholar 

  4. Wei, Z., Pham, V.T., Kapitaniak, T., Wang, Z.: Bifurcation analysis and circuit realization for multiple-delayed Wang–Chen system with hidden chaotic attractors. Nonlinear Dyn. 85(3), 1–16 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  5. Wei, Z., Yu, P., Zhang, W., Yao, M.: Study of hidden attractors, multiple limit cycles from Hopf bifurcation and boundedness of motion in the generalized hyperchaotic Rabinovich system. Nonlinear Dyn. 82(1–2), 31–141 (2015)

    MATH  MathSciNet  Google Scholar 

  6. Lawande, Q., Ivan, B., Dhodapkar, S.: Chaos based cryptography: a new approach to secure communications. BARC Newslett. 258, 258 (2005)

    Google Scholar 

  7. Banerjee, S., Kurths, J.: Chaos and cryptography: a new dimension in secure communications. Eur. Phys. J. Spec. Top. 223, 1441–1445 (2014)

    Article  Google Scholar 

  8. Banerjee, S., Mitra, M., Rondoni, L.: Applications of Chaos and Nonlinear Dynamics in Engineering, vol. 1. Springer, New York (2011)

    Book  MATH  Google Scholar 

  9. Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V.I.: Hidden attractor in smooth Chua systems. Phys. D Nonlinear Phenom. 241, 1482–1486 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V.I.: Localization of hidden Chuas attractors. Phys. Lett. A 375, 2230–2233 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  12. Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N.V., Leonov, G.A., Prasad, A.: Hidden attractors in dynamical systems. Phys. Rep. 637, 1–50 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  13. Leonov, G.A., Kuznetsov, N.V.: Hidden attractors in dynamical systems: from hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. Int. J. Bifurc. Chaos 23, 1330002–130071 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  14. Bragin, V.O., Vagaitsev, V.I., Kuznetsov, N.V., Leonov, G.A.: Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chuas circuits. J. Comput. Syst. Sci. Int. 50, 511–543 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Leonov, G.A., Kuznetsov, N.V., Kuznestova, O.A., Seledzhi, S.M., Vagaitsev, V.I.: Hidden oscillations in dynamical systems. Trans. Syst. Control. 6, 1–14 (2011)

    Google Scholar 

  16. Leonov, G.A., Kuznetsov, N.V.: Algorithms for searching for hidden oscillations in the Aizerman and Kalman problems. Dokl. Math. 84, 475–481 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. Li, Z., Park, J.B., Joo, Y.H., Zhang, B.: Bifurcations and chaos in a permanent-magnet synchronous motor. IEEE Trans. Circuit Syst. I Fundam. Theory Appl. 49, 383–387 (2002)

    Article  Google Scholar 

  18. Gao, Y., Chau, K.T.: Hopf bifurcation and chaos in synchronous reluctance motor drives. IEEE Trans. Energy Convers. 19, 296–302 (2004)

    Article  Google Scholar 

  19. Kundu, S., Kar, U., Chakrabarty, K.: Co-existence of multiple attractors in the PWM controlled DC drives. Eur. Phys. J. Spec. Top. 222, 699–709 (2013)

    Article  Google Scholar 

  20. Sangrody, R.A., Nazarzadeh, J., Nikravesh, K.Y.: Bifurcation and Lyapunovs exponents characteristics of electrical scalar drive systems. IET Power Electron. 5, 1236–1340 (2012)

    Article  Google Scholar 

  21. Ge, Z.-M., Chang, C.-M.: Chaos synchronization and parameters identification of single time scale brushless DC motors. Chaos Solitons Fractals 20, 883–903 (2004)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  23. Yuefeng, G.: Chaos control algorithm for redundant robotic obstacle avoidance. Inf. Technol. J. 12, 704–711 (2013)

    Article  Google Scholar 

  24. Ge, Y.X.: The space redundant robotic manipulator chaotic motion dynamics control algorithm. Sens. Transducers 175, 27–31 (2014)

