Numerical study of dynamical system generated by CABC vector field

Purpose of this study is to construct a helical vector field and analyze the dynamical system generated by it. Classic example of such field is the ABC (Arnold-Beltrami-Childress) flow, which is equations stationary solution of the dynamics of ideal incompressible fluid. The article numerically studies the structure of the phase space of a dynamical system determined by the constructed vector field under various assumptions. Methods. When constructing a dynamic system, the approach proposed for helical fields from the class of CABC (Compressible ABC) flows was used. Main research tool is numerical analysis based on the construction and study of Poincare map. For numerical solution of the Cauchy problem, the Runge-Kutta method of the 8th order of accuracy with a constant step is used. Results. For a new example of a helical vector field, analytical expressions are given for its components, and the structure of the phase space of a three-dimensional nonlinear dynamic system generated by it is studied. The integrable case and two types of its perturbation, called «compressible» and «incompressible», are considered. It is shown that the phase space in the presence of perturbations of the first type consists of stationary, periodic, and quasiperiodic trajectories, but has a complex structure - the Poincaré map contains a set of saddle singular points and periodic orbits separated by intertwining separatrices. In the case of «incompressible» perturbation, the dynamics develop according to the scenarios of the KAM theory with the appearance of chaotic regions. Conclusion. The paper presents a new example of a helical vector field, which, under additional conditions on the parameters, turns into a well-known ABC flow. The complex structure of the phase space discovered as a result of calculations can be interpreted as transitional from integrable to non-integrable, despite the absence of chaotic trajectories.