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November  2019, 24(11): 6053-6069. doi: 10.3934/dcdsb.2019130

## Coexisting hidden attractors in a 5D segmented disc dynamo with three types of equilibria

 1 School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510641, China 2 Guangxi Colleges and Universities Key Laboratory, of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, China 3 School of Electronic and Information Engineering, South China University of Technology, Guangzhou, Guangdong 510641, China

* Corresponding author: Jianghong Bao

Received  September 2017 Revised  April 2018 Published  November 2019 Early access  July 2019

Little seems to be known about coexisting hidden attractors in hyperchaotic systems with three types of equilibria. Based on the segmented disc dynamo, this paper proposes a new 5D hyperchaotic system which possesses the properties. This new system can generate hidden hyperchaos and chaos when initial conditions vary, as well as self-excited chaotic and hyperchaotic attractors when parameters vary. Furthermore, the paper proves that the Hopf bifurcation and pitchfork bifurcation occur in the system. Numerical simulations demonstrate the emergence of the two bifurcations. The MATLAB simulation results are further confirmed and validated by circuit implementation using NI Multisim.

Citation: Jianghong Bao, Dandan Chen, Yongjian Liu, Hongbo Deng. Coexisting hidden attractors in a 5D segmented disc dynamo with three types of equilibria. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6053-6069. doi: 10.3934/dcdsb.2019130
##### References:
 [1] A. Babloyantz and A. Destexhe, Low-dimensional chaos in an instance of epilepsy, Proc. Natl. Acad. Sci. USA, 83 (1986), 3513-3517.  doi: 10.1073/pnas.83.10.3513. [2] M.-F. Danca, Hidden transient chaotic attractors of Rabinovich-Fabrikant system, Nonlinear Dyn., 86 (2016), 1263-1270.  doi: 10.1007/s11071-016-2962-3. [3] D. Dudkowski, S. Jafari, T. Kapitaniak, N. V. Kuznetsov, G. A. Leonov and A. Prasad, Hidden attractors in dynamical systems, Phys. Rep., 637 (2016), 1-50.  doi: 10.1016/j.physrep.2016.05.002. [4] H. Erzgraber, D. Lenstra, B. Krauskopf, E. Wille, M. Peil, I. Fisher and W. Elsaer, Mutually delay-coupled semiconductor lasers: Mode bifurcation scenarios, Opt. Commun., 255 (2005), 286-296. [5] D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc., 8 (1902), 437-479.  doi: 10.1090/S0002-9904-1902-00923-3. [6] M. Hirota, M. Holmgren, E. H. Van Nes and M. Scheffer, Global resilience of tropical forest and savanna to critical transitions, Science, 334 (2011), 232-235.  doi: 10.1126/science.1210657. [7] S. Jafari and J. C. Sprott, Simple chaotic flows with a line equilibrium, Chaos, Solitons & Fractals, 57 (2013), 79-84.  doi: 10.1016/j.chaos.2013.08.018. [8] S. Jafari, J. C. Sprott and S. Golpayegani, Elementary quadratic chaotic flows with no equilibria, Phys. Lett. A, 377 (2013), 699-702.  doi: 10.1016/j.physleta.2013.01.009. [9] J. Kambhu, S. Weidman and N. Krishnam, New Directions for Understanding Systemic Risk: A Report on a Conference Cosponsored by the Federal Reserve Bank of New York and the National Academy of Sciences, The National Academies Press, Washington D.C., 2007. [10] N. V. Kuznetsov, The Lyapunov dimension and its estimation via the Leonov method, Phys. Lett. A, 380 (2016), 2142-2149.  