Bioenergy is one of the widespread renewable energy sources. Energy from biomass thermal conversion can reduce greenhouse emissions and have a positive effect on climate change. Biomass conversion is generally carried out in reactors of cylindrical shape. From a modelling point of view many factors have to be taken into account in order to optimize thermal efficiency of the conversion process. One of the important methods for the analysis of complex fluid flows is hydrodynamic stability theory. Base flow solution in classical hydrodynamic stability problems is usually found as a simple analytical solution of the equations of motion. Biomass conversion problems lead to nonlinear boundary value problems, which can be either solved numerically or analyzed using the bifurcation theory. In the present paper we analyze a mathematical model of heat transfer in the presence of nonlinear heat sources. This model includes the study of positive solutions to a nonlinear boundary value problem with certain boundary conditions. The equations in a problem contain several parameters, which essentially affect the behaviour and the number of solutions. Bifurcation analysis of the problem, conducted with respect to the parameters, allows obtaining somewhat precise results on the number of positive solutions. Generally, two, one and zero positive solutions are possible, depending on the values of the parameters. The obtained solutions represent base flow for the hydrodynamic stability problem, which can be solved with the objective to identify the factors affecting the conversion process.

We consider the three-dimensional gene regulatory network (GRN in short). This model consists of ordinary differential equations of a special kind, where the nonlinearity is represented by a sigmoidal function and the linear part is present also. The evolution of GRN is described by the solution vector X(t), depending on time. We describe the changes that system undergoes if the entries of the regulatory matrix are perturbed in some way.

<abstract> <p>A system of ordinary differential equations is considered, which arises in the modeling of genetic networks and artificial neural networks. Any point in phase space corresponds to a state of a network. Trajectories, which start at some initial point, represent future states. Any trajectory tends to an attractor, which can be a stable equilibrium, limit cycle or something else. It is of practical importance to answer the question of whether a trajectory exists which connects two points, or two regions of phase space. Some classical results in the theory of boundary value problems can provide an answer. Some problems cannot be answered and require the elaboration of new approaches. We consider both the classical approach and specific tasks which are related to the features of the system and the modeling object.</p> </abstract>

In the language of mathematics, the method of cognition of the surrounding world in which the description of the object is carried out the name is mathematical modeling. The study of the model is carried out using certain mathematical methods. The systems of the ordinary differential equations modeling artificial neuronal networks and the systems modeling the gene regulatory networks are considered. The one system consists of a sigmoidal function which depends on linear combinations of the arguments minus the linear part. The other system consists of a sigmoidal function which depends on the hyperbolic tangent function. The linear combinations and hyperbolic tangent functions of the arguments are described by one regulatory matrix. For the three-dimensional cases, two types of matrices are considered and the behavior of the solutions of the system is analyzed. The attracting sets are constructed for several cases. Illustrative examples are provided. The list of references consists of 19 items.

It is shown, how to construct a system of ordinary differential equations of arbitrary order, which has the periodic attractor and models some genetic network of arbitrary size. The construction is carried out by combining of multiple systems of lower dimensions with known periodic attractors. In our example the six-dimensional system is constructed, using two identical three-dimensional systems, which have stable periodic solutions.