Vol 60, No 6 (2024)

In the work of A. Grin and K. Schneider [1] two algebraic transversal ovals, which form the Poincare–Bendixson annulus 𝐴(𝜆), are analytically constructed. This annulus contains the unique limit cycle of the Rayleigh equation ¨ 𝑥+𝜆( ˙ 𝑥2/3−1) ˙ 𝑥+𝑥 = 0 for all values of the parameter 𝜆 > 0. In the constructed annulus 𝐴(𝜆) inner boundary consists of zero-level set of the Dulac–Cherkas function and outer boundary represents the diffeomorphic image corresponding boundary of such an annulus for the van der Pol system. In this paper new methods of construction two Dulac–Cherkas functions are worked out. With help of these functions a better inner boundary of the Poincare–Bendixson annulus 𝐴(𝜆) depending on the parameter is found. Also, for the Rayleigh system, a procedure for direct finding a polynomial whose zero-level set contains a transversal oval used as the outer boundary 𝐴(𝜆) is proposed. Then we determine the interval for 𝜆 where the better outer boundary of the annulus is a closed contour, which composed from two arcs of the constructed oval and two arcs of non-closed curves belonging to zerolevel set of one from Dulac–Cherkas functions. Thus, finally the specified global Poincare–Bendixson annulus for the limit cycle of the Rayleigh system is presented.

The paper considers a second-order quasilinear elliptic equation with an integrable right-hand side. Restrictions on the structure of the equation are formulated in terms of the generalized 𝑁-function. Unlike the author’s previous works, there is no sign condition for the low-order term of the equation. In non-reflexive Musielak–Orlicz–Sobolev spaces in an arbitrary unbounded strictly Lipschitz domain, the existence of a renormalized solution to the Dirichlet problem of this equation is proven.

The method for constructing a feedback matrix to solve the problem spectrum allocation (spectrum control, pole assignment) for linear dynamic system is given. A new proof of the well-known theorem about the connection between the complete controllability of a dynamic system with existence of the feedback matrix is formed in the process of construction by the cascade decomposition method. The entire set of arbitrary elements that influence the non-uniqueness of the matrix is identified. Examples of constructing a feedback matrix in the case of a real spectrum and in the presence of complex conjugate eigenvalues, as well as for the case of multiple eigenvalues, are given. The stability of the specified spectrum under small perturbations of the system parameters with a fixed feedback matrix is studied.

The article develops the theory of stability of linear operator schemes for operator inequalities and nonlinear nonstationary initial-boundary value problems of mathematical physics with nonlinearities of unlimited growth. Based on sufficient conditions for the stability of A.A. Samarsky’s two-layer and three-layer difference schemes, the corresponding a priori estimates for operator inequalities are obtained under the condition of the criticality of the difference schemes under consideration, i.e. when the difference solution and its first time derivative are non-negative at all nodes of the grid region. The results obtained are applied to the analysis of the stability of difference schemes that approximate the Fisher and Klein-Gordon equation with a nonlinear right-hand side.