doi: 10.1016/j.physleta.2016.04.036. [11] N. V. Kuznetsov, G. A. Leonov, T. N. Mokaev, A. Prasad and M. D. Shrimali, Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system, Nonlinear Dyn., 92 (2018), 267-285. [12] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, 1998. [13] T. Lauvdal, R. M. Murray and T. I. Fossen, Stabilization of integrator chains in the presence of magnitude and rate saturations: A gain scheduling approach, Proc. IEEE Control and Decision Conference, 4 (1997), 4404-4405.  doi: 10.1109/CDC.1997.652491. [14] G. A. Leonov and N. V. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, Int. J. Bifurcation Chaos, 23 (2013), 1330002, 69pp. doi: 10.1142/S0218127413300024. [15] G. A. Leonov, N. V. Kuznetsov and T. N. Mokaev, Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion, Eur. Phys. J. Special Topics, 224 (2015), 1421-1458. [16] Q. D. Li, S. Y. Hu, S. Tang and G. Zeng, Hyperchaos and horseshoe in a 4D memristive system with a line of equilibria and its implementation, Int. J. Circuit Theory Appl., 42 (2014), 1172-1188. [17] W. W. Lytton, Computer modelling of epilepsy, Nat. Rev. Neurosci., 9 (2008), 626-637.  doi: 10.1038/nrn2416. [18] R. M. May, G. Levin and S. A. Sugihara, Ecology for bankers, Nature, 451 (2008), 893-895.  doi: 10.1038/451893a. [19] P. E. McSharry, L. A. Smith and L. Tarassenko, Prediction of epileptic seizures: Are nonlinear methods relevant?, Nature Med., 9 (2003), 241-242.  doi: 10.1038/nm0303-241. [20] M. Molaie, S. Jafari, J. C. Sprott and S. Mohammad, Simple chaotic flows with one stable equilibrium, Int. J. Bifurcation Chaos, 23 (2013), 1350188, 7pp. doi: 10.1142/S0218127413501885. [21] H. K. Moffatt, A self-consistent treatment of simple dynamo systems, Geophys. Astrophys. Fluid Dyn., 14 (1979), 147-166.  doi: 10.1080/03091927908244536. [22] V.-T. Pham, C. Volos, S. Jafari and T. Kapitaniak, Coexistence of hidden chaotic attractors in a novel no-equilibrium system, Nonlinear Dyn., 87 (2017), 2001-2010. [23] A. N. Pisarchik and U. Feudel, Control of multistability, Phys. Rep., 540 (2014), 167-218.  doi: 10.1016/j.physrep.2014.02.007. [24] M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin, S. R. Carpenter, V. Dakos, H. Held, E. H. van Nes, M. Rietkerk and G. Sugihara, Early-warning signals for critical transitions, Nature, 461 (2009), 53-59.  doi: 10.1038/nature08227. [25] M. Scheffer, S. Carpenter, J. A. Foley, C. Folke and B. Walker, Catastrophic shifts in ecosystems, Nature, 413 (2001), 591-596.  doi: 10.1038/35098000. [26] J. P. Singh, K. Lochan, N. V. Kuznetsov and B. K. Roy, 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 (2017), 1277-1299.  doi: 10.1007/s11071-017-3726-4. [27] N. V. Stankevich, N. V. Kuznetsov, G. A. Leonov and L. O. Chua, Scenario of the birth of hidden attractors in the Chua circuit, Int. J. Bifurcation Chaos, 27 (2017), 1730038, 18pp. doi: 10.1142/S0218127417300385. [28] X. Wang and G. Chen, A chaotic system with only one stable equilibrium, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1264-1272.  doi: 10.1016/j.cnsns.2011.07.017. [29] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, 1990. doi: 10.1007/978-1-4757-4067-7. [30] R.-X. Zhang and S.-P., Yang, Adaptive synchronisation of fractional-order chaotic systems, Chin. Phys. B, 19 (2010), 020510.

