We consider questions of non-uniqueness of cycles in phase portraits of systems of ordinary differential equations of biochemical kinetics with block-linear right-hand sides considered as models of simplest molecular repressilators functioning, and that of other circular gene networks. For these models of different dimensions, conditions of existence of cycles were elaborated previously and stability of these cycles was studied. Now we describe a 3-dimensional dynamical system of this type and three piecewise linear cycles in its phase portrait, as well as their invariant neighborhoods, which are homeomorphic to a torus. This makes possible to localize these cycles and to determine their mutual arrangement.

The smallest of these three cycles is an elementary example of a hidden attractor of a nonlinear dynamical system. Two remaining cycles give examples of non-local oscillations in the phase portrait.

Numerical experiments with this dynamical system illustrate the results. In the previous publications, non-uniqueness of cycles was detected in higher-dimensional cases only, starting from dim = 5.

To study problems with the singular differential Bessel operator B_−γ with a negative parameter −γ ∈ (−1, 0), the paper introduces an integral transformation based on the solution u=Jµ of the singular Bessel equation B_−γ u+u=0, which is expressed through the Bessel function of the first kind with a positive parameter µ= γ+1 . An even and an odd K-Bessel (Hankel–Kipriyanov–Katrakhov) transform as well as a class of singular K-pseudodifferential operators are constructed. The main theorems on the orders of singular K-pseudodifferential operators with symbol from '3m (Sobolev–Kipriyanov function spaces) and a theorem on products and commutators are proved.

The works of F. Tricomi, A. V. Bitsadze, M. M. Smirnov and many other authors are devoted to the study of various boundary value problems for equations of mixed type of second order. In these works, the theory of singular integral equations was used. Since the 1970s, functional methods and methods associated with functional analysis began to be applied to the study of boundary value problems for mixed type equations. The construction of a general theory of boundary value problems for equations of mixed type with an arbitrary variety of changing type began. In particular, under certain assumptions and the sign of the coefficient of the second derivative with respect to time near the bases of the cylindrical region, the existence and uniqueness of a regular solution to the enemy boundary value problem and the first boundary value problem for a second order mixed type equation is proved using the regularization method.

In 2019 A. N. Artyushin proved the existence and uniqueness of a generalized and regular solution to Vragov’s boundary value problem in the weighted Sobolev space, when the coefficient of the second derivative with respect to time can change sign on the bases of a cylindrical domain.

In this work, we will establish the existence of a generalized solution and the unique regular solvability of the first boundary value problem for a second order mixed type equation in the weighted Sobolev space, when the coefficient of the highest derivative of the equation with respect to time can change sign on the lower base and negative on the upper base of the cylindrical domain.

Considering the differential equations of any order with variable coefficients, we study the solvability of nonlocal boundary value problems with the Ionkin classical condition in Sobolev spaces. We prove the unique existence of regular solutions, i.e., those that enter the equations with all weak derivatives.

The work is devoted to investigating the solvability in Sobolev spaces of nonlinear inverse problems of determination, along with the solution u(x, t) of a parabolic equation, the unknown coefficient dependent on time. The studied problems are unique since the original parabolic equation is degenerate. As the integral overdetermination conditions, we use domain-wide integral overdetermination conditions or integral boundary overdetermination conditions. The existence and uniqueness theorems are proved for regular solutions, i.e. the solutions having all generalized derivatives included in the corresponding equation.

We introduce and study the structured pseudospectrum and the essential pseudospectrum of closed linear operator pencils on ultrametric Banach spaces. We establish a characterization of the structured pseudospectrum of closed linear operator pencils and relationship between the structured pseudospectrum and the structured pseudospectrum of closed linear operator pencils on ultrametric Banach spaces. Many characterizations of structured essential pseudospectra of operators, such as the structured essential pseudospectrum of closed linear operator pencils, is invariant under per- turbation of completely continuous linear operators on ultrametric Banach spaces over Qp. Finally, we give some illustrative examples.

We obtain sharp bounds in the generalized Zalcman conjecture for the initial coefficients and the second Hankel determinants H_2,1,k(f ), H_2,2,k(f ) for the k^th-root transformation to the subclass of analytic functions.

We study the numerical solution to the nonlinear heat conduction problem for a plate with a nonlinear heat source (thermal conductivity coefficient and internal heat source are exponential functions of temperature). In particular, for the nonlinear problem, the phenomena of self-similarity, inertia, and heat localization were found, which also manifest themselves in solutions of linear hyperbolic heat equations. With a self-similar change in temperature in some ranges of spatial and temporal variables, similarity (self-similarity) of temperature curves is observed. When heat is localized in a certain range of spatial variable, the temperature does not change over time. The inertia of heat is revealed in the finite speed of its propagation, despite the solution of the parabolic heat equation. The listed phenomena are also observed in solutions of linear hyperbolic heat equations, the derivation of which takes into account the time dependence of the heat flow in the formula of Fourier’s law, leading to a finite rate of heat propagation. In nonlinear problems, a similar effect manifests itself due to the dependence of the physical properties and heat source on temperature, leading to a similar delay in heat flow.

Studying the dynamics of carbon stocks in wetland ecosystems will allow us to more accurately assess the contribution of wetlands to global climate change. This work proposes a zero-dimensional mathematical model that describes the carbon dynamics of a local (at the watershed scale) wetland ecosystem, taking the ambient temperature into account. The proposed model identifies two carbon reservoirs: plant phytomass and organic carbon in mortmass. The main processes of the model include photosynthesis, respiration, phytomass die-off, and carbon leaching by groundwater. Numerical experiments were carried out to show how changes in ambient temperature affect the dynamics of carbon stocks in wetland ecosystems.