    Google Scholar 

  25. Duarte, M., Fernando, B.M., Machado, J.A.T.: Chaos dynamics in the trajectory control of redundant manipulators. In: Proceedings of the IEEE International Conference on Robotics and Automation, ICRA 2000, pp. 4109–4114 (2000)

  26. Sofroniou, A., Bishop, S.: Dynamics of a parametrically excited system with two forcing terms. Mathematics 2, 172–195 (2014)

    Article  MATH  Google Scholar 

  27. Luo, A.C.J., Guo, Y.: Periodic motions to chaos in pendulum. Int. J. Bifurc. Chaos. 26, 1650159–165070 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  28. Kiseleva, M.A., Kuznetsov, N.V., Leonov, G.A.: Hidden attractors in electromechanical systems with and without equilibria. IFAC Papers OnLine 49, 051–055 (2016)

    Article  Google Scholar 

  29. Zang, X., Iqbal, S., Zhu, Y., Liu, X., Zhao, J.: Applications of chaotic dynamics in robotics. Int. J. Adv. Robot. Syst. 13(2), 1–17 (2016)

    Article  Google Scholar 

  30. Lochan, K., Roy, B.K., Subudhi, B.: A review on two-link flexible manipulators. Annu. Rev. Control 42, 346–367 (2016)

    Article  Google Scholar 

  31. Liu, H., Zhaohui, L., Zhang, L.: Control chaos in a spatially redundant manipulator with flexible Link. In: 12th IFToMM World Congress Besanon, Besanon (2007)

  32. Dwivedy, S.K., Kar, R.C.: Nonlinear dynamics of a cantilever beam carrying an attached mass with 1:3:9 internal resonances. Nonlinear Dyn. 31, 49–72 (2003)

    Article  MATH  Google Scholar 

  33. Zhihua, F., Haiyan, H.: Principal parametric and three-to-one internal resonances of flexible beams undergoing a large linear motion. Acta Mech. Sin. 19, 355–364 (2003)

    Article  Google Scholar 

  34. Dwivedy, S.K., Kar, R.C.: Simultaneous combination and 1:3:5 internal resonances in a parametrically excited beam-mass system. Int. J. Non. Linear. Mech. 38, 585–596 (2003)

    Article  MATH  Google Scholar 

  35. Pratiher, B., Dwivedy, S.K.: Nonlinear vibration of a magneto-elastic cantilever beam with tip mass. J. Vib. Acoust. 131, 21011 (2009)

    Article  Google Scholar 

  36. Hamed, Y.S., Elagan, S.K.: On the vibration behavior study of a nonlinear flexible composite beam under excitation forces via nonlinear active vibration controller. Int. J. Basic Appl. Sci. 13, 09–18 (2013)

    Google Scholar 

  37. Sadri, M., Younesian, D., Esmailzadeh, E.: Nonlinear harmonic vibration and stability analysis of a cantilever beam carrying an intermediate lumped mass. Nonlinear Dyn. 84, 1–16 (2016)

    Article  MathSciNet  Google Scholar 

  38. Dupac, M., Beale, D.G.: Dynamic analysis of a flexible linkage mechanism with cracks and clearance. Mech. Mach. Theory. 45, 1909–1923 (2010)

    Article  MATH  Google Scholar 

  39. Dwivedy, S.K., Kar, R.C.: Non-linear dynamics of a slender beam carrying a lumped mass under principal parametric resonance with three-mode interactions. Int. J. Non. Linear. Mech. 36, 927–945 (2001)

    Article  MATH  Google Scholar 

  40. Dwivedy, S.K., Kar, R.C.: Simultaneous combination, principal parametric and internal resonances in a slender beam with a lumped mass: three-mode interactions. J. Sound Vib. 242, 27–46 (2001)

    Article  MATH  Google Scholar 

  41. Dwivedy, S., Kar, R.: Dynamics of a slender beam with an attached mass under combination parametric and internal resonances, Part II: periodic and chaotic responses. J. Sound Vib. 222, 281–305 (1999)

    Article  Google Scholar 

  42. Feng, Z.H., Lan, X.J., Zhu, X.D.: Principal parametric resonances of a slender cantilever beam subject to axial narrow-band random excitation of its base. Int. J. Non. Linear. Mech. 42, 1170–1185 (2007)

