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We introduce a multistable subordinator, which generalizes the stable subordinator to the case of time-varying stability index. This enables us to define a multifractional Poisson process. We study properties of these processes and…
Due to the processes that occur during the functioning of modern electromechanical systems, these systems can be considered complex nonlinear dynamic systems from the point of view of the theory of dynamic systems. The movement of such…
In this work we provide conditions for the existence of periodic solutions to nonlinear, second-order difference equations of the form \begin{equation*} y(t+2)+by(t+1)+cy(t)=g(t,y(t)) \end{equation*} where $c\neq 0$, and…
Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…
This paper deals with the solution of large classes of systems of nonlinear partial differential equations (PDEs) in spaces of generalized functions that are constructed as the completion of uniform convergence spaces. The existence result…
Consider a periodically forced nonlinear system which can be presented as a collection of smaller subsystems with pairwise interactions between them. Each subsystem is assumed to be a massive point moving with friction on a compact surface,…
For a class of polynomial non-autonomous differential equations of degree n, we use phase plane analysis to show that each equation in this class has n periodic solutions. The result implies that certain rigid two-dimensional systems have…
We study the homogenization property of systems of quasi-linear PDEs of parabolic type with periodic coefficients, highly oscillating drift and highly oscillating nonlinear term. To this end, we propose a probabilistic approach based on the…
Recently, various systems of nonlinear difference equations, of different forms, were studied. In this existing work, two earlier published papers, due respectively to Bayram and Das. [Appl. Math. Sci. (Ruse), 4(7) (2010) pp. 817-821] and…
We consider a discrete time dynamic system described by a difference equation with periodic coefficients and with additive stochastic noise. We investigate the possibility of the periodicity for the solution. In particular, we found…
A nonlinear inequality is formulated in the paper. An estimate of the rate of growth/decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations. It can…
We derive bifurcation test equations for A-series singularities of nonlinear functionals and, based on these equations, we propose a numerical method for detecting high codimensional bifurcations in parameter-dependent PDEs such as…
Can neural networks learn to solve partial differential equations (PDEs)? We investigate this question for two (systems of) PDEs, namely, the Poisson equation and the steady Navier--Stokes equations. The contributions of this paper are…
We study a class of nonlinear elliptic problems driven by a double-phase operator with variable exponents, arising in the modeling of heterogeneous materials undergoing phase transitions. The associated Poisson problem features a…
We consider the problem of numerically approximating the solutions to a partial differential equation (PDE) when there is insufficient information to determine a unique solution. Our main example is the Poisson boundary value problem, when…
For the study of highly nonlinear, conservative dynamic systems, finding special periodic solutions which can be seen as generalization of the well-known normal modes of linear systems is very attractive. However, the study of…
The present work deals with the resolution of the Poisson equation in a bounded domain made of a thin and periodic layer of finite length placed into a homogeneous medium. We provide and justify a high order asymptotic expansion which takes…
Green's functions characterize the fundamental solutions of partial differential equations; they are essential for tasks ranging from shape analysis to physical simulation, yet they remain computationally prohibitive to evaluate on…
This paper investigates the dynamical behavior of periodic solutions for a class of second-order non-autonomous differential equations. First, based on the Lyapunov-Schmidt reduction method for finite-dimensional functions, the…
We study function-valued solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider semilinear equations under suitable parabolicity hypotheses. We provide…