We construct a one-dimensional model of conflict dynamics with regime of chaotic behavior. It is shown that this type of behavior appears as a consequence of the nonlinear law of interaction between a pair of abstract opponents. As the main factor in the emergence of chaos, we can mention the influence of the parameter of external support given to the minimal element of the system. The properties of the chaotic behavior are graphically illustrated.
The paper consists of two parts. In the first part, we establish sufficient and necessary conditions for the integral representation of even positive-definite hyperbolically convex (h.c.) functions k(x), x ∈ H. These functions are continuous in the j-topology. The positive definiteness of a function is understood in the pointwise sense. The analyzed theorem is a modified version of the Berezansky theorem presented in [Yu. M. Berezansky and I. M. Gali, Ukr. Math. J., 24, 351–372 (1972)]. The integral representations for some other positive-definite kernels were considered in [O. V. Lopotko and I. I. Rudinskii, Ukr. Math. J., 34, 310–312 (1982)] and [O. V. Lopotko, Dop. Akad. Nauk Ukr., Ser. A, 8, 11–13 (1991)]. In the second part, we prove the integral representation for a family of commutative self-adjoint operators 𝒜_x connected by algebraic relations. To this end, we construct a rigging (chain) Hκ = H0 ⊃ H+ = L2 ⊃ 𝒟 for x ∈ H. Our proof is based on the integral representation of even positive definite hyperbolically constant functions of infinitely many variables (see [Yu. M. Berezanskii and Yu. G. Kondrat’ev, Spectral Methods in Infinite-Dimensional Analysis, Naukova Dumka, Kiev (1988)] and [O. V. Lopotko, Bukov. Mat. Zh., 11, 26–38 (2023)]). Some other forms of generalizations of this kind were considered in [Yu. M. Berezanskii and A. A. Kalyuzhnyi, Ukr. Math. J., 36, 334–338 (1984); A. A. Kurepa, Canad. Math. J., 12, 45–50 (1960)], and [Yu. S. Samoilenko, Spectral Theory of the Sets of Self-Adjoint Operators, Naukova Dumka, Kiev (1984)].
The Dirichlet problem is studied for a one-dimensional isotropic superdiffusion equation of order α ∈ (1; 2) on the semiaxis. The analytic solutions of this problem are obtained by the mapping method in the classes of continuous Hölder functions, which may have an integrable singularity with respect to time. The probability interpretation of the solutions is clarified and the uniqueness of its regular solution is proved for an infinite time interval.
For the hyperbolic systems of first-order differential equations on the semiaxis, we study a scattering problem under the assumption that all its solutions have asymptotic behaviors at infinity corresponding either to incident or to scattered waves. The problem is reduced to finding the solutions of the system in the case where all incident waves and a certain homogeneous boundary condition are given. Under these assumptions, the solution exists and is unique. Therefore, we observe the appearance of a scattering operator on the semiaxis, which transforms all incident waves into scattered waves. For a special case in which the numbers of incident and scattered waves are identical, we determine the detailed properties of the scattering operator, including its factorization into the Volterra factors with the opposite polarities. This factorization makes it possible to solve the inverse scattering problem on the semiaxis, i.e., to unambiguously reconstruct all potentials in the system according to the two known scattering operators corresponding to two different boundary conditions.
We construct the solution of a one-dimensional nonlinear nonstationary heat-conduction problem for a multilayer plate in the case where one of its bounding surfaces is heated by a heat flux whose intensity varies as a sinusoidal function of time, whereas the opposite surface is cooled down in the process of convective heat exchange with the medium whose temperature varies according to the exponential law. Moreover, we take into account the effect of thermal emission from the heated surface. The solution is constructed by using generalized functions, an appropriate Green function for a multilayer plate, and linear splines. The exact sums of series required to improve the convergence rate of slowly convergent series are found. The results of numerical analyses of the temperature fields formed in a five-layer plate are also presented.
