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For a linear ordinary differential equation (ODE in short) of the third order, results are presented that supplement the theory of conjugate points and extremal solutions by W. Leighton, Z. Nehari, M. Hanan. It is especially noted the sensitivity of solutions to the initial data, which makes their numerical study difficult. Similar results were obtained for the third-order nonlinear equations of the Emden-Fowler type.

The comparative analysis of systems of ordinary differential equations, modeling gene regulatory networks and neuronal networks, is provided. In focus of the study are asymptotical behavior of solutions, types of attractors. Emphasis is made on the chaotic behavior of solutions.

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.

In the theory of gene networks, the mathematical apparatus that uses dynamical systems is fruitfully used. The same is true for the theory of neural networks. In both cases, the purpose of the simulation is to study the properties of phase space, as well as the types and the properties of attractors. The paper compares both models, notes their similarities and considers a number of illustrative examples. A local analysis is carried out in the vicinity of critical points and the necessary formulas are derived.

This paper introduces a new 3D chaotic attractor in a gene regulatory network. The proposed model has eighteen parameters. Formulas for characteristic numbers of critical points for three-dimensional systems were considered. We show that the three equilibrium points of the new chaotic 3D system are unstable and deduce that the three-dimensional system exhibits chaotic behavior. The possible outcomes of this 3D model were compared with the results of the Chua circuit. The bifurcation structures of the proposed 3D system are investigated numerically, showing periodic solutions and chaotic solutions. Lyapunov exponents and Kaplan-Yorke dimension are calculated. For calculations, the Wolfram Mathematica is used.

The linear stability of a convective flow in a vertical fluid layer caused by nonlinear heat sources in the presence of cross-flow through the walls of the channel is investigated in this paper. This study is relevant to the analysis of factors that affect the effectiveness of biomass thermal conversion. The nonlinear problem for the base flow temperature is investigated in detail using the Krasnosel’skiĭ–Guo cone expansion/contraction theorem. It is shown that a different number of solutions can exist depending on the values of the parameters. Estimates for the norm of the solutions are obtained. The linear stability problem is solved numerically by a collocation method based on Chebyshev polynomials. It is shown that the increase in the cross-flow intensity stabilizes the flow, but there is also a small region of the radial Reynolds numbers where the flow is destabilized.

The question of targeted control over trajectories of systems of differential equations encountered in the theory of genetic and neural networks is considered. Examples are given of transferring trajectories corresponding to network states from the basin of attraction of one attractor to the basin of attraction of the target attractor. This article considers a system of ordinary differential equations that arises in the theory of gene networks. Each trajectory describes the current and future states of the network. The question of the possibility of reorienting a given trajectory from the initial state to the assigned attractor is considered. This implies an only partial control of the network. The difficulty lies in the selection of parameters, the change of which leads to the goal. Similar problems arise when modeling the response of the body’s gene networks to serious diseases (e.g., leukemia). Solving such problems is the first step in the process of applying mathematical methods in medicine and pharmacology.

Linear stability analysis of a combined convective flow in an annulus is performed in the paper. The base flow is generated by two factors: (a) different constant wall temperatures and (b) heat release as a result of a chemical reaction that takes place in the fluid. The nonlinear boundary value problem for the distribution of the base flow temperature is analyzed using bifurcation analysis. The linear stability problem is solved numerically using a collocation method. Two separate cases are considered: Case 1 (non-zero different constant wall temperatures) and Case 2 (zero wall temperatures). Numerical calculations show that the development of instability is different for Cases 1 and 2. Multiple minima on the marginal stability curves are found for Case 1 as the Prandtl number increases. Concurrence between local minima leads to the selection of the global minimum in such a way that a finite jump in the value of the wave number is observed for some values of the Prandtl number. All marginal stability curves for Case 2 have one minimum in the range of the Prandtl numbers considered. The corresponding critical values of the Grashof number decrease monotonically as the Prandtl number grows.

We consider a third order system ${\boldsymbol x}'''={\boldsymbol f}({\boldsymbol x})$ with the two-point boundary conditions ${\boldsymbol x}(0)={\boldsymbol 0}$, ${\boldsymbol x}'(0)={\boldsymbol 0}$, ${\boldsymbol x}(1)={\boldsymbol 0}$, where ${\boldsymbol f}({\boldsymbol 0})={\boldsymbol 0}$ and the vector field ${\boldsymbol f}\in C^1(\mathbb{R}^n,\mathbb{R}^n)$ is asymptotically linear with the derivative at infinity ${\boldsymbol f}'(\infty)$. We introduce an asymptotically linear vector field ${\boldsymbol \phi}$ such that its singular points (zeros) are in a one-to-one correspondence with the solutions of the boundary value problem. Using the vector field rotation theory, we prove that under the non-resonance conditions for the linearized problems at zero and infinity the indices of ${\boldsymbol \phi}$ at zero and infinity can be expressed in the terms of the eigenvalues of the matrices ${\boldsymbol f}'({\boldsymbol 0})$ and ${\boldsymbol f}'(\infty)$, respectively. This proof constitutes an essential part of our article. If these indices are different, then standard arguments of the vector field rotation theory ensure the existence of at least one nontrivial solution to the boundary value problem. At the end of the article we consider the consequences for the scalar case.