show all references

##### References:
 [1] A. Babloyantz and A. Destexhe, Low-dimensional chaos in an instance of epilepsy, Proc. Natl. Acad. Sci. USA, 83 (1986), 3513-3517.  doi: 10.1073/pnas.83.10.3513. [2] M.-F. Danca, Hidden transient chaotic attractors of Rabinovich-Fabrikant system, Nonlinear Dyn., 86 (2016), 1263-1270.  doi: 10.1007/s11071-016-2962-3. [3] D. Dudkowski, S. Jafari, T. Kapitaniak, N. V. Kuznetsov, G. A. Leonov and A. Prasad, Hidden attractors in dynamical systems, Phys. Rep., 637 (2016), 1-50.  doi: 10.1016/j.physrep.2016.05.002. [4] H. Erzgraber, D. Lenstra, B. Krauskopf, E. Wille, M. Peil, I. Fisher and W. Elsaer, Mutually delay-coupled semiconductor lasers: Mode bifurcation scenarios, Opt. Commun., 255 (2005), 286-296. [5] D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc., 8 (1902), 437-479.  doi: 10.1090/S0002-9904-1902-00923-3. [6] M. Hirota, M. Holmgren, E. H. Van Nes and M. Scheffer, Global resilience of tropical forest and savanna to critical transitions, Science, 334 (2011), 232-235.  doi: 10.1126/science.1210657. [7] S. Jafari and J. C. Sprott, Simple chaotic flows with a line equilibrium, Chaos, Solitons & Fractals, 57 (2013), 79-84.  doi: 10.1016/j.chaos.2013.08.018. [8] S. Jafari, J. C. Sprott and S. Golpayegani, Elementary quadratic chaotic flows with no equilibria, Phys. Lett. A, 377 (2013), 699-702.  doi: 10.1016/j.physleta.2013.01.009. [9] J. Kambhu, S. Weidman and N. Krishnam, New Directions for Understanding Systemic Risk: A Report on a Conference Cosponsored by the Federal Reserve Bank of New York and the National Academy of Sciences, The National Academies Press, Washington D.C., 2007. [10] N. V. Kuznetsov, The Lyapunov dimension and its estimation via the Leonov method, Phys. Lett. A, 380 (2016), 2142-2149.  doi: 10.1016/j.physleta.2016.04.036. [11] N. V. Kuznetsov, G. A. Leonov, T. N. Mokaev, A. Prasad and M. D. Shrimali, Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system, Nonlinear Dyn., 92 (2018), 267-285. [12] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, 1998. [13] T. Lauvdal, R. M. Murray and T. I. Fossen, Stabilization of integrator chains in the presence of magnitude and rate saturations: A gain scheduling approach, Proc. IEEE Control and Decision Conference, 4 (1997), 4404-4405.  doi: 10.1109/CDC.1997.652491. [14] G. A. Leonov and N. V. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, Int. J. Bifurcation Chaos, 23 (2013), 1330002, 69pp. doi: 10.1142/S0218127413300024. [15] G. A. Leonov, N. V. Kuznetsov and T. N. Mokaev, Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion, Eur. Phys. J. Special Topics, 224 (2015), 1421-1458. [16] Q. D. Li, S. Y. Hu, S. Tang and G. Zeng, Hyperchaos and horseshoe in a 4D memristive system with a line of equilibria and its implementation, Int. J. Circuit Theory Appl., 42 (2014), 1172-1188. [17] W. W. Lytton, Computer modelling of epilepsy, Nat. Rev. Neurosci., 9 (2008), 626-637.  doi: 10.1038/nrn2416. [18] R. M. May, G. Levin and S. A. Sugihara, Ecology for bankers, Nature, 451 (2008), 893-895.  doi: 10.1038/451893a. [19] P. E. McSharry, L. A. Smith and L. Tarassenko, Prediction of epileptic seizures: Are nonlinear methods relevant?, Nature Med., 9 (2003), 241-242.  doi: 10.1038/nm0303-241. [20] M. Molaie, S. Jafari, J. C. Sprott and S. Mohammad, Simple chaotic flows with one stable equilibrium, Int. J. Bifurcation Chaos, 23 (2013), 1350188, 7pp. doi: 10.1142/S0218127413501885. [21] H. K. Moffatt, A self-consistent treatment of simple dynamo systems, Geophys. Astrophys. Fluid Dyn., 14 (1979), 147-166.  doi: 10.1080/03091927908244536. [22] V.-T. Pham, C. Volos, S. Jafari and T. Kapitaniak, Coexistence of hidden chaotic attractors in a novel no-equilibrium system, Nonlinear Dyn., 87 (2017), 2001-2010. [23] A. N. Pisarchik and U. Feudel, Control of multistability, Phys. Rep., 540 (2014), 167-218.  