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Qualitative properties and two strong resonances of a discrete reduced Lorenz system
Journal of Difference Equations and Applications ( IF 1.1 ) Pub Date : 2021-06-29 , DOI: 10.1080/10236198.2021.1944124
Jiyu Zhong 1
Affiliation  

In this paper, we discuss the dynamics of a discrete reduced Lorenz system. At first, applying the centre manifold reduction, computing normal form, using Takens's theorem and the equivalence between the mappings and the time 1 mappings of flows, we investigate the topological types of a fixed point for the system, including hyperbolic and non-hyperbolic. Then, we prove that the system undergoes the 1:2 resonance and 1:3 resonance and present all bifurcations as the parameters vary near the 1:2 resonance point and the 1:3 one, respectively. At last, by our results, we numerically simulate the scenes of bifurcations as the parameters vary near the 1:2 resonance point and the 1:3 resonance point, respectively. Furthermore, we show by numerical simulation that near the 1:3 resonance point the system possesses more plentiful dynamical properties, including invariant sets formed by three circles, 12-periodic cycles with four points on each circle, 3-coexisting chaotic attractors and full chaotic attractors.



中文翻译:

离散约简洛伦兹系统的定性性质和两个强共振

在本文中,我们讨论离散约简洛伦兹系统的动力学。首先,应用中心流形约简,计算范式,利用Takens定理以及流的映射和时间1映射之间的等价性,我们研究了系统不动点的拓扑类型,包括双曲和非双曲。然后,我们证明系统经历了 1:2 共振和 1:3 共振,并且当参数分别在 1:2 共振点和 1:3 共振点附近变化时呈现所有分叉。最后,根据我们的结果,当参数分别在 1:2 共振点和 1:3 共振点附近变化时,我们数值模拟了分叉的场景。此外,我们通过数值模拟表明,在 1:3 共振点附近,系统具有更丰富的动力学特性,

更新日期:2021-08-15
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