Abstract:
The subject of the research of this work is at the intersection of two directions in the qualitative theory of differential equations — the theory of Lyapunov exponents and the theory of oscillation. In the present work, we investigate various types of oscillation exponents (strict and non-strict) of the signs of solutions of linear homogeneous differential equations of the third order with coefficients continuous on the positive semi-axis. Structurally, a multiparameter family of third-order differential equations is constructed in the work, on which various relationships between the main values of the oscillation exponents are realized. For fixed values of the sequence of parameters, points are obtained from the specified family of equations, in which all the main values of the oscillation exponents are not invariant with respect to infinitesimal perturbations (i.e., vanishing at infinity). In addition, on the set of all non-zero solutions of the specified family of equations, all oscillation exponents coincide with each other. When constructing the specified equation and proving the required results, analytical methods of the qualitative theory of differential equations and methods of perturbation theory of solutions of linear differential equations, in particular, the method of equation variation, were used.
Keywords:differential equation, linear system, oscillation, number of zeros, oscillation exponent, characteristic frequency, Lyapunov exponent
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