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March  2020, 25(3): 1097-1108. doi: 10.3934/dcdsb.2019210

## Homoclinic orbits and chaos in the generalized Lorenz system

 School of Information Technology, Jiangxi University of Finance and Economics, Nanchang, Jiangxi 330013, China

Received  October 2018 Revised  May 2019 Published  September 2019

This paper investigates the homoclinic orbits and chaos in the generalized Lorenz system. Using center manifold theory and Lyapunov functions, we get non-existence conditions of homoclinic orbits associated with the origin. The existence conditions of the homoclinic orbits are obtained by Fishing Principle. Therefore, sufficient and necessary conditions of existence of homoclinic orbits associated with the origin are given. Furthermore, with the broken of the homoclinic orbits, we show that the chaos is in the sense generalized Shil'nikov homoclinic criterion.

Citation: Ting Yang. Homoclinic orbits and chaos in the generalized Lorenz system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1097-1108. doi: 10.3934/dcdsb.2019210
##### References:

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##### References:
Two symmetrical homoclinic orbits of system (1)
Chaotic attractor $\mathcal{A}$ of system (1) with $s_1 = 5$, $s_2 = 4$, $d = 1.5$, $q = 2$, $R = 3.08435$
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