Two-dimensional exactly and completely integrable dynamical systems

1983 ◽  
Vol 89 (1) ◽  
pp. 59-75 ◽  
Author(s):  
A. N. Leznov ◽  
M. V. Saveliev
Keyword(s):  
Dynamical Systems ◽  
Two Dimensional ◽  
Completely Integrable ◽  
Integrable Dynamical Systems

scholarly journals Qualitative Theory of Two-Dimensional Polynomial Dynamical Systems

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1884
Author(s):  
Yury Shestopalov ◽  
Azizaga Shakhverdiev
Keyword(s):  
Dynamical Systems ◽  
Feasible Solution ◽  
Qualitative Theory ◽  
Single Parameter ◽  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Two Dimensional ◽  
Polynomial Dynamical Systems ◽  
Specific Solution ◽  
Integrable Dynamical Systems

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.

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