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New dynamics coined in a 4-D quadratic autonomous hyper-chaotic system. (English) Zbl 1428.34060

Summary: This note revisits a 4-D quadratic autonomous hyper-chaotic system in [A. Zarei and S. Tavakoli [ibid. 291, 323–339 (2016; Zbl 1410.34119)] and mainly considers some of its rich dynamics not yet investigated: global boundedness, invariant algebraic surface, singularly degenerate heteroclinic cycle and limit cycle. The main contributions of the work are summarized as follows: firstly, we prove that for \(4a \geq c > 2a > 0, d > 0\) and \(e > 0\) the solutions of that system are globally bounded by constructing a suitable Lyapunov function. Secondly, \(Q = x_3 - \frac{1}{2 a} x_1^2 = 0\) is found to be one of invariant algebraic surfaces with the cofactor \(-4a\) for the model. Thirdly, numerical simulations for \(c = 0\) not only illustrate different types of infinitely many singularly degenerate heteroclinic cycles near which chaotic attractors or limit cycles generate, but also that some of more degenerate (in term of a pure imaginary pair, one zero and one negative eigenvalue) or stable (in sense of three negative eigenvalues and one null eigenvalue) non-isolated equilibria \((0, 0, x_3, 0) (x_3 \in \mathbb{R})\) directly change into the limit cycles or chaotic attractors with a small perturbation of \(c > 0\), which is in the absence of singularly degenerate heteroclinic cycles and degenerate pitchfork bifurcation at the non-isolated equilibria. In particular, some kind of forming mechanism of Lorenz attractor and the hyper-chaotic attractor of that system with \((a, b, c, d, e) = (10, 28, \frac{8}{3}, 1, 16)\) is revealed, which are collapses of singularly degenerate heteroclinic cycles and explosions of stable non-isolated equilibria. Finally, circuit experiment implements the aforementioned hyper-chaotic attractor, showing very good agreement with the simulation results.

MSC:

34C28 Complex behavior and chaotic systems of ordinary differential equations
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
34D45 Attractors of solutions to ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations

Citations:

Zbl 1410.34119
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References:

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