Painlevé Property and Integrability of Polynomial Dynamical Systems

The main purpose of this paper is to investigate the connection between the Painlev′e property and the integrability of polynomial dynamical systems. We show that if a polynomial dynamical system has Painlev′e property, then it admits certain class of first integrals. We also present some relationships between the Painlev′e property and the structure of the differential Galois group of the corresponding variational equations along some complex integral curve.     

Compatibility,Multi-brackets and Integrability of Systems of PDEs

We establish an efficient compatibility criterion for a system of generalized complete intersection type in terms of certain multi-brackets of differential operators. These multi-brackets generalize the higher Jacobi-Mayer brackets, important in the study of evolutionary equations and the integrability problem. We also calculate Spencer δ-cohomology of generalized complete intersections and evaluate the formal functional dimension of the solutions space. The results are used to establish new integration methods.     

Martelli’s Chaos in Inverse Limit Dynamical Systems and Hyperspace Dynamical Systems

In this paper we investigate Martelli’s chaos of inverse limit dynamical systems and hyperspace dynamical systems which are both induced from dynamical systems on a compact metric space. We give the implication of Martelli’s chaos among those systems. More precisely, we show that inverse limit dynamical system is Martelli’s chaos if and only if so is original system, and we prove that hyperspace dynamical system is Martelli’s chaos implies original system is Martelli’s chaos if the orbit of every single point set of original system is unstable in hyperspace dynamical system.     

On Polynomial Solutions of Generalized Moisil-Théodoresco Systems and Hodge-de Rham Systems

The aim of the paper is to study relations between polynomial solutions of generalized Moisil-Théodoresco (GMT) systems and polynomial solutions of Hodge-de Rham systems and, using these relations, to describe polynomial solutions of GMT systems. We decompose the space of homogeneous solutions of GMT system of a given homogeneity into irreducible pieces under the action of the group O(m) and we characterize individual pieces by their highest weights and we compute their dimensions.     

Nonsmooth Lur’e Dynamical Systems in Hilbert Spaces

In this paper, we study the well-posedness and stability analysis of set-valued Lur’e dynamical systems in infinite-dimensional Hilbert spaces. The existence and uniqueness results are established under the so-called passivity condition. Our approach uses a regularization procedure for the term involving the maximal monotone operator. The Lyapunov stability as well as the invariance properties are considered in detail. In addition, we give some sufficient conditions ensuring the robust stability of the system in finite-dimensional spaces. The theoretical developments are illustrated by means of two examples dealing with nonregular electrical circuits and an other one in partial differential equations. Our methodology is based on tools from set-valued and variational analysis.     

On a Class of Lur’e Dynamical Systems with State-Dependent Set-Valued Feedback

Set-Valued and Variational Analysis - Using a new implicit discretization scheme, we study in this paper the existence and uniqueness of strong solutions for a class of Lur’e dynamical...     

Isochronous and Strongly Isochronous Foci of Polynomial Liénard Systems

Differential Equations - The real Liénard system $$\dot x=-y$$ , $$\dot y=x+A(x)-B(x)y$$ , where the polynomials $$A(x) $$ and $$B(x) $$ and the derivative $$A^{\prime }(x) $$ satisfy the...     

Number and Location of Limit Cycles in a Class of Perturbed Polynomial Systems

In this paper,we investigate the number,location and stability of limit cycles in a class of perturbedpolynomial systems with (2n 1) or (2n 2)-degree by constructing detection function and using qualitativeanalysis.We show that there are at most n limit cycles in the perturbed polynomial system,which is similar tothe result of Perko in [8] by using Melnikov method.For n=2,we establish the general conditions dependingon polynomial's coefficients for the bifurcation,location and stability of limit cycles.The bifurcation parametervalue of limit cycles in [5] is also improved by us.When n=3 the sufficient and necessary conditions for theappearance of 3 limit cycles are given.Two numerical examples for the location and stability of limit cycles areused to demonstrate our theoretical results.     

On Ω-limit Sets of Discrete-time Dynamical Systems

The paper investigates z -limit sets for discrete-time dynamical systems of the form x n +1 = f n +1 ( x n ), n S 0, with each f n mapping an interval I of R into itself. For autonomous systems, i.e. f n = f for all n , and f continuous on I =[ a , b ], the case that all z -limit sets consist of one point only is characterized by several equivalent conditions, one being that f has no 2-periodic points. The non-autonomous case assumes that the functions f n converge uniformly to a continuous function f X that has no 2-periodic points. It is shown that the z -limit sets are closed intervals consisting of fixed points of f X only. Under certain conditions these closed intervals contain exactly one point each. This allows a treatment of certain discrete-time dynamical systems in R n .     

