Cascade of Invariant Curve Doubling Bifurcations and Quasi-Periodic Hénon Attractor in the Discrete Lorenz-84 Model

Background and Objectives: Chaotic behavior is one of the fundamental properties of nonlinear dynamical systems, including maps. Chaos can be most easily and reliably diagnosed using the largest Lyapunov exponent, which will be positive for the chaotic mode. Unlike flow dynamical systems, the presence of zero Lyapunov exponent in the spectrum is not an obligatory condition for maps. The zero exponent in the spectrum of a map will indicate the possibility of embedding such a map in a flow. In the framework the present paper, using as an example a three-dimensional discrete Lorenz-84 model, it is shown that there can appear chaotic attractors whose Lyapunov exponent spectrum contains one positive, one zero, and one negative exponents. Such a specific attractor represents the production of a two-dimensional torus and the Hénon attractor and was called the Quasi-periodic Hénon attractor. A scenario of development of such kind behavior is an open problem. Materials and Methods: The discrete Lorenz-84 oscillator obtained by discretizing the three-dimensional flow Lorenz-84 model is considered as an object of the present research. The dynamics of the map is studied numerically. The main analysis is carried out using the charts of dynamical regimes based on the calculation of Lyapunov exponents. Lyapunov exponents were calculated by the Benettin method with Gramm-Schmidt orthogonalization. Results: The scenario of occurrence of the Quasi-periodic Hénon attractor via a cascade of invariant curve doubling bifurcations is described. Conclusion: We study the discrete Lorenz-84 oscillator, which is a three-dimensional map (three-dimensional diffeomorphism). In the map the possibility of implementing a steady state of equilibrium, an invariant curve, a torus-attractor, and chaos with zero Lyapunov exponent in the spectrum was shown. It is also demonstrated that the chaotic mode with zero Lyapunov exponent, the Quasi-periodic Hénon attractor, can appear as a result of a cascade of doubling bifurcations of the invariant curve. Typical structures on the parameter plane, such as CrossRoad-Area, Spring-Area, whose base mode is an invariant curve, are illustrated. These structures and chaotic oscillations can arise on the basis of various invariant curves.