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

  • * Corresponding author: Jianghong Bao

    * Corresponding author: Jianghong Bao 
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  • 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.

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

    Citation:

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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] $

    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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