Fig 1.
Phase portraits of the 10-D hyperchaotic system (1): (a) x1 − x2 attractor, (b) x1 − x3 attractor, (c) x3 − x4, (d) x5 − x6 attractor, (e) x9 − x10 attractor and (f) x3 − x9 − x10 attractor.
Fig 2.
Lyapunov exponents of the new 10-D hyperchaotic system (1) with a = 0.1, b = 0.1, c = 1.1 and d = 0.01.
Table 1.
Kaplan-Yorke fractal dimension of ten high dimensional chaotic system.
Fig 3.
Bifurcation diagram (a) and Lyapunov exponents spectrum (b) of the new 10-D system (1) when: b = 0.1, c = 1.8, d = 0.01 and a ∈ [0;0.2].
Fig 4.
Phase portraits of the new 10-D system (1) for different values of a.
(a) x3 − x2 Quasi-periodic attractor, (b) x3 − x2 chaotic attractor and (c) x3 − x2 hyperchaotic attractor.
Table 2.
Lyapunov exponents, Kaplan-Yorke dimensin and dynamics of the new 10D system (1) with parameter a varying.
Fig 5.
Bifurcation diagram (a) and Lyapunov exponents spectrum (b) of the new 10-D system (1) when: a = 0.1, c = 1.8, d = 0.01 and b ∈ [0.1;2].
Fig 6.
Phase portraits of the new 10-D system (1) for different values of b.
(a)x3 − x4Periodic attractor, (b) x3 − x4 chaotic attractor and (c) x3 − x4 hyperchaotic attractor.
Table 3.
Lyapunov exponents, Kaplan-Yorke dimension and dynamics of the new 10D system (1) with parameter b varying.
Fig 7.
Bifurcation diagram (a) and Lyapunov exponents spectrum (b) of the new 10-D system (1) when: a = 0.1, b = 0.1, d = 0.01 and c ∈ [0; 3].
Fig 8.
Phase portraits of the new 10-D system (1) for different values of c.
(a) x4 − x5 Periodic attractor, (b) (a) x4 − x5 chaotic attractor and (c) (a) x4 − x5 hyperchaotic attractor.
Table 4.
Lyapunov exponents, Kaplan-Yorke dimension and dynamics of the new 10D system (1) with parameter c varying.
Fig 9.
Bifurcation diagram (a) and Lyapunov exponents spectrum (b) of the new 10-D system (1) when: a = 0.1, b = 0.1, c = 1.8 and d ∈ [0; 1].
Fig 10.
Phase portraits of the new 10-D system (1) for various values of d.
(a) x7 − x10 Periodic attractor, (b) x7 − x10 chaotic attractor and (c) x7 − x10 hyperchaotic attractor.
Table 5.
Lyapunov exponents, Kaplan-Yorke dimension and dynamics of the new 10-D hyperchaotic system (1) with parameter d varying.
Fig 11.
Bifurcation diagram of system (1) versus a starting from:ξ1 (blue), ξ2(red), ξ3(green), ξ4(magenta), ξ5(yellow) and ξ6(cyan).
Fig 12.
Coexistence of six different attractors projected on the x1 − x5 plane.
(a) Coexistence of one hyperchaotic attractor and five quasi-periodic attractors when a = 0.01. (b) Coexistence of two hyperchaotic attractors and four quasi-periodic attractors when a = 0.1.
Table 6.
Lyapunov exponents, Kaplan-Yorke dimensin and dynamics of system (1) coexisisting attractors with parameter a varying.
Fig 13.
Bifurcation diagram of system (1) versus b starting from:ξ1 (blue), ξ2(red), ξ3(green), ξ4(magenta), ξ5(yellow) and ξ6(cyan).
Fig 14.
Coexistence of six different attractors projected on the x1 − x3 plane.
(a) Coexistence of one hyperchaotic attractor (blue), one chaotic attractor (magenta) and four periodic attractors when b = 0.5. (b) Coexistence of one chaotic attractor and five periodic attractors when b = 1.5. (c) Coexistence of three chaotic attractors and three periodic attractors when b = 2.5.
Table 7.
Lyapunov exponents, Kaplan-Yorke dimensin and dynamics of system (1) coexisisting attractors with parameter b varying.
Fig 15.
Bifurcation diagram of system (1) versus c starting from:ξ1 (blue), ξ2(red), ξ3(green), ξ4(magenta), ξ5(yellow) and ξ6(cyan).
Fig 16.
Coexistence of six different attractors projected on the x1 − x2 plane.
(a) Coexistence of two chaotic attractor and four quasi-periodic attractors when c = 0.3. (b) Coexistence of one hyperchaotic attractor starting from ξ1 (blue) and five quasi-periodic attractors when c = 0.85. (c) Coexistence of one chaotic attractor starting from ξ4 (magenta), one periodic attractor (blue) and four quasi-periodic attractors when c = 2.9.
Table 8.
Lyapunov exponents, Kaplan-Yorke dimensin and dynamics of system (1) coexisisting attractors with parameter c varying.
Fig 17.
Phase portraits of the hyperchaotic system (17): (a) x1 − x2 attractor, (b) x1 − x3 attractor.
Fig 18.
Phase portraits of the hyperchaotic system (18): (a) x1 − x2 attractor, (b) x1 − x3 attractor.
Fig 19.
Phase portraits of the hyperchaotic system (19): (a) x1 − x2 attractor, (b) x1 − x3 attractor.
Fig 20.
Time evolution of the synchronization errors with controllers deactivated (t < 200s) and activated (t > 200s).
Fig 21.
Electronic circuit schematic of the proposed 10-D hyperchaotic system (1).
Fig 22.
Experimental phase portraits of the system (1) x3 − x4 plane.
(a): Periodic orbit, (b): Chaotic attractor and (c): Hyperchaotic attractor.
Fig 23.
Experimental phase portraits of four coexisting attractors: (a), (b) and (c) three chaotic attractors with initial points (0,0,0,0,0,0,0,0,0,0.5), (0,0,0,0,0,0,0,0,0,±1); (d) periodic attractor with initial point (1,0,0,0,0,0,0,0,0,0).