Smooth equivalence of differential equations and linear automorphisms

For systems of ordinary differential equations admitting linear automorphisms, we consider the problems of smooth equivalence and linearization preserving these automorphisms. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 567–578, October, 1999.     

On Periodic Solutions of One Class of Systems of Differential Equations

We study the problem of the existence of periodic solutions of two-dimensional linear inhomogeneous periodic systems of differential equations for which the corresponding homogeneous system is Hamiltonian. We propose a new numerical-analytic algorithm for the investigation of the problem of the existence of periodic solutions of two-dimensional nonlinear differential systems with Hamiltonian linear part and their construction. The results obtained are generalized to systems of higher orders. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 483–495, April, 2005.     

Contact-equivalence problem for linear hyperbolic equations

We consider the local equivalence problem for the class of linear second-order hyperbolic equations in two independent variables under an action of the pseudo-group of contact transformations. é. Cartan’s method is used for finding the Maurer-Cartan forms for symmetry groups of equations from the class and computing structure equations and complete sets of differential invariants for these groups. The solution of the equivalence problem is formulated in terms of these differential invariants. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 119–142, 2005.     

Finite-Dimensional Sturm–Liouville Vessels and Their Tau Functions

We introduce a theory of a class of finite-dimensional vessels, a concept originating from the pioneering work of Livšic (Soobshch Akad Nauk Gruzin SSSR 91(2):281–284, 1978). Our work may be considered as a first step toward analyzing and constructing Lax Phillips scattering theory for Sturm–Liouville differentiable equations on the half axis (0,∞) with singularity at 0. We also develop a rich and interesting theory of vessels with deep connections to the notion of the τ function, arising in non linear differential equations (LDE), and to the Galois differential theory for LDEs.     

Systems of essentially infinite-dimensional differential equations

We investigate systems of differential equations with essentially infinite-dimensional elliptic operators (of the Laplace–Lévy type). For nonlinear systems, we prove theorems on the existence and uniqueness of solutions. For a linear system, we give an explicit formula for the solution.     

Invariant cones and stability of linear dynamical systems

We present a method for the investigation of the stability and positivity of systems of linear differential equations of arbitrary order. Conditions for the invariance of classes of cones of circular and ellipsoidal types are established. We propose algebraic conditions for the exponential stability of linear positive systems based on the notion of maximal eigenpairs of a matrix polynomial. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1446–1461, November, 2006.     

Asymptotic equivalence of differential equations and asymptotically almost periodic solutions

In this paper we use Rab’s lemma [M. Ráb, Über lineare perturbationen eines systems von linearen differentialgleichungen, Czechoslovak Math. J. 83 (1958) 222–229; M. Ráb, Note sur les formules asymptotiques pour les solutions d’un systéme d’équations différentielles linéaires, Czechoslovak Math. J. 91 (1966) 127–129] to obtain new sufficient conditions for the asymptotic equivalence of linear and quasilinear systems of ordinary differential equations. Yakubovich’s result [V.V. Nemytskii, V.V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, New Jersey, 1966; V.A. Yakubovich, On the asymptotic behavior of systems of differential equations, Mat. Sb. 28 (1951) 217–240] on the asymptotic equivalence of a linear and a quasilinear system is developed. On the basis of the equivalence, the existence of asymptotically almost periodic solutions of the systems is investigated. The definitions of biasymptotic equivalence for the equations and biasymptotically almost periodic solutions are introduced. Theorems on the sufficient conditions for the systems to be biasymptotically equivalent and for the existence of biasymptotically almost periodic solutions are obtained. Appropriate examples are constructed.     

Versions of the Feynman-Kac formula

Some new versions of the Feynman-Kac formula for Brownian motion are considered. An interesting generalization of the formula is related to solutions of systems of linear differential equations. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 46–60. Translated by T. Safonova     

On a product formula for the Conley–Zehnder index of symplectic paths and its applications

Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, we prove a product formula for the Conley–Zehnder index of continuous paths with arbitrary endpoints in the symplectic group. We discuss two applications of the formula, to the metaplectic group and to periodic solutions of Hamiltonian systems.      

