Abstract:
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 $n\ge3$ 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.
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