Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations

The article presents a new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations, including autonomous and nonautonomous ordinary differential equations (ODE), partial differential equations, and delay differential equations. The theory relies on four remarkable results: Feigenbaum’s period doubling theory for cycles of one-dimensional unimodal maps, Sharkovskii’s theory of birth of cycles of arbitrary period up to cycle of period three in one-dimensional unimodal maps, Magnitskii’s theory of rotor singular point in two-dimensional nonautonomous ODE systems, acting as a bridge between one-dimensional maps and differential equations, and Magnitskii’s theory of homoclinic bifurcation cascade that follows the Sharkovskii cascade. All the theoretical propositions are rigorously proved and illustrated with numerous analytical examples and numerical computations, which are presented for all classical chaotic nonlinear dissipative systems of differential equations.     

Numerical solutions of index-1 differential algebraic equations can be computed in polynomial time

The cost of solving an initial value problem for index-1 differential algebraic equations to accuracy ɛ is polynomial in ln(1/ɛ). This cost is obtained for an algorithm based on the Taylor series method for solving differential algebraic equations developed by Pryce. This result extends a recent result by Corless for solutions of ordinary differential equations. The results of the standard theory of information-based complexity give exponential cost for solving ordinary differential equations, being based on a different model.     

Tracking a reference solution of a control system of phase field equations

We consider the problem of tracking a reference solution of a dynamical system described by a pair of distributed differential equations, the phase field equations. To solve this problem, we propose an algorithm based on Yu.S. Osipov’s theory of dynamic inversion and on N.N. Krasovskii’s extremal shift method developed in the theory of positional differential games.     

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.     

Approximated solutions in rational form for systems of differential equations

In this paper we present a technique to study the existence of rational solutions for systems of differential equations — for an ordinary differential equation, in particular. The method is relatively straightforward; it is based on a rationality characterisation that involves matrix Padé approximants. It is important to note that, when the solution is rational, we use formal power series “without taking into account” their circle of convergence; at the end of this paper we justify this. We expound the theory for systems of linear first-order ordinary differential equations in the general case. However, the main ideas are applied in numerical resolution of partial differential equations. This revised version was published online in June 2006 with corrections to the Cover Date.     

Introduction to the theory of functional differential equations and their applications. Group approach

In this work, we give an introduction to the theory of nonlinear functional differential equations of pointwise type on a finite interval, semi-axis, or axis. This approach is based on the formalism using group peculiarities of such differential equations. For the main boundary-value problem and the Euler-Lagrange boundary-value problem, we consider the existence and uniqueness of the solution, the continuous dependence of the solution on boundary-value and initial-value conditions, and the “roughness” of functional differential equations in the considered boundary-value problems. For functional differential equations of pointwise type we also investigate the pointwise completeness of the space of solutions for given boundary-value conditions, give an estimate of the rank for the space of solutions, describe types of degeneration for the space of solutions, and establish conditions for the “smoothness” of the solution. We propose the method of regular extension of the class of ordinary differential equations in the class of functional differential equations of pointwise type. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 8, Functional Differential Equations, 2004.     

Some problems of the linear theory of systems of ordinary differential equations

We consider problems of the linear theory of systems of ordinary differential equations related to the investigation of invariant hyperplanes of these systems, the notion of equivalence for these systems, and the Floquet–Lyapunov theory for periodic systems of linear equations. In particular, we introduce the notion of equivalence of systems of linear differential equations of different orders, propose a new formula of the Floquet form for periodic systems, and present the application of this formula to the introduction of amplitude–phase coordinates in a neighborhood of a periodic trajectory of a dynamical system.     

Differential equations satisfied by Eisenstein series of level 2

Ramanujan’s differential equations for the classical Eisenstein series are of great importance to many areas in number theory and special functions. H.H. Chan recently demonstrated that these differential equations can be derived from the triple product identity and the quintuple product identity in an elementary manner. In this article, we extend this method in a uniform manner to derive corresponding differential equations for the Eisenstein series of level 2. Several applications of these differential equations are also given.     

Forward-backward stochastic differential equations and quasilinear parabolic PDEs

This paper studies, under some natural monotonicity conditions, the theory (existence and uniqueness, a priori estimate, continuous dependence on a parameter) of forward–backward stochastic differential equations and their connection with quasilinear parabolic partial differential equations. We use a purely probabilistic approach, and allow the forward equation to be degenerate. Received: 12 May 1997 / Revised version: 10 January 1999     

Nonlinear oscillation of fourth order functional differential equations

Summary In the oscillation theory of nonlinear differential equations one of the important problems is to find necessary and sufficient conditions for the equations under consideration to be oscillatory. Beginning with the pionearing work of F. V. Atkinson, there have been a number of papers. Recently, Kusano and Naito proved the interesting results to the jourth order nonlinear ordinary differential equations of the from [r(t)y″(t)]″+y(t)F(y(t) ₂ ,t)=0. In the present paper, we will extend them to the more general functional differential equations and improve the not clear parts of them. Also, we will propose a new simple definition of nonlinearity of the functional differential equations. Entrata in Redazione il 5 settembre 1977.     