On asymptotic equivalence of linear and quasilinear difference equations

The asymptotic equivalence of systems of difference equations of linear and quasilinear type is investigated. The first result on the asymptotic equivalence of linear systems is a discrete analog of an improved version of the Levinson's well-known theorem on asymptotic equivalence of linear differential equations, while the second one providing conditions for asymptotic equivalence of linear and quasilinear systems is related to that of Yakubovich in differential equations case.     

On Favard's theorems

In this paper, we improve and extend the classical Favard's theorems, i.e. Favard's theorem of the module containment, Favard's theorem of linear differential equations. We study Favard's theory of linear differential equations with piecewise constant argument. An example shows that the new module containment is necessary in the study of differential equations with piecewise constant argument. The equivalence between almost automorphic functions and N-almost periodic ones is studied.     

On linear homogeneous almost periodic systems that satisfy the Favard condition

We prove the existence of a linear homogeneous almost periodic system of differential equations that has nontrivial bounded solutions and is such that all systems from a certain neighborhood of it have no nontrivial almost periodic solutions. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 3. pp. 409–413, March, 1998.     

A Newton-Picard Collocation Method for Periodic Solutions of Delay Differential Equations

This paper presents a collocation method with an iterative linear system solver to compute periodic solutions of a system of autonomous delay differential equations (DDEs). We exploit the equivalence of the linearized collocation system and the discretization of the linearized periodic boundary value problem (BVP). This linear BVP is solved using a variant of the Newton-Picard method [Int. J. Bifurcation Chaos, 7 (1997), pp. 2547–2560]. This method combines a direct method in the low-dimensional subspace of the weakly stable and unstable modes with an iterative solver in the high-dimensional orthogonal complement. As a side effect, we also obtain good estimates for the dominant Floquet multipliers. We have implemented the method in the DDE-BIFTOOL environment to test our algorithm. AMS subject classification (2000) 65J15, 65P30, 65Q05     

Construction of a periodic system of moment equations

We construct a system of moment equations for a system of linear differential equations with periodic coefficients. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 774–780, June, 1998.     

Systems of first order differential inclusions with maximal monotone terms

In this paper, we establish the existence of solutions to systems of first order differential inclusions with maximal monotone terms satisfying the periodic boundary condition. Our proofs rely on the theory of maximal monotone operators, and the Schauder and the Kakutani fixed point theorems. A notion of solution-tube to these problems is introduced. This notion generalizes the notion of upper and lower solutions of first order differential equations.     

Maurer–Cartan forms and the structure of Lie pseudo-groups

This paper begins a series devoted to developing a general and practical theory of moving frames for infinite-dimensional Lie pseudo-groups. In this first, preparatory part, we present a new, direct approach to the construction of invariant Maurer–Cartan forms and the Cartan structure equations for a pseudo-group. Our approach is completely explicit and avoids reliance on the theory of exterior differential systems and prolongation. The second paper [60] will apply these constructions in order to develop the moving frame algorithm for the action of the pseudo-group on submanifolds. The third paper [61] will apply Gr?bner basis methods to prove a fundamental theorem on the freeness of pseudo-group actions on jet bundles, and a constructive version of the finiteness theorem of Tresse and Kumpera for generating systems of differential invariants and also their syzygies. Applications of the moving frame method include practical algorithms for constructing complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, analyzing invariant variational principles, and solving equivalence and symmetry problems arising in geometry and physics.     

Analytic representation of periodic solutions of linear differential systems

We study the problem of periodic solutions of linear differential systems with small parameter. We establish new conditions for the existence and uniqueness of periodic solutions of these systems, which can be efficiently verified. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 731–735, May, 1997.     

Hydrodynamic profiles and constants of motion fromd-Branes

Various hydrodynamic systems, governed by nonlinear differential equations, have a hidden higher-dimensional dynamic Poincaré symmetry because the governing equations descend from a Nambu-Goto action. For the same reason, there are also equivalence transformations between different models. We discuss these interconnections and summarize them in a simple diagram. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, pp. 215–226. August, 2000     

On the asymptotic equilibrium and asymptotic equivalence of differential equations in Banach spaces

We present some conditions for the asymptotic equilibrium of nonlinear differential equations and, in particular, a linear inhomogeneous equation in Banach spaces. We also discuss analogous problems for a linear equation with unbounded operator. Some obtained results are applied to problems of asymptotic equivalence. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 626–635, May, 2008.