ANALYTICAL PROPERTIES OF SOLUTIONS OF A FAMILY OF THREE-DIMENSIONAL DYNAMIC DISSIPATIVE FIVE-LEMENT SYSTEMS WITH ONE QUADRATIC NONLINEARITY

Abstract

The object of the study is a family of three-dimensional dynamic five-element dissipative systems with one quadratic nonlinearity, an arbitrary parameter A and a parameter e, e² = 1. In systems of the specified family, the parameter A is included as a multiplier with a linear element (systems of the first class), or as a separate constant element (systems of the second class). A characteristic feature (from a qualitative point of view) of this family is the presence in it of systems with chaotic behavior, in particular, with strange attractors. The purpose of the study is to determine the nature of the moving singular points of solutions of the specified family. To analyze solutions to systems of the family under consideration, the Painlevé test was used, as well as reducing the systems to equivalent second- or third-order equations and comparing the latter with known nonlinear P‑type equations. Solutions of systems of the first class do not have the Painlevé property (despite the fact that the components of the solutions of some of them do not have moving singular points at all), or do not satisfy the Painlevé test. Similarly, solutions of systems of the second class either do not satisfy the Painlevé test or do not possess the Painlevé property, despite the fact that the components of the solutions of some systems do not have moving singular points at all. The presence of systems with chaotic behavior among the systems under consideration allows us to indicate autonomous third-order differential equations with chaotic behavior.