The global stability conditions for solutions of nonstationary differential equations with instant impulsive effects in pseudolinear form

We obtained the sufficient conditions for the stability of solutions of a class of nonlinear differential equations with fixed instant impulsive effects in the Banach space. With the use of the Slyusarchuk’s condition and methods of the theory of operators in a partially ordered Banach space, the problem is reduced to the study of the stability of a linear system of second-order impulsive differential equations.     

Algebraic independence of the values ofE-functions at singular points and Siegel’s conjecture

We prove a generalization of Shidlovskii’s theorem on the algebraic independence of the values ofE-functions satisfying a system of linear differential equations that is well known in the theory of transcendental numbers. We consider the case in which the values ofE-functions are taken at singular points of these systems. Using the obtained results, we prove Siegel’s conjecture that, for the case of first-order differential equations, anyE-function satisfying a linear differential equation is representable as a polynomial in hypergeometricE-functions. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 174–190, February, 2000.     

Foreword

Steve Smale set the agenda for FoCM in his call for the 1995 conference in Park City, Utah. No stranger he to ambitious agendas and extraordinary accomplishments. He is one of the dominant figures in the mathematics of the second half of the twentieth century. Smale’s theory of immersions, the generalized Poincare conjecture, and H-cobordism theorems with their far-reaching consequences are the bedrock of differential topology. His horseshoe is the hallmark of chaos, and his hyperbolic dynamics the rejuvenation of the geometric theory of dynamical systems. He is one of the pioneers in the introduction of infinite-dimensional manifolds for the study of nonlinear problems in the calculus of variations and partial differential equations. The list goes on: the systematic use of differential techniques in microeconomics, electrical circuit theory, chaos in predator–prey equations and, finally, for the twentieth century, the foundations of computational mathematics and complexity theory, and now, in the twenty-first century, the theory of learning. It has been our privilege to be among his collaborators and students in the broadest sense of the word. With these issues (Volume 5 Number 4 and Volume 6 Number 1, as well as an earlier article appearing in Volume 5 Number 2, are dedicated to Steve Smale’s 75th Birthday) we celebrate Steve’s 75th birthday and continuing vitality. He sets the bar high. We do our best.     

Application of the method of straight lines to determining the stressed-strained state and dynamic characteristics of a hollow noncircular cylinder

Using the method of straight lines, problems of elasticity theory for a noncircular hollow cylinder are investigated. By means of separation of variables along the generatrix and finite-difference approximation across the thickness, the input partial differential equations are reduced to a system of ordinary differential equations, which is solvable by the stable numerical method of discrete orthogonalization. Specific features of application of the method proposed to static and dynamic problems of a hollow noncircular cylinder are illustrated by way of examples. Bibliography: 5 titles. Translated from,Obchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 89–87.     

Some modern aspects of the theory of impulsive differential equations

We give a brief survey of the main results obtained in recent years in the theory of impulsive differential equations. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 81–94, January, 2008.     

Clunie theorems for difference and q-difference polynomials

The main purpose of this paper is to prove difference and q-differencecounterparts of the Clunie and Mohon’ko lemmas from theNevanlinna theory of differential polynomials. We also giveapplications to the value distribution theory of meromorphicsolutions of some complex difference equations.     

Limiting subgradients of minimal time functions in Banach spaces

The paper mostly concerns the study of generalized differential properties of the so-called minimal time functions associated, in particular, with constant dynamics and arbitrary closed target sets in control theory. Functions of this type play a significant role in many aspects of optimization, control theory, and Hamilton–Jacobi partial differential equations. We pay the main attention to computing and estimating limiting subgradients of the minimal value functions and to deriving the corresponding relations for Fréchet type ε-subgradients in arbitrary Banach spaces.     

Geometric theory of differential systems: Linearization criterion for systems of second-order ordinary differential equations with a 4-dimensional solvable symmetry group of the Lie–Petrov type VI 1

In the framework of projective-geometric theory of systems of differential equations developed by the authors, this paper studies the group properties of systems of two (resolved with respect to the second derivatives) second-order ordinary differential equations whose right-hand sides are polynomials of the third degree with respect to the derivatives of the unknown functions. A classification of such systems admitting four-dimensional symmetry group of the Lie–Petrov type VI ₁ is given. For each of the systems, a necessary and sufficient linearization criterion is obtained, i.e., the authors find the necessary and sufficient conditions under which, by a change of variables, the system can be reduced to a differential system whose integral curves are straight lines and are expressed by three linear parametric equations or two linear equations with constant coefficients. For all linearizable systems, the linearizing changes of variables are indicated. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 54, Suzdal Conference–2006, Part 2, 2008.     

A new method for solving the nonlinear second-order boundary value differential equations

In this paper we use measure theory to solve a wide range of second-order boundary value ordinary differential equations. First, we transform the problem to a first order system of ordinary differential equations (ODE’s) and then define an optimization problem related to it. The new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; the optimal measure is then approximated by a finite combination of atomic measures and the problem converted approximatly to a finite-dimensional linear programming problem. The solution to this problem is used to construct the approximate solution of the original problem. Finally we get the error functionalE (we define in this paper) for the approximate solution of the ODE’s problems.