CHAOTIC REGIMES GENERATED BY SPECIAL POINTS AND SINGULAR SOLUTIONS

The role of singular solutions and some singular points of ordinary differential equations with respect to the occurrence of chaotic regimes in physical systems is considered. It is shown that in those physical systems in which singular solutions exist, they can cause the occurrence of chaotic regimes. Examples of such systems are given. Of particular interest are the results showing that chaotic regimes can occur even in systems with one degree of freedom and even in completely integrable systems. These results, at first glance, violate known mathematical theorems (Poincare-Bendixson), and also destroy the paradigms of dynamic chaos. It is shown that this contradiction is an apparent contradiction. The reason for the appearance of chaotic dynamics in these cases are special solutions. At the points of this solution, the conditions of the uniqueness theorem are violated. In numerical calculations, this is manifested in the fact that a random component (numerical noise) appears in the calculations. As a result, the system under study becomes a system with 1.5 degrees of freedom. It is shown that in almost all cases, increasing the accuracy of the calculations allows suppressing chaotic dynamics in systems with one degree of freedom.