Qualitative properties and two strong resonances of a discrete reduced Lorenz system

Abstract

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.