Complex Dynamics of a Simple 3d Autonomous Chaotic System With Four-Wing

A true three-scroll chaotic attractor coined

“…In fact, the aforementioned chaotic attractor corresponds the solution of system the normally hyperbolic stable foci E z as t → ∞, forming singularly degenerate heteroclinic cycles, which further also collapse into three-scroll chaotic attractors when c = 0.11, as shown in Figure 7. This also indicates that the bidirectional forming mechanism of singularly degenerate heteroclinic cycles exists in system (1), which is different from the unilateral one in most other Lorenz-like systems [6,9,15,18,22,36,37,38,39,40,41,42,43,44,45,46].…”

“…Moreover, Figure 1-2 illustrate that the aforementioned three-scroll chaotic attractor coexists with the single saddle-node E 0 . Secondly, since Kokubu and Roussarie [9] introduced the concept of a singularly degenerate heteroclinic cycle that consists of an invariant set formed by a line of equilibria together with a heteroclinic orbit connecting two of the equilibria, many other researchers [6,15,18,22,36,37,38,39,40,41,42,43,44,45,46] have begun to study it, due to the broken of it could create classical and conjugate Lorenzlike attractors, four-wing attractors and so on. When studying a four-dimensional hyperchaotic Lorenz-like system, Wang and Dong [44] recently found that simplex explosions of stable non-isolated equilibria is also one route to chaos/hyperchaos, especially in the case of the original Lorenz attractor and some hyperchaotic attractors.…”

“…Suppose that (x 0 , ξ 0 ) is an equilibrium point of (15) where the Jacobian matrix A has a pair of purely imaginary eigenvalues λ 2,3 = ±ω 0 i, ω 0 > 0, and admits no other eigenvalue with zero real part. Let T c be the generalized eigenspace of A corresponding to λ 2,3 .…”