On a set of linear homogeneous differential equations of higher than second order with continuous coefficients on the positive semi-axis, all possible relationships between the principal values of the oscillation exponents (strict and non-strict) of the signs were established, and a study was also conducted on the stability of all principal values with respect to infinitesimal perturbations (i.e., vanishing at infinity) of the equation coefficients. In the work, a multiparameter family of differential equations of a given order n ≥ 3 is constructed, on which strict inequalities between the main values of the characteristic frequencies and oscillation exponents are realized. For fixed values of the sequence of parameters, we highlight points from the indicated family of equations in which all the main values of the oscillation exponents are not invariant under infinitesimal perturbations. In addition, on the set of all non-zero solutions of the specified family of equations all oscillation exponents are exact, absolute and coincide with the exact characteristic frequency of signs. In constructing the specified family of equations and proving the required results, analytical methods of the qualitative theory of differential equations and methods of the theory of perturbations of solutions of linear differential equations were used. In particular, the method of varying an equation, which allows the original differential equation to be transformed in a special way so that it has predetermined properties. Examples of the transition from one differential equation to another are also given in order to generalize the properties of the characteristic frequencies of signs and to exponents of the oscillation of signs.
Keywords: differential equation, linear system, oscillation, number of zeros, exponents of oscillation, Characteristic frequency, stability, Lyapunov’s exponent
Received September 8, 2025
Revised September 27, 2025
Accepted October 6, 2025
Funding Agency: The work was carried out with the financial support of the Ministry of Science and Higher Education of the Russian Federation within the framework of the state assignment No. 075-03-2024-074/5 for the project “Study of asymptotic characteristics of the oscillation of differential equations and systems, as well as optimization methods”.
Angela Evgenievna Artisevich, Adyghe state University, Maykop, 385000 Russia, e-mail: artisevichangela@gmail.com
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Cite this article as: A.E. Artisevich. On the mobility of the main values of the oscillation exponents of the signs of linear differential equations under infinitesimal perturbations. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 4, pp. 26–38.