    Article  Google Scholar 

  43. Yin, C., Zhong, S.-M., Chen, W.-F.: Design of sliding mode controller for a class of fractional-order chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 17(1), 356–366 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  44. Yin, C., Cheng, Y., Chen, Y.Q., Stark, B., Zhong, S.: Adaptive fractional-order switching-type control method design for 3D fractional-order nonlinear systems. Nonlinear Dyn. 82(1–2), 39–52 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  45. Yin, C., Cheng, Y., Zhong, S.-M., Bai, Z.: Fractional-order switching type control law design for adaptive sliding mode technique of 3D fractional-order nonlinear systems. Complexity 21(6), 363–373 (2015)

    Article  MathSciNet  Google Scholar 

  46. Rostami Kandroodi, M., Farivar, F., Zamani Pedram, M., Aliyari Shoorehdeli, M.: Variable structure control and anti-control of flexible joint manipulator with experimental validation. In: Proceedings of the 2011 IEEE International Conference Mechatronics (ICM ), IEEE, Istanbul, pp. 294–299 (2011)

  47. Kandroodi, M.R., Farivar, F., Shoorehdeli, M.A.: Lyapunov based control of flexible joint manipulator with experimental validation by using chaotic gyroscope synchronization. Int. J. Mech. Syst. Eng. 2, 169–175 (2012)

    Google Scholar 

  48. Zamani, M., Aliyari, M., Farivar, F.: Hybrid concepts of the control and anti-control of flexible joint manipulator. Int. J. Robot. 2, 35–44 (2011)

    Google Scholar 

  49. Wang, F., Bajaj, A.K.: Nonlinear dynamics of a three-beam structure with attached mass and three-mode interactions. Nonlinear Dyn. 62, 461–484 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  50. Farahan, S.B., Ghazavi, M., Rahmanian, S.: Nonlinear dynamic analysis of a four-bar mechanism having revolute joint with clearance. J. Theor. Appl. Vib. Acoust. 2, 91–106 (2016)

    Google Scholar 

  51. Pomares, J., Perea, I., Jara, C.A., Garca, G.J., Torres, F.: Dynamic visual servo control of a 4-axis joint tool to track image trajectories during machining complex shapes. Robot. Comput. Integr. Manuf. 29, 261–270 (2013)

    Article  Google Scholar 

  52. de la Fraga, L.G., Tlelo-Cuautle, E.: Optimizing the maximum Lyapunov exponent and phase space portraits in multi-scroll chaotic oscillators. Nonlinear Dyn. 76(2), 503–1515 (2014)

    Google Scholar 

  53. Carbajal-Gmez, V.H., Tlelo-Cuautle, E., Fernndez, F.V., de la Fraga, L.G., Snchez-Lpez, C.: Maximizing Lyapunov exponents in a chaotic oscillator by applying differential evolution. Int. J. Nonlinear Sci. Numer. Simul. 15(1), 11–17 (2014)

    MathSciNet  Google Scholar 

  54. Pham, V.T., Volos, C., Jafari, S., Kapitaniak, T.: Coexistence of hidden chaotic attractors in a novel no-equilibrium system. Nonlinear Dyn. 87(3), 2001–2010 (2016)

    Article  Google Scholar 

  55. Pham, V.T., Vaidyanathan, S., Volos, C., Jafari, S., Kingni, S.T.: A no-equilibrium hyperchaotic system with a cubic nonlinear term. Optik 127, 3259–3265 (2016)

    Article  Google Scholar 

  56. Singh, J.P., Roy, B.K.: Analysis of an one equilibrium novel hyperchaotic system and its circuit validation. Int. J. Control Theory Appl. 8, 1015–1023 (2015)

    Google Scholar 

  57. Wei, Z., Zhang, W.: Hidden hyperchaotic attractors in a modified Lorenz–Stenflo system with only one stable equilibrium. Int. J. Bifurc. Chaos. 24, 1450127–1450132 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  58. Bao, B.C., Li, Q.D., Wang, N., Xu, Q.: Multistability in Chuas circuit with two stable node-foci. Chaos Interdiscip. J. Nonlinear Sci. 26, 43111–43120 (2016)

