Searcharxiv⌕ Search

arXiv subjects

Igor Parasyuk

Publications and source records attributed to Igor Parasyuk.

8 recordsLinked to original sources
arXiv2018

On asymptotic phase of dynamical system hyperbolic along attracting invariant manifold

We consider a dynamical system which has the hyperbolic structure along an attracting invariant manifold $M$. The problem is whether every motion starting in a neighborhood of $M$ possesses an asymptotic phase, i.e. eventually approaches a particular motion on $M$. Earlier, positive solutions to the problem were obtained under the condition that the decay rate of solutions toward the manifold exceeds the decay rate of the solutions within the manifold. We show that in our case the above condition is not necessary. To prove that a neighborhood of $M$ is filled with motions for each of which there exists an asymptotic phase we apply the Brouwer fixed point theorem. An invariant foliation structure which appears in the neighborhood of $M$ is discussed.

math.DS↗
arXiv2017

Hyperbolic quasiperiodic solutions of U-monotone systems on Riemannian manifolds

We consider a second order non-autonomous system which can be interpreted as the Newtonian equation of motion on a Riemannian manifold under the action of time-quasiperiodic force field. The problem is to find conditions which ensures: (a) the existence of a solution taking values in a given bounded domain of configuration space and possessing a bounded derivative; (b) the hyperbolicity of such a solution; (c) the uniqueness and, as a consequence, the quasiperiodicity of such a solution. Our approach exploits ideas of Ważewski topological principle. The required conditions are formulated in terms of an auxiliary convex function $U$. We use this function to establish the Landau type inequality for the derivative of solution, as well as to introduce the notion of U-monotonicity for the system. The U-monotonicity property of the system implies the uniqueness and the quasiperiodicity of its bounded solution. We also find the bounds for magnitude of perturbations which do not destroy the quasiperiodic solution. The results obtained are applied to study the motion of a charged particle on a unite sphere under the action of time-quasiperiodic electric and magnetic fields.

math.DS↗
arXiv2017

A counterpart of Landau -- Hadamard type inequality for manifold-valued mappings

We obtain a Landau -- Hadamard type inequality for mappings defined on the whole real axis and taking values in Riemannian manifolds. In terms of an auxiliary convex function, we find conditions under which the boundedness of covariant derivative along the curve under consideration ensures the boundedness of the corresponding tangent vector field. As an example we obtain a Landau -- Hadamard type inequality for curves on 2D unite sphere.

math.CA↗
arXiv2010

Existence of V-bounded solutions for nonautonomous nonlinear systems via the Wazewski topological principle

We establish a number of new sufficient conditions for the existence of global (defined on the entire time axis) solutions of nonlinear nonautonomous systems by means of the Wazewski topological principle. The systems under consideration are characterized by the monotonicity property with respect to a certain auxiliary guiding function W(t,x) depending on time and phase coordinates. Another auxiliary spatially coercive function V(t,x) is used to estimate the location of global solutions in the extended phase space. The approach developed is applied to Lagrangian systems, and in particular, to establish new sufficient conditions for the existence of almost periodic solutions.

math.CA↗
arXiv2009

Waźewski Topological Principle and V-bounded Solutions of Nonlinear Systems

We use the Waźewski topological principle to establish a number of new sufficient conditions for the existence of proper (defined on the entire time axis) solutions of essentially nonlinear nonautonomous systems. The systems under consideration are characterized by the monotonicity property with respect to a certain auxiliary guiding function $W(t,x)$ depending on time and phase coordinates. Another auxiliary function $V(t,x)$, which is positively defined in the phase variables $x$ for any $t$, is used to estimate the deviation of the proper solutions from the origin.

math.CA↗
arXiv2008

The integral representation of solutions to a boundary value problem on the half-line for a system of linear ODEs with singularity of first kind

We consider a problem of finding vanishing at infinity $C^1([0,\oo))$-solutions to non-homogeneous system of linear ODEs which has the pole of first order at $x=0$. The resonant case where the corresponding homogeneous problem has nontrivial solutions is of main interest. Under the conditions that the homogeneous system is exponentially dichotomic on $[1,\oo)$ and the residue of system's operator at $x=0$ does not have eigenvalues with real part 1, we construct the so called generalized Green function. We also establish conditions under which the main non-homogeneous problem can be reduced to the Noetherian one with nonzero index.

math.CA↗