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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.

We consider boundary value problems of the type x'' = f(t, x, x'), (∗) x(a) = A, x(b) = B. A solution ξ(t) of the above BVP is said to be of type i if a solution y(t) of the respective equation of variations y'' = fx(t, ξ(t), ξ' (t))y + fx' (t, ξ(t), ξ' (t))y' , y(a) = 0, y' (a) = 1, has exactly i zeros in the interval (a, b) and y(b) 6= 0. Suppose there exist two solutions x1(t) and x2(t) of the BVP. We study properties of the set S of all solutions x(t) of the equation (∗) such that x(a) = A, x'1(a) ≤ x' (a) ≤ x'2(a) provided that solutions extend to the interval [a, b].

We study the structure of attractors in the two-dimensional dynamical system that appears in the network control theory. We provide description of the attracting set and follow changes this set suffers under the changes of positive parametersµ andΘ.&nbsp

Boundary value problems of the formare considered, whereIn our considerations functionsfandgare generally nonlinear. We give a description of a solution set of the problem (i), (ii). It consist of all triples () such that (λ,μ,x(t)) nontrivially ′solves the problem(i),(ii) and |x(z)| = α at zero pointszof the functionx(t) (iii). We show that this solution set is a union of solution surfaces which are centro-affine equivalent. Each solution surface is associated with nontrivial solutions with definite nodal type. Properties of solution surfaces are studied. It is shown, in particular, that solution surface associated with solutions with exactlyizeroes in the interval (a,b) is centro-affne equivalent to a solution surface of the Dirichlet problem (i),x(a) = 0 =x(b), (iii) corresponding to solutions with odd number of zeros 2j− 1 (i≠ 2j)in the interval (a,b).

We consider the second-order nonlinear boundary value problems (BVPs) with Sturm–Liouville boundary conditions. We define types of solutions and show that if there exist solutions of different types then there exist intermediate solutions also.

The Liénard type equationx''+f(x, x')x'+g(x)= 0 (i) is considered. We claim that if the associated conservative equationx'' + g(x)= 0 has period annuli then a dissipationf(x, x')exists such that a limit cycle of equation (i) exists in a selected period annulus. Moreover, it is possible to definef(x, x')so that limit cycles appear in all period annuli. Examples are given. A particular example presents two limit cycles of non-convex shape in two disjoint period annuli.

We state the conditions of geometrical nature which guarantee the existence of a solution to the boundary value problemx′′+ 2δx′+ λf(x+) −µg(x−) =h(t, x, x′),x(0) = 0 =x(1) with a damping term 2δx′and nonnegative parameters λ, µ, provided thatf(x+) −g(x−) is a sector-bounded nonlinearity.

The boundary value problemis considered provided thatf: [0, +∞) → [0, +∞) is Lipschitzian andis continuous and Lipschitzian inxandx′. We assume thatfis bounded by two linear functionskxandlx, wherek>l> 0, andhis bounded. We find the conditions on (λ, µ) which guarantee the existence of a solution to the problem. These conditions are of geometrical nature.

We consider two types of nonlinear boundary value problems involving parameters. The second type of problems includes the Fučík problem. Properties of spectra are discussed in connection with the number of solutions to BVP. First published online:14 Oct 2010

Eigenvalue problems of the formx” = −λf(x+)+ μg(x−),x‘(a) = 0,x'(b) = 0 are considered. We are looking for (λ,μ) such that the problem (i), (ii) has a nontrivial solution. This problem generalizes the famous Fučík problem for piece‐wise linear equations. In our considerations functionsfandgmay be nonlinear. Consequently spectra may differ essentially from those for the Fučík equation. First published online:14 Oct 2010

Eigenvalue problems of the formx″=-λf(x+) + μg(x-), x(0)= 0,x(1)= 0 are considered, wherex+andx−are respectively the positive and the negative parts ofx. We are looking for (λ, μ) such that the problem has a nontrivial solution. This problem generalizes the famous Fučik problem for piece‐wise linear equations. In order to show that nonlinear Fučik spectra may differ essentially from the classical ones, we consider piece‐wise linear functionsf(x)andg(x). We show that the first branches of the Fučik spectrum may contain bounded components. First Published Online:14 Oct 2010

We consider two second order autonomous differential equations with critical points, which allow the detection of an exact number of solutions to the Dirichlet boundary value problem. Non‐autonomous equations with similar behaviour of solutions also are considered. Estimations from below of the number of solutions to the Dirichlet boundary value problem are given. First Published Online: 14 Oct 2010

We consider the Emden‐Fowler equationx” = ‐q(t)|x|2εx,ε> 0, in the interval [a,b]. The coefficientq(t)is a positive valued continuous function. The Nehari characteristic number Anassociated with the Emden‐Fowler equation coincides with a minimal value of the functional [] over all solutions of the boundary value problemx” = ‐q(t)|x|2εx, x(a) = x(b)= 0,x(t)has exactly (n ‐ 1) zeros in (a,b). The respective solution is called the Nehari solution. We construct an example which shows that the Nehari extremal problem may have more than one solution. First Published Online: 14 Oct 2010