We consider systems of the form $$\displaylines{ x_1''+ g_1(x_1) = h_1(x_1,x_2,\ldots,x_n),\cr x_2''+ g_2(x_2) = h_2(x_1,x_2,\ldots,x_n),\cr \cdots \cr x_n''+ g_n(x_n) = h_n(x_1,x_2,\ldots,x_n) }$$ along with the boundary conditions $$ x_1(0)=x_2(0)=\dots=x_n(0)=0=x_1(1)=x_2(1)=\dots=x_n(1)\,. $$ We assume that right sides are bounded continuous functions, and satisfy $h_i(0,0,\ldots,0)=0$. Also we assume that $g_i(x_i)$ are asymptotically asymmetric functions. By using vector field rotation theory, we provide the existence of solutions.

We consider period annuli (continua of periodic solutions) in equations of the type $x''+g(x)=0$ and $x''+f(x) x'^2 + g(x)= 0,$ where $g$ and $f$ are polynomials. The conditions are provided for existence of multiple nontrivial (encircling more than one critical point) period annuli. The conditions are obtained (by phase-plane analysis of period annuli) for existence of families of solutions to the Neumann boundary value problems.

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 investigate the three-dimensional dynamical system occurring in the network regulatory systems theory for specific choices of regulatory matrix { { 0, 1, 1 } { 1, 0, 1 } { 1, 1, 0 } } and sigmoidal regulatory function f(z) = 1 / (1 + e-μz), where z = ∑ Wij xj - θ. The description of attracting sets is provided. The attracting sets consist of respectively one, two or three critical points. This depends on whether the parameters (μ,θ) belong to a set Ω or to the complement of Ω or to the boundary of Ω, where Ω is fully defined set.

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 parameters µ and Θ.

Eigenvalue problems of the form $x'' = -\lambda f(x^+) + \mu g(x^-),$ $\quad (i),$ $x(0) = 0, \; x(1) = 0,$ $\quad (ii)$ are considered, where $x^+$ and $x^-$ are the positive and negative parts of $x$ respectively. We are looking for $(\lambda, \mu)$ such that the problem $(i), (ii)$ has a nontrivial solution. This problem generalizes the famous Fu\v{c}\'{i}k problem for piece-wise linear equations. In our considerations functions $f$ and $g$ may be nonlinear functions of super-, sub- and quasi-linear growth in various combinations. The spectra obtained under the normalization condition $|x'(0)|=1$ are sometimes similar to usual Fu\v{c}\'{i}k spectrum for the Dirichlet problem and sometimes they are quite different. This depends on monotonicity properties of the functions $\xi t_1 (\xi)$ and $\eta \tau_1 (\eta),$ where $t_1 (\xi)$ and $\tau_1 (\eta)$ are the first zero functions of the Cauchy problems $x''= -f(x),$ $\: x(0)=0,$ $\: x'(0)=\xi> 0,$ $y''= g(y),$ $\: y(0)=0,$ $\: y'(0)=-\eta,$ $(\eta > 0)$ respectively.

We consider Dirichlet boundary value problem for systems of two second-order differential equations with nonlinear continuous and bounded functions in right-hand sides. We prove the existence of a nontrivial solution to the problem comparing behaviors of solutions of auxiliary Cauchy problems at zero solution and at infinity.

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.

We review the results concerning types of solutions of boundary value problems for the second order nonlinear equation (l2x)(t)=f(t,x,x′), where (l2x)(t) is the second order linear differential form. The existence results and the multiplicity results are stated in terms of types of solutions.

The Liénard type equation x'' + f(x, x')x' + g(x) = 0 (i) is considered. We claim that if the associated conservative equation x'' + g(x) = 0 has period annuli then a dissipation f(x, x') exists such that a limit cycle of equation (i) exists in a selected period annulus. Moreover, it is possible to define f(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 problem x ′′ + 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 that f (x +) − g(x −) is a sector-bounded nonlinearity.

Properties of asymmetric oscillator described by the equation (i), where and , are studied. A set of such that the problem (i), (ii), and (iii) have a nontrivial solution, is called α-spectrum. We give full description of α-spectra in terms of solution sets and solution surfaces. The exact number of nontrivial solutions of the two-parameter Dirichlet boundary value problem (i), and (ii) is given.

The boundary value problem is considered provided that f : [0, +∞) → [0, +∞) is Lipschitzian and is continuous and Lipschitzian in xand x′. We assume that f is bounded by two linear functions kx and lx, where k > l > 0, and h is bounded. We find the conditions on (λ, µ) which guarantee the existence of a solution to the problem. These conditions are of geometrical nature.

An equation is considered together with the boundary conditions , . This problem under appropriate conditions can be reduced to quasilinear problem for two-dimensional differential system. The conditions for existence of multiple solutions to the original problem are obtained by multiply applying the quasilinearization technique.

Boundary value problems of the form are considered, where In our considerations functions f and g are 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 points z of the function x(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 exactly i zeroes 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 equation x′′+g(x)=0, where g(x) is a polynomial, allowing the equation to have multiple period annuli. We detect the maximal number of possible period annuli for polynomials of odd degree and show how the respective optimal polynomials can be constructed.

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 form x” = −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 functions f and g may be nonlinear. Consequently spectra may differ essentially from those for the Fučík equation. First published online: 14 Oct 2010

Eigenvalue problems of the form x″ = -λf(x+) + μg(x-), x(0) = 0, x(1) = 0 are considered, where x+ and x− are respectively the positive and the negative parts of x. 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 functions f(x) and g(x). We show that the first branches of the Fučik spectrum may contain bounded components. First Published Online: 14 Oct 2010