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.

在本文中，我们通过提供新的见解来分析和几何研究 Maxwell-Bloch 系统的复杂性。首先，用广义梅尔尼科夫方法严格证明了同宿轨道的存在。更准确地说，对于 6 a -2 b > c和d>0，分析证明麦克斯韦-布洛赫系统有两个非横向同宿轨道。其次，从 Kosambi-Cartan-Chern 理论（KCC-theory）的角度考察了 Maxwell-Bloch 系统轨道上的 Jacobi 稳定性。换句话说，根据与重新制定的 Maxwell-Bloch 系统相关的五个对应不变量的偏差曲率张量，我们进一步证明了在适当参数下所有平衡的 Jacobi 稳定性。此外，偏差矢量以及平衡点附近的偏差矢量曲率集中于解释麦克斯韦-布洛赫系统在芬斯勒几何中的混沌行为。