Singularly Degenerate Heteroclinic Cycles with Nearby Apple-Shape Attractors

Abstract

Compared with most known singularly degenerate heteroclinic cycles consisting of two different equilibria of a line or a curve, or two parallel lines of semi-hyperbolic equilibria, little seems to be noticed about the one that connects two perpendicular lines of semi-hyperbolic equilibria, i.e. Ez = (0, 0,z) and Ex = (x,ex,k + ef), z,x , which is found in the mathematical chaos model: = a(y - x) + dxz, = kx + fy - xz, z = -ex2 + xy + cz when c = 0 and e(a + fd) = (a - kd). Surprisingly, apple-shape attractors are also created nearby that kind of singularly degenerate heteroclinic cycles in the case of small c > 0. Further, some other rich dynamics are uncovered, i.e. global boundedness, Hopf bifurcation, limit cycles coexisting with one chaotic attractor, etc. We not only prove that the ultimate bound sets and globally exponentially attracting sets perfectly coincide under the same parameters, but also illustrate four limit cycles coexisting with one chaotic attractor, the saddle in the origin, and other two stable nontrivial node-foci, which are also trapped in the obtained globally exponentially attracting set, extending the recently reported results of the Lü-type subsystem. In addition, combining theoretical analysis and numerical simulation, the bidirectional forming mechanism of that kind of singularly degenerate heteroclinic cycles is illustrated, and their collapses may create three-scroll/apple-shape attractors, or limit cycles, etc. Finally, conservative chaotic flows are numerically found in the new system. We expect that the outcome of the study may provide a reference for subsequent research.