Abstract
A qualitative theory of two-dimensional quadratic-polynomial integrable dynamical systems (DSs) is constructed on the basis of a discriminant criterion elaborated in the paper. This criterion enables one to pick up a single parameter that makes it possible to identify all feasible solution classes as well as the DS critical and singular points and solutions. The integrability of the considered DS family is established. Nine specific solution classes are identified. In each class, clear types of symmetry are determined and visualized and it is discussed how transformations between the solution classes create new types of symmetries. Visualization is performed as series of phase portraits revealing all possible catastrophic scenarios that result from the transition between the solution classes.
Highlights
dynamical systems (DSs), in particular two-dimensional ones, are widely used to model various processes described by several dependent quantities varying in time [1,2]
We prove the integrability of the addressed DS family and show that qualitatively, the solutions of the DSs under study can be divided into nine classes defined using one real parameter and that all the DS evolution scenarios are associated with the transitions between these solution classes
Visualization is performed as series of phase portraits revealing all possible catastrophic scenarios that result from the transition between the solution classes
Summary
DSs, in particular two-dimensional ones, are widely used to model various processes described by several dependent quantities varying in time [1,2]. A mission of particular interest is to distinguish families of DSs, integrable, for which one can construct a complete qualitative theory including all possible solution scenarios and end up with a complete collection (atlas) of phase portraits This task may be solved by considering autonomous two-dimensional polynomial DSs [8,9] which still produce a variety of unsolved problems [9,10]. One can sort out among them a set of DSs integrable in elementary functions with the right-hand sides comprising quadratic trinomials This will be a six-parameter DS family with several possibilities of polynomial factorization which generates a rich variety of solutions possessing comprehensive qualitative behavior.
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