doi: 10.1016/j.physrep.2014.02.007. [24] M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin, S. R. Carpenter, V. Dakos, H. Held, E. H. van Nes, M. Rietkerk and G. Sugihara, Early-warning signals for critical transitions, Nature, 461 (2009), 53-59.  doi: 10.1038/nature08227. [25] M. Scheffer, S. Carpenter, J. A. Foley, C. Folke and B. Walker, Catastrophic shifts in ecosystems, Nature, 413 (2001), 591-596.  doi: 10.1038/35098000. [26] J. P. Singh, K. Lochan, N. V. Kuznetsov and B. K. Roy, 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 (2017), 1277-1299.  doi: 10.1007/s11071-017-3726-4. [27] N. V. Stankevich, N. V. Kuznetsov, G. A. Leonov and L. O. Chua, Scenario of the birth of hidden attractors in the Chua circuit, Int. J. Bifurcation Chaos, 27 (2017), 1730038, 18pp. doi: 10.1142/S0218127417300385. [28] X. Wang and G. Chen, A chaotic system with only one stable equilibrium, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1264-1272.  doi: 10.1016/j.cnsns.2011.07.017. [29] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, 1990. doi: 10.1007/978-1-4757-4067-7. [30] R.-X. Zhang and S.-P., Yang, Adaptive synchronisation of fractional-order chaotic systems, Chin. Phys. B, 19 (2010), 020510.
For parameters $\left( {m, g, {k_1}, {k_2}, {k_3}, {k_4}, {k_5}, {k_6}} \right)$ = (0.02, 12, 0, 11.9, 0.01, 0, $-0.01$, $-99$) and initial condition (0, 0, 0, 0, 0), the finite-time local Lyapunov exponents spectrum in system (2.2) versus $r \in (0, 3]$
Parameters $\left( {m, r, g, {k_1}, {k_2}, {k_3}, {k_4}, {k_5}, {k_6}} \right)$ = (0.02, 0.02, 12, 0, 12.3, 0.0001, 0.01, 0.01, $-100$) and initial condition (0.8147, 0.9058, 0.1270, 0.9134, 0.6324); (a) chaotic attractor of system(2.2); (b) Poincaré map on the $x$-$z$ plane; (c) time series of $x$; (d) finite-time local Lyapunov dimension
Chaotic attractor of system (2.2) for initial condition (21, 0.1, 1, 0, 0) and parameters $\left( {m, r, g, {k_1}, {k_2}, {k_3}, {k_4}, {k_5}, {k_6}} \right)$ = (1.1, 6.1, 12, 0, 12, 0, 0, 0, $-100$)
Initial condition (0, $-10$, 1, $-100$, $-10$) and parameters $\left( {m, r, g, {k_1}, {k_2}, {k_3}, {k_4}, {k_5}, {k_6}} \right)$ = (0.01, 0.02, 12, 0, 13, 0, 0, 0.1, $-100$); (a) chaotic attractor of system (2.2); (b) Poincaré map on the $x$-$y$ plane
Hyperchaotic attractor of system (2.2) for parameters $\left( {m, r, g, {k_1}, {k_2}, {k_3}, {k_4}, {k_5}, {k_6}} \right)$ = (0.02, 0.02, 12, 0, 12, 0.001, 0.01, 0.01, $-100$) and initial condition (0.5268, 7.3786, 2.6912, 4.2284, 0.8147)
Parameters $\left( {m, r, g, {k_1}, {k_2}, {k_3}, {k_4}, {k_5}, {k_6}} \right)$ = (0.02, 0.02, 12, 0, 12, 0.001, 0.01, 0.01, $-100$) and initial condition $(0, -1000,100, -10, 0)$; (a) chaotic attractor of system (2.2); (b) Poincaré map on the $y$-$z$ plane
Parameters $\left( {m, r, g, {k_1}, {k_2}, {k_3}, {k_4}, {k_5}, {k_6}} \right)$ = (0.02, 0.02, 12, 0, 12, 0.001, 0.01, 0.01, -100) and initial condition $(0.5, 0.5, 0.5, 0.5, 0.5)$; (a) trajectory for $t \in [0, 10000 ]$; (b) trajectory for $t \in [0, 20000 ]$; (c) time series of $x$; (d) finite-time local Lyapunov dimension
Initial condition $(0.0026, -0.3011, 0.2967, 0, -0.7291)$ and $\left({m, r, g, {k_1}, {k_3}, {k_4}, {k_5}, {k_6}} \right)$ = (1.3, 0.1, 0.1, 0, 0.1, 0, 0.1, $-2$), a stable limit cycle of system (2.2) for ${k_2}$ = 0.0614
Pitchfork bifurcation diagram in system (2.2) near ${k_6} = 0$
Circuit diagram for system (2.2)
Hyperchaotic attractor of system (2.2) obtained using NI Multisim circuit implementation for $\left( {m, r, g, {k_1}, {k_2}, {k_3}, {k_4}, {k_5}, {k_6}} \right)$ = (0.02, 0.02, 12, 0, 12, 0.001, 0.01, 0.01, $-100$)
2D projections of hyperchaotic attractor of system (2.2) with parameters $\left( {m, r, g, {k_1}, {k_2}, {k_3}, {k_4}, {k_5}, {k_6}} \right)$ = (0.02, 0.02, 12, 0, 12, 0.001, 0.01, 0.01, $-100$) and initial condition (0.5268, 7.3786, 2.6912, 4.2284, 0.8147)
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