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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
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##### References:
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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