Ukrainian Mathematical Journal最新文献

Our main result can be formulated as follows: Consider the set of natural numbers in which the following relation is introduced: n₁ precedes n₂ (n₁ ⪯ n₂) if, for any continuous map of the real line into itself, the existence of a cycle of order n₂ follows from the existence of a cycle of order n₁. The following theorem is true:

Theorem. The introduced relation turns the set of natural numbers into an ordered set with the following ordering:

This descriptive text is essentially based on Sharkovsky’s and Myrberg’s publications on the ordering of periodic solutions (cycles) generated by a Dim 1 unimodal smooth map f(x, λ). Taking f(x, λ) = x²−λ as an example, it was shown in a paper published in 1975 that the bifurcations are organized in the form of a sequence of well-defined fractal embedded “boxes” (parameter λ intervals) each of which is associated with a basic cycle of period k and a symbol j permitting to distinguish cycles with the same period k. Without using the denominations Intermittency (1980) and Attractors in Crisis (1982), this new text shows that the notion of fractal embedded “boxes” describes the properties of each of these two situations as the limit of a sequence of well-defined boxes (k, j) as k → ∞.

We consider the problem of existence of the solution of a weakly nonlinear boundary-value problem for the Hammerstein-type integral equation with unbounded kernel, which turns, for ε = 0, into one of solutions of the generating problem. The necessary and sufficient conditions for the existence of this solution are obtained and the iterative procedure is proposed for its construction.

By using the apparatus of Lyapunov’s direct method with a function from the class of quadratic forms, we establish algebraic sufficient conditions for the stability of trivial solutions to the nonlinear systems of differential equations of the second and third orders.

Motivated by the recent work by Ma and Magal [Proc. Amer. Math. Soc. (2021); https://doi.org/10.1090/proc/15629] on the global stability property of the Gurtin–MacCamy’s population model, we consider a family of scalar nonlinear convolution equations with unimodal nonlinearities. In particular, we relate the Ivanov and Sharkovsky analysis of singularly perturbed delay differential equations in [https://doi.org/10.1007/978-3-642-61243-5_5] to the asymptotic behavior of solutions of the Gurtin–MacCamy’s system. According to the classification proposed in [https://doi.org/10.1007/978-3-642-61243-5_5], we can distinguish three fundamental kinds of continuous solutions of our equations, namely, solutions of the asymptotically constant type, relaxation type, and turbulent type. We present various conditions assuring that all solutions belong to the first of these three classes. In the setting of unimodal convolution equations, these conditions suggest a generalized version of the famous Wright’s conjecture.

Despite having low rates of tuberculosis (TB) mortality in many countries, like China, Europe, and the United States, some other countries, such as India continue to struggle to contain the epidemic. Our aim is to examine the effects of vaccinations and treatments on the dynamics of TB in two countries, Ukraine and Algeria, with contrasted demographic profiles. A mathematical model called the VSLIT model is considered for this purpose. The stability of both disease-free and endemic equilibrium is discussed qualitatively. For numerical simulations, the parameters are evaluated by the least-squares approach according to the TB-reported data for Algeria and Ukraine in 1990–2020.

Differential equations with state-dependent delays specify a semiflow of continuously differentiable solution operators, in general, only on an associated submanifold of the Banach space C¹([−h, 0],ℝⁿ). We extend a recent result on the simplicity of these solution manifolds to systems in which the delay is given by the state only implicitly in an extra equation. These algebraic delay systems appear in various applications.

We consider the balanced pantograph equation (BPE) (y{prime}left(xright)+yleft(xright)={sum }_{k=1}^{m}{p}_{k}yleft({a}_{k}xright)), where a_k, p_k > 0 and ({sum }_{k=1}^{m}{p}_{k}=1). It is known that if (K={sum }_{k=1}^{m}{p}_{k}{text{ln}}{a}_{k}le 0) then, under mild technical conditions, the BPE does not have bounded solutions that are not constant, whereas for K > 0 these solutions exist. In the present paper, we deal with a BPE of mixed type, i.e., a₁ < 1 < a_m, and prove that, in this case, the BPE has a nonconstant solution y and that y(x) ~ cx^σ as x → ∞, where c > 0 and σ is the unique positive root of the characteristic equation (Pleft(sright)=1-sum_{k=1}^{m} {p}_{k}{a}_{k}^{-s}=0). We also show that y is unique (up to a multiplicative constant) among the solutions of the BPE that decay to zero as x → ∞.

We study homoclinic bifurcations in an interval map associated with a saddle-focus of (2, 1)-type in ℤ₂-symmetric systems. Our study of this map reveals a homoclinic structure of the saddle-focus, with bifurcation unfolding guided by the codimension-two Belyakov bifurcation. We consider three parameters of the map corresponding to the saddle quantity, splitting parameter, and the focal frequency of the smooth saddle-focus in a neighborhood of homoclinic bifurcations. We symbolically encode the dynamics of the map in order to find stability windows and locate homoclinic bifurcation sets in a computationally efficient manner. The organization and possible shapes of homoclinic bifurcation curves in the parameter space are examined, taking into account the symmetry and discontinuity of the map. Sufficient conditions for stability and local symbolic constancy of the map are presented. This study provides insights into the structure of homoclinic bifurcations of the saddle-focus map, furthering comprehension of low-dimensional chaotic systems.

We consider a 1D continuous piecewise smooth map, which depends on seven parameters. Depending on the values of parameters, it may have up to six branches. This map was proposed by Matsuyama [Theor. Econ., 8, 623 (2013); Sec. 5]. It describes the macroeconomic dynamics of investment and credit fluctuations in which three types of investment projects compete in the financial market. We introduce a partitioning of the parameter space according to different branch configurations of the map and illustrate this partitioning for a specific parameter setting. Then we present an example of the bifurcation structure in a parameter plane, which includes periodicity regions related to superstable cycles. Several bifurcation curves are obtained analytically; in particular, the border-collision bifurcation curves of fixed points. We show that the point of intersection of two curves of this kind is an organizing center, which serves as the origin of infinitely many other bifurcation curves.