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

Jianghong Bao; Dandan Chen; Yongjian Liu; Hongbo Deng

doi:10.3934/dcdsb.2019130

Article Contents

2019, Volume 24, Issue 11: 6053-6069. Doi: 10.3934/dcdsb.2019130

This issue Previous Article Applying battelli-fečkan's method to transversal heteroclinic bifurcation in piecewise smooth systems Next Article Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term

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
^* Corresponding author: Jianghong Bao

Received: September 2017

Revised: April 2018

Early access: July 2019

Published: August 2019

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 34C28, 37D45; Secondary: 74H65.

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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] $

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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

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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 $)

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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

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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)

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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

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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

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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

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Figure 9. Pitchfork bifurcation diagram in system (2.2) near $ {k_6} = 0 $

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Figure 10. Circuit diagram for system (2.2)

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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 $)

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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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