    Article  MathSciNet  Google Scholar 

  59. Singh, J.P., Roy, B.K.: A novel asymmetric hyperchaotic system and its circuit validation. Int. J. Control Theory Appl. 8, 1005–1013 (2015)

    Google Scholar 

  60. Chen, Y., Yang, Q.: Dynamics of a hyperchaotic Lorenz-type system. Nonlinear Dyn. 77, 569–581 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  61. Yang, Q., Liu, Y.: A hyperchaotic system from a chaotic system with one saddle and two stable node-foci. J. Math. Anal. Appl. 360, 293–306 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  62. Tang, L., Zhao, L., School, Q.Z.: A novel four-dimensional autonomous hyperchaotic system ICAIC Part III, vol. 2011. Springer, Berlin (2011)

    Google Scholar 

  63. Li, Q., Zeng, H., Li, J.: Hyperchaos in a 4D memristive circuit with infinitely many stable equilibria. Nonlinear Dyn. 79, 2295–2308 (2014)

    Article  MathSciNet  Google Scholar 

  64. Li, C., Sprott, J.C., Thio, W.: Bistability in a hyperchaotic system with a line equilibrium. J. Exp. Theor. Phys. 118, 494–500 (2014)

    Article  Google Scholar 

  65. Singh, J.P., Roy, B.K.: A Novel hyperchaotic system with stable and unstable line of equilibria and sigma shaped Poincare map. IFAC PapersOnLine 49, 526–531 (2016)

    Article  Google Scholar 

  66. Zhang, H., Li, Z.: Research on nonlinear dynamics of two-degrees-of-freedom robot. In: Xang, C., et al. (eds.) ICIRA, pp. 420–426. Springer, Berlin (2008)

    Google Scholar 

  67. Ravishankar, A., Ghosal, A.S.: Nonlinear dynamics and chaotic in feedback controlled two and three degree of freedom robot. Int. J. Rob. Res. 18, 93–108 (1999)

    Google Scholar 

  68. Spong, M.W., Hutchinson, S., Vidyasagar, M.: Robot Modeling and Control, 1st edn. Wiley, New York (2002)

    Google Scholar 

  69. Thoma, M., Allgwer, F., Morari, M.: Robot Motion and Control. Scientific Publishing Services Pvt. Ltd, Chennai (2009)

    Google Scholar 

  70. Spong, M.W.: Adaptive control of flexible joint manipulators. Syst. Control Lett. 13, 15–21 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  71. Shen, C., Yu, S., Lu, J., Chen, G.: Designing hyperchaotic systems with any desired number of positive Lyapunov exponents via a simple model. IEEE Trans. Circuits Syst. I Regul. Pap. 61(8), 2380–2389 (2014)

    Article  Google Scholar 

  72. Shen, C., Yu, S., Lu, J., Chen, G.: A systematic methodology for constructing hyperchaotic systems with multiple positive Lyapunov exponents and circuit implementation. IEEE Trans. Circuits Syst. I Reg. Pap. 61, 854–864 (2014)

    Article  Google Scholar 

  73. Ma, J., Wu, X., Chu, R., Zhang, L.: Selection of multi-scroll attractors in Jerk circuits and their verification using Pspice. Nonlinear Dyn. 76(4), 1951–1962 (2014)

    Article  Google Scholar 

  74. Chen, J.L.G.: Generating multiscroll chaotic attractors: theories, methods and applications. Int. J. Bifurc. Chaos 16(4), 775–858 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  75. Jafari, S., Pham, V.-T., Kapitaniak, T.: Multiscroll chaotic sea obtained from a simple 3D system without equilibrium. Int. J. Bifurc. Chaos 26(2), 1650031–1650036 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  76. Zhang, H., Liu, X., Shen, X., Liu, J.: Chaos entanglement: a new approach to generate chaos. Int. J. Bifurc. Chaos 23(5), 1330014–1330031 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  77. Sahab, A.R., Modabbernia, M.R.: Backstepping method for a single-link flexible-joint manipulator using genetic algorithm. Int. J. Innov. Comput. Inf. Control. 7, 4161–4170 (2011)

