Abstract:
The oscillatory properties of solutions of an arbitrary nonlinear differential system (with a zero solution) are considered. For such a system, the lower, upper, lowest, and upperest indices of oscillation, wandering, and rotation are determined. The connections of the numerical values of those indices with both the corresponding complete and completest oscillatory properties of that system and with the properties opposite to them: non-oscillation, non-wandering, and non-rotation are studied. The logical connections of all the listed properties with each other are also studied, and the absence of individual such connections is confirmed by concrete counterexamples. In addition, a similar connection is studied between the presence of any oscillatory property in a system and the unit or zero value of the measure of the corresponding property for the system itself (the concepts of such measures of a probabilistic nature for a differential system have only recently been introduced into consideration).
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