Homoclinic orbits and Jacobi stability on the orbits of Maxwell–Bloch system

Abstract

In this paper, we analytically and geometrically investigate the complexity of Maxwell–Bloch system by giving new insight. In the first place, the existence of homoclinic orbits is rigorously proved by means of the generalized Melnikov method. More precisely, for 6a−2b>c and d>0, it is certified analytically that Maxwell–Bloch system has two nontransverse homoclinic orbits. Secondly, Jacobi stability on the orbits of Maxwell–Bloch system is examined in view point of Kosambi–Cartan–Chern theory (KCC-theory). In other words, in the light of the deviation curvature tensor of the five corresponding invariant associated to the reformulated Maxwell–Bloch system, we further proved that Jacobi stability of all equilibria under appropriate parameters. Moreover, the deviation vector, as well as the curvature of the deviation vector near equilibrium points, is focused to interpret the chaotic behavior of Maxwell–Bloch system in Finsler geometry.

Keywords:

• Maxwell–Bloch system
• homoclinic orbit
• KCC-theory
• Jacobi stability
• chaos

2010 Mathematics Subject Classifications:

• 34D35
• 53B40
• 53C60
• 37C29

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 11961074), Natural Science Foundation of Guangxi Province (Grant Nos. 2018GXNSFDA281028, 2017GXNSFAA198234), the High Level Innovation Team Program from Guangxi Higher Education Institutions of China (Document No. [2018] 35), the Youth Project of Hunan Provincial Education Department(Grant Nos. 18B518, 18B082) and the Senior Talent Research Foundation of Yulin Normal University (Grant No. G2019ZK51).

Data Availability

All data generated or analyzed during this study are included in this article.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant number 11961074] and Natural Science Foundation of Guangxi Province [grant numbers 2017GXNSFAA198234, 2018GXNSFDA281028], the High Level Innovation Team Program from Guangxi Higher Education Institutions of China (Document No.[[2018] 35) and Youth Project of Hunan Provincial Education Department [grant numbers 18B082, 18B518] and Senior Talent Research Foundation of Yulin Normal University [grant number G2019ZK51].