Related papers: Some Applications of the Extended Bendixson-Dulac …
In this paper, we consider the family of planar piecewise linear differential systems with two zones separated by a straight line without sliding regions, that is, differential systems whose flow transversally crosses the switching line…
Van der Pol and Rayleigh oscillators are two traditional paradigms of nonlinear dynamics. They can be subsumed into a general form of Li\'enard--Levinson--Smith(LLS) system. Based on a recipe for finding out maximum number of limit cycles…
We derive the multi-field, micropolar-type continuum theory for the two-dimensional model of crystal having finite-size particles. Continuum theories are usually valid for waves with wavelength much larger than the size of primitive cell of…
In this paper, using our bifurcational geometric approach, we solve the problem on the maximum number and distribution of limit cycles in the Kukles system representing a planar polynomial dynamical system with arbitrary linear and cubic…
We prove that the number of limit cycles, which bifurcate from a two-saddle loop of a planar quadratic Hamiltonian system, under an arbitrary quadratic deformation, is less than or equal to three.
The paper is devoted to vector fields on the spaces R^2 and R^3, their flow and invariants. Attention is plaid on the tensor representations of the group GL(2,R) and on fundamental vector fields. The rotation group on R^3 is generalized to…
We provide a simple proof of the existence of a planar separator by showing that it is an easy consequence of the circle packing theorem. We also reprove other results on separators, including: (A) There is a simple cycle separator if the…
We study periodic infinite billiards in the plane. We show that for rational models, some particular obstacles can be added periodically, so that the billiard flow in the resulting table is recurrent in almost every direction.
We consider oscillons - localized, quasiperiodic, and extremely long-living classical solutions in models with real scalar fields. We develop their effective description in the limit of large size at finite field strength. Namely, we note…
Maxwell's equation, Dirac's equation and the equation of gravito-electromagnetism are shown to be particular instances of the extended Maxwell system. The equations are discussed in the framework of the theory of evolutionary equations.…
In recent decades, several mathematical models describing the pacemaker activity of the rabbit sinoatrial node have been developed. We demonstrate that it is not possible to establish the existence, uniqueness, and stability of a limit…
We develop elliptic regularity theory for Dirac operators in a very general framework: we consider Dirac operators linear over $C^*$-algebras, on noncompact manifolds, and in families which are not necessarily locally trivial fibre bundles.
We study the distribution modulo one of linear recurrent sequences of real numbers. We prove criteria for the finiteness of the set of limit values of the fractional parts of such a sequence and give lower bounds for the maximal distance…
Formalism of extended Lagrangian represent a systematic procedure to look for the local symmetries of a given Lagrangian action. In this work, the formalism is discussed and applied to a field theory. We describe it in detail for a field…
The purpose of this note is to study the existence of a nontrivial solution for an elliptic system which comes from a newly introduced mathematical problem so called Field-Road model. Specifically, it consists of coupled equations set in…
We study the set ${\cal D}(\Phi)$ of limit directions of a vector cocycle $(\Phi_n)$ over a dynamical system, i.e., the set of limit values of $\Phi_n(x) /\|\Phi_n(x)\|$ along subsequences such that $\|\Phi_n(x)\|$ tends to $\infty$. This…
An analysis of possible extension of the Painlev\'e test, to encompass the one-dimensional Vlasov equation, is performed. The extending requires a nontrivial generalization of the test. The proposed singularity analysis provides…
For Young systems, i.e. for hyperbolic systems without/with singularities satisfying Lai-Sang Young's axioms (which imply exponential decay of correlation and the CLT) a local CLT is proven. In fact, a unified version of the local CLT is…
Taylor's theorem (and its variants) is widely used in several areas of mathematical analysis, including numerical analysis, functional analysis, and partial differential equations. This article explains how Taylor's theorem in its most…
We provide an up-to-date review of the recent constructive program for field theories of the vector, matrix and tensor type, focusing not on the models themselves but on the mathematical tools used.