Nonlinear Dynamics

, Volume 80, Issue 1–2, pp 969–981 | Cite as

On singular orbits and a given conjecture for a 3D Lorenz-like system

  • Haijun Wang
  • Xianyi Li
Original Paper


We revisit a 3D chaotic system in Dias et al. (Nonlinear Anal Real World Appl 11(5): 3491–3500, 2010) and mainly consider its singular orbits not yet investigated: homoclinic and heteroclinic orbits and singularly degenerate heteroclinic cycles. We first consider the existence of homoclinic and heteroclinic orbits. Our results, one of which shows the existence of two heteroclinic orbits for \(c \ge 2a > 0\) and \(b > 0\), not only further supplement the ones obtained in this literature, but also give something new to theoretically helpfully understand the occurrence of chaos. Further, numerical simulations show that this system has not only two heteroclinic orbits for \(a \le c < 2a, b > 0\) or \(a > c > 0\) and some \( b_{0} \in (0, \frac{a+c}{a-c})\), but also chaotic attractor when heteroclinic orbits disappear. Then, by utilizing a known conclusion, we demonstrate the existence of singularly degenerate heteroclinic cycles in this system. Combining analytical and numerical techniques, it is shown that for the parameter value \(c = 0\) the system presents an infinite set of singularly degenerate heteroclinic cycles, which completely solves a conjecture presented in the above literature for the existence of infinitely many singularly degenerate heteroclinic cycles in the system.


3D Lorenz-like system Boundedness Homoclinic and heteroclinic orbit Singularly degenerate heteroclinic cycle  Lyapunov function 

Mathematics Subject Classification

34C23 34C37 34D08 34D20 



This work is partly supported by NSF of China (grant: 61473340, 10771094), the Postgraduate Innovation Project of Jiangsu Province (grant: KYZZ\(_{-}\)0361) and the NSF of Yangzhou University.


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.College of Mathematical ScienceYangzhou UniversityYangzhouPeople’s Republic of China

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