Structure-Preserving Algorithms for a Class of Dynamical Systems

In this paper,we study structure-preserving algorithms for dynamical systems defined by ordinarydifferential equations in R~n.The equations are assumed to be of the form y~·=A(y) D(y) R(y),where A(y)is the conservative part subject to     

On Complete Integrability of a Large Class of Finite-dimensional Hamiltonian Systems

Let J be a Hamiltonian operator and ut = JδH/δu be an infinite-dimensional integrable Hamiltonian equation. It is shown that under certain broad assumptions the corresponding stationary equation δH/δu = 0, viewing H as a Lagrangian, can be transformed to a classical Hamiltonian systems qi' = (?)h/(?)pi,pi' = -(?)h/(?)qi, (i= I, …, n) , which is Liouville integrable in the sense that it possesses n first integrals hi which are in involution in pairs. Moreover, a constructive method for calculating the integrals hi is proposed. This connection between finite and infinite-dimensional integrable systems paves a way for constructing a large number of Liouville integrable Hamiltonian systems of finite dimensions.     

Characterization of Polynomial Systems Satisfying Generalized Convolution Differential-Difference Equations

I. Introduction. The present paper has been motivated by the desire to find all polynomial solutions of the convolution type differential -difference equation (1.1) D_xg_n(x)=sum from i=1 to n-1 (g_i(x)g_(n-i)(x),n≥2,) where g_1(x) is assumed to be a constant. This problem arose in work by one of the authors (Kerr) with a differential equation arising in a coal research project     

Crossed Product Algebras and the Homology of Certain p-Adic and Adélic Dynamical Systems

We identify the Hochschild, cyclic, and periodic cyclic homology groups of dynamical systems algebras arising from the action of Q on the spaces of finite and infinite adéles of Q. In the process, we establish several results on the homology of the space of functions on a locally compact, totally disconnected space and its crossed products. Then we use these results to compute the homology groups of the Bost–Connes algebra.     

Aspects of Poincaré’s Program for Dynamical Systems and Mathematical Physics

This article is mainly historical, except for the discussion of integrability and characteristic exponents in Sect.?2. After summarising the achievements of Henri Poincaré, we discuss his theory of critical exponents. The theory is applied to the case of three degrees-of-freedom Hamiltonian systems in (1:2:n)-resonance (n>4). In addition we discuss Poincaré??s mathematical physics, in particular the theory of partial differential equations, rotating fluid masses and relativity. Attention is given to the priority question of Special Relativity.     

“Invariance-Inducing” Control of Nonlinear Discrete-Time Dynamical Systems

The present study aims at the derivation of model-based control laws that attain the invariance objective for nonlinear skew-product discrete-time dynamical systems. The problem under consideration naturally arises in a variety of control problems pertaining to physical/chemical systems, and in the present study, it is conveniently formulated and addressed in the context of functional equations theory. In particular, the mathematical formulation of the problem of interest is realized via a system of nonlinear functional equations (NFEs), and a rather general set of necessary and sufficient conditions for solvability is derived. The solution to the above system of NFEs can be proven to be a unique locally analytic one, and this enables the development of a series solution method that is easily programmable with the aid of a symbolic software package such as MAPLE. It is also shown that, on the basis of the solution to the above system of NFEs, a locally analytic manifold and a nonlinear control law can be explicitly derived that renders the manifold invariant for the class of skew-product systems considered. Furthermore, the restriction of the system dynamics on the aforementioned invariant manifold represents exactly the target controlled system dynamics. Finally, the proposed method is applied to the HF molecular system classically modeled as a rotationless Morse oscillator in the presence of an external laser-field, where the primary objective is molecular dissociation.     

An Algorithm for Finding All Isolated Zeros of Polynomial Systems

Finding all zeros of polynomial systems is very interesting and it is also useul for many applied science problems.In this paper,based on Wu‘s method,we give an algorithm to find all isolated zeros of polynomial systems (or polynomial equations).By solving Lorenz equations,it is shown that our algo-rithm is efficient and powerful.     

Dynamical Behavior in Behavior in Simple and Coupled Systems of Continuous Josephson－Junctions

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Fractional-Linear Transformations of Operator Balls; Applications to Dynamical Systems

The operator sets, which are the subject of this paper, have been studied in many papers where, under different restrictions on the generating operators, convexity, compactness in the weak operator topology, and nonemptiness were proved for sets of different classes under study. Then the results obtained were used in these papers to solve several applied problems. Namely, they played the key role in establishing the dichotomy of nonautonomous dynamical systems, with either continuous or discrete time. In the present paper, we generalize and sharpen the already known criteria and obtain several new criteria for convexity, compactness, and nonemptiness of several special operator sets. Then, using the assertions obtained, we construct examples of sets of the form under study which are nonconvex, noncompact in the weak operator topology, as well as empty, and are generated by ＂smooth＂ operators of a special class. The existence problem for such sets remained open until the authors of this paper announced some of its results.     

The Baire Class of Topological Entropy of Non–Autonomous Dynamical Systems

We prove that the lower topological entropy considered as a function on the space of sequences of continuous self–maps of a metric compact space belongs to the second Baire class and the upper one belongs to the fourth Baire class.