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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  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 & 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.  Google Scholar

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[16]

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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.  Google Scholar

[21]

H. K. Moffatt, A self-consistent treatment of simple dynamo systems, Geophys. Astrophys. Fluid Dyn., 14 (1979), 147-166.  doi: 10.1080/03091927908244536.  Google Scholar

[22]

V.-T. PhamC. VolosS. Jafari and T. Kapitaniak, Coexistence of hidden chaotic attractors in a novel no-equilibrium system, Nonlinear Dyn., 87 (2017), 2001-2010.   Google Scholar

[23]

A. N. Pisarchik and U. Feudel, Control of multistability, Phys. Rep., 540 (2014), 167-218.  doi: 10.1016/j.physrep.2014.02.007.  Google Scholar

[24]

M. SchefferJ. BascompteW. A. BrockV. BrovkinS. R. CarpenterV. DakosH. HeldE. H. van NesM. Rietkerk and G. Sugihara, Early-warning signals for critical transitions, Nature, 461 (2009), 53-59.  doi: 10.1038/nature08227.  Google Scholar

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M. SchefferS. CarpenterJ. A. FoleyC. Folke and B. Walker, Catastrophic shifts in ecosystems, Nature, 413 (2001), 591-596.  doi: 10.1038/35098000.  Google Scholar

[26]

J. P. SinghK. LochanN. 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.  Google Scholar

[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.  Google Scholar

[28]

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[29]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, 1990. doi: 10.1007/978-1-4757-4067-7.  Google Scholar

[30]

R.-X. Zhang and S.-P., Yang, Adaptive synchronisation of fractional-order chaotic systems, Chin. Phys. B, 19 (2010), 020510. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

D. DudkowskiS. JafariT. KapitaniakN. V. KuznetsovG. A. Leonov and A. Prasad, Hidden attractors in dynamical systems, Phys. Rep., 637 (2016), 1-50.  doi: 10.1016/j.physrep.2016.05.002.  Google Scholar

[4]

H. ErzgraberD. LenstraB. KrauskopfE. WilleM. PeilI. Fisher and W. Elsaer, Mutually delay-coupled semiconductor lasers: Mode bifurcation scenarios, Opt. Commun., 255 (2005), 286-296.   Google Scholar

[5]

D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc., 8 (1902), 437-479.  doi: 10.1090/S0002-9904-1902-00923-3.  Google Scholar

[6]

M. HirotaM. HolmgrenE. 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.  Google Scholar

[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.  Google Scholar

[8]

S. JafariJ. 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.  Google Scholar

[9] J. KambhuS. 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.   Google Scholar
[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.  Google Scholar

[11]

N. V. KuznetsovG. A. LeonovT. N. MokaevA. Prasad and M. D. Shrimali, Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system, Nonlinear Dyn., 92 (2018), 267-285.   Google Scholar

[12]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, 1998.  Google Scholar

[13]

T. LauvdalR. 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.  Google Scholar

[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.  Google Scholar

[15]

G. A. LeonovN. 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.   Google Scholar

[16]

Q. D. LiS. Y. HuS. 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.   Google Scholar

[17]

W. W. Lytton, Computer modelling of epilepsy, Nat. Rev. Neurosci., 9 (2008), 626-637.  doi: 10.1038/nrn2416.  Google Scholar

[18]

R. M. MayG. Levin and S. A. Sugihara, Ecology for bankers, Nature, 451 (2008), 893-895.  doi: 10.1038/451893a.  Google Scholar

[19]

P. E. McSharryL. A. Smith and L. Tarassenko, Prediction of epileptic seizures: Are nonlinear methods relevant?, Nature Med., 9 (2003), 241-242.  doi: 10.1038/nm0303-241.  Google Scholar

[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.  Google Scholar

[21]

H. K. Moffatt, A self-consistent treatment of simple dynamo systems, Geophys. Astrophys. Fluid Dyn., 14 (1979), 147-166.  doi: 10.1080/03091927908244536.  Google Scholar

[22]

V.-T. PhamC. VolosS. Jafari and T. Kapitaniak, Coexistence of hidden chaotic attractors in a novel no-equilibrium system, Nonlinear Dyn., 87 (2017), 2001-2010.   Google Scholar

[23]

A. N. Pisarchik and U. Feudel, Control of multistability, Phys. Rep., 540 (2014), 167-218.  doi: 10.1016/j.physrep.2014.02.007.  Google Scholar

[24]

M. SchefferJ. BascompteW. A. BrockV. BrovkinS. R. CarpenterV. DakosH. HeldE. H. van NesM. Rietkerk and G. Sugihara, Early-warning signals for critical transitions, Nature, 461 (2009), 53-59.  doi: 10.1038/nature08227.  Google Scholar

[25]

M. SchefferS. CarpenterJ. A. FoleyC. Folke and B. Walker, Catastrophic shifts in ecosystems, Nature, 413 (2001), 591-596.  doi: 10.1038/35098000.  Google Scholar

[26]

J. P. SinghK. LochanN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, 1990. doi: 10.1007/978-1-4757-4067-7.  Google Scholar

[30]

R.-X. Zhang and S.-P., Yang, Adaptive synchronisation of fractional-order chaotic systems, Chin. Phys. B, 19 (2010), 020510. Google Scholar

Figure 1.  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] $
Figure 2.  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
Figure 3.  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 $)
Figure 4.  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
Figure 5.  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)
Figure 6.  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
Figure 7.  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
Figure 8.  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
Figure 9.  Pitchfork bifurcation diagram in system (2.2) near $ {k_6} = 0 $
Figure 10.  Circuit diagram for system (2.2)
Figure 11.  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 $)
Figure 12.  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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