MMN-4489
Conjoined Lorenz-like attractors coined
Haijun Wang; Guiyao Ke; Jun Pan; Qifang Su;Abstract
In contrast to most other periodically forced chaotic systems with infinitely many isolated coexisting strange attractors, little seems to be known about the ones that possess conjoined Lorenz-like attractors with potential existence of infinitely many pairs of wings/scrolls. To achieve this target, this note proposes a new periodically forced extended Lorenz-like system: ˙x = y, ˙y =asin(2x) − by + c1sin(2x)z + c2sin(2x)sin2(x), ˙z = −d1z + d2sin2(x). For d1 = 0, that system generates infinitely many singularly degenerate heteroclinic cycles or heteroclinic orbits to any two equilibria of a family of semi-hyperbolic lines Ez = (kπ, 0, z) (z∈R, k∈N), the collapses of which create not only the desirable conjoined Lorenz-like attractors, but also infinitely many isolated coexisting ones. What is more, the state variable x of that conjoined Lorenz-like attractor presents stochastic behaviors, confirming the links between long period and chaos. This also generalizes the classical concept of boundedness of chaos, i.e., the system orbits beginning from one sub-two-scroll of that conjoined Lorenz-like attractor might tend to the ones at infinity. Apart from those, the existence of an invariant algebraic surface and a family of infinitely many pairs of symmetrical heteroclinic orbits is proved by utilizing the Lyapunov function, the definitions of both α-limit set and ω-limit set.
Vol. 26 (2025), No. 1, pp. 527-546