    Google Scholar 

  78. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D 16, 285–317 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  79. Singh, J.P., Roy, B.K.: The nature of Lyapunov exponents is \((+, +, -, -)\). Is it a hyperchaotic system? Chaos Solitons Fractals 92, 73–85 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  80. Leonov, G.A., Kuznetsov, N.V.: Time-varying linearization and the Perron effects. Int. J. Bifurc. Chaos. 17, 1079–1107 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  81. Sabarathinam, S., Thamilmaran, K.: Transient chaos in a globally coupled system of nearly conservative Hamiltonian Duffing oscillators. Chaos Solitons Fractals 73, 129–140 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  82. Singh, J.P., Roy, B.K.: Crisis and inverse crisis route to chaos in a new 3-D chaotic system with saddle, saddle foci and stable node foci nature of equilibria. Optik 127, 11982–12002 (2016)

    Article  Google Scholar 

  83. Gottwald, B.G.A., Melbourne, I.: A new test for chaos in deterministic systems. Proc. Math. Phys. Eng. Sci. 460, 603–611 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  84. Gottwald, G.: On the implementation of the 0–1 test for chaos. Arxiv Prepr. arXiv:0906.1418. vol. 1367, pp. 1–22 (2009)

  85. Sabarathinam, S., Thamilmaran, K., Borkowski, L., Perlikowski, P., Brzeski, P., Stefanski, A., et al.: Transient chaos in two coupled, dissipatively perturbed Hamiltonian Duffing oscillators. Commun. Nonlinear Sci. Numer. Simul. 18, 3098–3107 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  86. Xiong, L., Lu, Y.-J., Zhang, Y.-F., Zhang, X.-G., Gupta, P.: Design and hardware implementation of a new chaotic secure communication technique. PLoS ONE 11(8), 1–19 (2016)

    Google Scholar 

  87. Zhang, R.-X., Yang, S.-P.: Adaptive synchronisation of fractional-order chaotic systems. Chin. Phys. B 19(2), 1–7 (2010)

    MathSciNet  Google Scholar 

  88. Lao, S.-K., Tam, L.-M., Chen, H.-K., Sheu, L.-J.: Hybrid stability checking method for synchronization of chaotic fractional-order systems. Abstr. Appl. Anal. 2014, 1–11 (2014)

    Article  MathSciNet  Google Scholar 

  89. Kenfack, G., Tiedeu, A.: Secured transmission of ECG signals: numerical and electronic simulations. J. Signal Inf. Process. 04(02), 158–169 (2013)

    Google Scholar 

  90. Chen, D., Liu, C., Wu, C., Liu, Y., Ma, X., You, Y.: A new fractional-order chaotic system and its synchronization with circuit simulation. Circuits Syst. Signal Process. 31(5), 1599–1613 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Prof. N. V. Kuznetsov is supported by the RSF Project No. 14-21-00041p. This support is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Department of EE, National Institute of Technology Silchar, Silchar, Assam, 788010, India

    Jay Prakash Singh, K. Lochan & B. K. Roy

  2. Faculty of Mathematics and Mechanics, Saint-Petersburg State University, Saint-Petersburg, Russia, 198504

    Nikolay V. Kuznetsov

  3. Department of Mathematical Information Technology, University of Jyväkylä, 40014, Kokkola, Finland

    Nikolay V. Kuznetsov

Authors
  1. Jay Prakash Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Lochan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nikolay V. Kuznetsov
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. B. K. Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jay Prakash Singh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Singh, J.P., Lochan, K., Kuznetsov, N.V. et al. Coexistence of single- and multi-scroll chaotic orbits in a single-link flexible joint robot manipulator with stable spiral and index-4 spiral repellor types of equilibria. Nonlinear Dyn 90, 1277–1299 (2017). https://doi.org/10.1007/s11071-017-3726-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3726-4

Keywords

Search

Navigation