| Citation: |
Jiaopeng Yang, Pengxian Zhu. A NOVEL 5D SYSTEM GENERATED INFINITELY MANY HYPERCHAOTIC ATTRACTORS WITH THREE POSITIVE LYAPUNOV EXPONENTS[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2843-2873. doi: 10.11948/20230016
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A NOVEL 5D SYSTEM GENERATED INFINITELY MANY HYPERCHAOTIC ATTRACTORS WITH THREE POSITIVE LYAPUNOV EXPONENTS
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Abstract
Little seems to be known about the five-dimensional (5D) differential dynamical system with infinitely many hyperchaotic attractors, which have three positive Lyapunov exponents under no or infinitely many equilibria. This article presents a 5D dynamical system that can generate infinitely many hyperchaotic attractors. Of particular interest is the system exists not only infinitely many hyperchaotic attractors but also infinitely many periodic attractors in the following three cases: (ⅰ) no equilibria, (ⅱ) only infinitely many non-hyperbolic equilibria, (ⅲ) only infinitely many hyperbolic equilibria. By numerical analysis, one finds the 5D system could generate infinitely many coexisting hyperchaotic or chaotic or periodic attractors in the three kinds of equilibria cases. And one obtains the global dynamical behavior of the system, such as the Lyapunov exponential spectrum, bifurcation diagram. To study the hyperchaotic complexity of the 5D system, we rigorously show the stability of hyperbolic equilibria and some mathematical characterization for 5D Hopf bifurcation. In particular, the existence of an infinite number of isolated bifurcated periodic orbits is strictly proven. These complex dynamics studies in this paper may further contribute to a deep understanding of the hyperchaotic systems with infinitely many attractors.
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Keywords:
- Hyperchaos /
- multistability /
- equilibria and stability /
- coexisting attractors /
- Hopf bifurcation
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References
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Jiaopeng Yang, Pengxian Zhu. A NOVEL 5D SYSTEM GENERATED INFINITELY MANY HYPERCHAOTIC ATTRACTORS WITH THREE POSITIVE LYAPUNOV EXPONENTS[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2843-2873. doi: 10.11948/20230016
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Figure 1.
System (2.1) with parameter set
$ (A_1) $ $ \mathcal{A}_0 $ $ x-v $ $ y=0 $ -
Figure 2.
System (2.1) with parameters set
$ (A_2) $ $ \mathcal{B}_0 $ $ x-v $ $ y=0 $ -
Figure 3.
System (2.1) with parameter set
$ (A_3) $ $ \mathcal{C}_0 $ $ x-v $ $ y=0 $ -
Figure 4.
Coexistence of hidden hyperchaotic attractors
$ \mathcal{A}_m $ $ m=0, \pm1, \pm2, \pm3 $ $ x-z-v $ $ z-x $ $ v=0 $ -
Figure 5.
Projection of attractor on
$ z-y-v $ $ \mathcal{D}_0^{(1)} $ $ \mathcal{D}_m^{(1)} $ -
Figure 6.
Projection of attractor on
$ z-y-v $ $ \mathcal{D}_0^{(2)} $ $ \mathcal{D}_m^{(2)} $ -
Figure 7.
System (2.1) with parameter set
$ (A_2) $ $ \mathcal{B}_m $ $ z-x $ $ v=0 $ -
Figure 8.
Projection of attractor on
$ z-y-v $ $ \mathcal{E}_0^{(1)} $ $ \mathcal{E}_m^{(1)} $ -
Figure 9.
Projection of attractor on
$ z-y-v $ $ \mathcal{E}_0^{(2)} $ $ \mathcal{E}_m^{(2)} $ -
Figure 10.
System (2.1) with parameters set
$ (A_3) $ $ \mathcal{C}_m $ $ z-v $ $ y=0 $ -
Figure 11.
Projection of attractor on
$ z-y-v $ $ \mathcal{F}_0^{(1)} $ $ \mathcal{F}_m^{(1)} $ -
Figure 12.
Projection of attractor on
$ z-y-v $ $ \mathcal{F}_0^{(2)} $ $ \mathcal{F}_m^{(2)} $ -
Figure 13.
Projection of periodic attractor: (a) coexistence of periodic attractor
$ \mathcal{G}_m $ $ z-y-v $ $ \mathcal{G}_m $ $ z-y-x $ -
Figure 14.
System (2.1) with parameters
$ a=1 $ $ c=5 $ $ d=-3 $ $ e=10 $ $ f=-10 $ $ g=3 $ $ h_1=-2 $ $ h_2=1 $ $ l=-0.2 $ $ k=-4.5 $ $ b\in [-10, -1] $ -
Figure 15.
System (2.1) with parameters
$ a=1 $ $ b=-1.12 $ $ d=-3 $ $ e=5 $ $ f=-10 $ $ g=3 $ $ h_1=-2 $ $ h_2=1 $ $ k=-4.5 $ $ l\in [0.5, 5] $ $ c=\frac{1}{gl}(egk-dfk) $ -
Figure 16.
System (2.1) with parameters
$ b=-1.12 $ $ c=5 $ $ d=-3 $ $ e=10 $ $ f=-10 $ $ g=3 $ $ h_1=-2 $ $ h_2=1 $ $ l=8 $ $ k=-4.5 $ $ a\in [4.7, 13] $ -
Figure 17.
System (2.1) with parameters
$ ( a, c, d, e, f, g, h_1, h_2, l, k) =(1, 5, -2, 2, -1, 2, 1, -1, 1, -2) $ $ b=-1.92 $ $ b=-2 $