Bifurcations, ultimate boundedness and singular orbits in a unified hyperchaotic Lorenz-type system

Abstract

In this note, by using the theory of bifurcation and Lyapunov function, one performs a qualitative analysis on a novel four-dimensional unified hyperchaotic Lorenz-type system (UHLTS), including stability, pitchfork bifurcation, Hopf bifurcation, singularly degenerate heteroclinic cycle, ultimate bound estimation, global exponential attractive set, heteroclinic orbit and so on. Numerical simulations not only are consistent with the results of theoretical analysis, but also illustrate singularly degenerate heteroclinic cycles with distinct geometrical structures and nearby hyperchaotic attractors in the case of small \begin{document}$ b > 0 $\end{document} , i.e. conjugate hyperchaotic Lorenz-type attractors (CHCLTA) and nearby a short-duration transient of singularly degenerate heteroclinic cycles approaching infinity or singularly degenerate heteroclinic cycles consisting of normally hyperbolic saddle-foci and stable node-foci, etc. In particular, by a linear scaling, a possibly new forming mechanism behind the creation of well-known hyperchaotic attractor with \begin{document}$ (a_{1}, a_{2}, c, d, e, f, b, p, q) = (-12, 12, 23,-1, -1, 1, 2.1, -6, -0.2) $\end{document} , consisting of occurrence of degenerate pitchfork bifurcation at \begin{document}$ S_{z} $\end{document} , the change in the stability index of the saddle at the origin as \begin{document}$ b $\end{document} crosses the null value, explosion of normally hyperbolic stable node-foci, collapse of singularly degenerate heteroclinic cycles consisting of normally hyperbolic saddle-foci or saddle-nodes, and stable node-foci, is revealed. The findings and results of this paper may provide theoretical support in some future applications, since they improve and complement the known ones.