New dynamics coined in a 4-D quadratic autonomous hyper-chaotic system

This note revisits a 4-D quadratic autonomous hyper-chaotic system in Zarei and Tavakoli (2016) 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 >= 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) - 1/2a 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) is an element of 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, 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. (C) 2018 Elsevier Inc. All rights reserved.