Complex dynamics of a novel 3D autonomous system without linear terms having line of equilibria: coexisting bifurcations and circuit design

Abstract

In recent years, an increasing interest has been devoted to the discovery of new chaotic systems with special properties. In this paper, a novel and singular 3D autonomous system without linear terms is introduced. The singularity of the model is that it is dissipative, possesses rotation symmetry and line of equilibria thus displays complex dynamics. The nonlinear behaviour of the introduced model is studied in terms of bifurcation diagrams, Lyapunov exponent plots, time series, frequency spectra two-parameter diagrams, Lyapunov stability diagrams as well as basins of attraction. Some interesting phenomena are found including, for instance, periodic oscillations, chaotic oscillations, periodic windows, symmetry restoring crises scenario, the coexistence of multiple bifurcations and offset-boosting property while monitoring the system parameters. Coexistence of attractors discovered in this work includes up to six competing attractors with one point attractor. Compared to some few cases previously reported (system without linear terms) (Xu and Wang in Opt Int J Light Electron Opt 125:2526–2530, 2014. https://doi.org/10.1007/s11071-016-3170-x; Kengne in Commun Nonlinear Sci Numer Simul 52:62–76, 2017. https://doi.org/10.1016/j.cnsns.2017.04.017; Mobayen et al. in Int J Syst Sci 49:1–15, 2018. https://doi.org/10.1080/00207721.2017.1410251; Zhang et al. in Int J Non-linear Mech 106:1–12, 2018. https://doi.org/10.1016/j.ijnonlinmec.2018.08.012; Pham et al. in Chaos Solitons Fractals 120:213–221, 2019. https://doi.org/10.1016/j.chaos.2019.02.003), the model considered in this work is the only one in which such type of complex dynamics has already been found. A suitable analog simulator (electronics circuit) is designed and used to support the theoretical analysis.