Abstract
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.
Similar content being viewed by others
References
Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)
Sparrow, C.: The Lorenz equations: bifurcation, chaos, and strange attractor. Springer, New York (1982)
Ottino, J.M., Leong, C.W., Rising, H., Swanson, P.D.: Morphological structures produced by mixing in chaotic flows. Nature 333(6172), 419–425 (1988)
Alvarez, G., Li, S., Montoya, F., Pastor, G., Romera, M.: Breaking projective chaos sychronization secure communication using filtering and generalized synchronization. Chaos Solitons Fractals 24(3), 775–783 (2005)
Sprott, J.C.: Some simple chaotic flows. Phys. Rev. E 50(2), R647–R650 (1994)
Asakura, H., Takemura, K., Yoshida, Z., Uchida, T.: Collisionless heating of electrons by meandering chaos and its application to a low-pressure plasma source. Jpn. J. Appl. Phys. 36(1), 4493–4496 (1997)
Chen, G.: Controlling chaos and bifurcations in engineer system. CRC Press, Boca Raton (1999)
Dias, F.S., Mello, L.F., Zhang, J.G.: Nonlinear analysis in a Lorenz-like system. Nonlinear Anal. RWA 11(5), 3491–3500 (2010)
Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurcat. Chaos 9(7), 1465–1466 (1999)
Rössler, E.: An equation for continuous chaos. Phys. Lett. A 57(5), 397–398 (1976)
Rikitake, T.: Oscillation of a system of disk dynamos. Proc. Camb. Philos. Soc. 54, 89–105 (1958)
Liu, C., Liu, T., Liu, L.: A new chaotic attractor. Chaos Solitons Fractals 22(5), 1031–1038 (2004)
Qiao, Z., Li, X.: Dynamical analysis and numerical simulation of a new Lorenz-type chaotic system. Math. Comput. Model. Dyn. Syst. 20(3), 264–283 (2014)
Li, X., Zhou, Z.: Hopf bifurcation of codimension one and dynamical simulation for a 3D autonomous chaotic system. Bull. Korean Math. Soc. 51(2), 457–478 (2014)
Ilyashenko, Y.: Finiteness theorems for limit cycles. American Mathematical Society, Providence (1993)
Ferragut, A., Llibre, J., Pantazi, C.: Polynomial vector fields in \({\mathbb{R}}^3\) with infinitely many limit cycles. Int. J. Bifurcat. Chaos 23(2), 1350029 (2013)
Li, X., Chu, Y., Zhang, J., Chang, Y.: Nonliear dynamics and circuit implementation for a new Lorenz-like attractor. Chaos Solitons Fractals 41(5), 2360–2370 (2009)
Kokubu, H., Roussarie, R.: Existence of a singularly degenerate heteroclinic cycle in the Lorenz system and its dynamical consequences: part I. J. Dyn. Differ. Equ. 16(2), 513–557 (2004)
Llibre, J., Messias, M.: Global dynamics of the Rikitake system. Phys. D 238(3), 241–252 (2009)
Messias, M.: Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system. J. Phys. A Math. Theor. 42(11), 115101 (2009)
Llibre, J., Messias, M., Silva, P.R.: On the global dynamics of the Rabinovich system. J. Phys. A Math. Theor. 41(27), 275210 (2008)
Messias, M.: Dynamics at infinity of a cubic Chuas system. Int. J. Bifurcat. Chaos 21(1), 333–340 (2011)
Liu, Y., Yang, Q.: Dynamics of the Lü system on the invariant algebraic surface and at infinity. Int. J. Bifurcat. Chaos 21(9), 2559–2582 (2011)
Zhou, T., Cheng, G.: Classification of chaos in 3-D autonomous quadratic system-I: basic framework and methods. Int. J. Bifurcat. Chaos 16(9), 2459–2479 (2006)
Wang, X., Chen, G.: A chaotic system with only one stable equilibrium. Commun. Nonlinear Sci. Numer. Simul. 17(3), 1264–1272 (2012)
Kuzenetsov, Y.A.: Elements of applied bifurcation theory, 3rd edn. Springer, New-York (2004)
Mees, A.I., Chapman, P.B.: Homoclinic and heteroclinic orbits in the double scroll attractor. IEEE Trans. Circuits Syst. 34(9), 1115–1120 (1987)
Tigan, G., Constantinescu, D.: Heteroclinic orbits in the \(T\) and the Lü system. Chaos Solitons Fractals 42(1), 20–23 (2009)
Wiggins, S.: Global bifurcations and chaos: analytical methods. Springer, New York (1988)
Li, T., Chen, G., Chen, G.: On homoclinic and heteroclinic orbits of Chen’s system. Int. J. Bifurcat. Chaos 16(10), 3035–3041 (2006)
Li, X., Wang, H.: Homoclinic and heteroclinic orbits and bifurcations of a new Lorenz-type system. Int. J. Bifurcat. Chaos 21(9), 2695–2712 (2011)
Liu, Y., Yang, Q.: Dynamics of a new Lorenz-like chaotic system. Nonlinear Anal. Real World Appl. 11(4), 2563–2572 (2010)
Li, X., Ou, Q.: Dynamics of a new Lorenz-like chaotic system. Nonlinear Dyn. 65(3), 255–270 (2011)
Li, X., Wang, P.: Hopf bifurcation and heteroclinic orbit in a 3D autonomous chaotic system. Nonlinear Dyn. 73(1–2), 621–632 (2013)
Yang, Q., Wei, Z.: An unusual 3D autonomous quadratic chaotic system with two stable node-foci. Int. J. Bifurcat. Chaos 20(4), 1061–1083 (2010)
El-Dessokya, M.M., Yassen, M.T., Saleh, E., Aly, E.S.: Existence of heteroclinic and homoclinic orbits in two different chaotic dynamical systems. Appl. Math. Comput. 218(24), 11859–11870 (2012)
Hale, J.K.: Ordinary diferential equations. Wiley, New York (1969)
Silva, C.P.: Shil’nikov’s theorem—a tuitorial. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 40(10), 675–682 (1993)
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, H., Li, X. On singular orbits and a given conjecture for a 3D Lorenz-like system. Nonlinear Dyn 80, 969–981 (2015). https://doi.org/10.1007/s11071-015-1921-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-015-1921-8
Keywords
- 3D Lorenz-like system
- Boundedness
- Homoclinic and heteroclinic orbit
- Singularly degenerate heteroclinic cycle
- Lyapunov function