Algebraic aspects of integrability for polynomial differential systems

In this article we summarize the results on algebraic aspects of integrability for polynomial differential systems and its application, which include the Darboux, elementary and Liouvelle integrability. Darboux theory of integrability was found by Darboux in 1878, and it becomes extremely useful in study of the center focus problem, of bifurcation, of limit cycle problem and of global dynamics. The importance of Darboux theory of integrability is also presented by the Singer's theorem for planar polynomial differential system. That is, if a polynomial system is Liouville integrable, then it is Darboux integrable, i.e. the system has a Darboux first integral or a Darboux integrating factor.     

An improvement to Darboux integrability theorem for systems having a center

We consider planar polynomial differential systems of degree m with a center at the origin and with an arbitrary linear part. We show that if the system has m(m + 1)/2 − [(m + 1)/2] algebraic solutions or exponential factors then it has a Darboux integrating factor. This result is an improvement of the classical Darboux integrability theorem and other recent results about integrability.     

INTEGRABILITY VIA INVARIANT ALGEBRAIC CURVES FOR PLANAR POLYNOMIAL DIFFERENTIAL SYSTEMS

1 IntroductionBy definition a complex (respectively reao planar polynomial differentialsystem or simply a polynomial system is a differential system of the formdx. da.~ = i = p(x,y), ac = i = Q(x,y), (l)dt dt ~ y = Q(x,y), (l)where the dependent variables x and y are complex (respectively real), theindependent one (the time) t is real, and P and Q are polynomials in thevariables x and y with complex (respectively real) coefficients. In all thispaper m = ma-c{deg P, deg Q} denote the degree o…     

Integrability and solvability of polynomial Liénard differential systems

We provide the necessary and sufficient conditions of Liouvillian integrability for Liénard differential systems describing nonlinear oscillators with a polynomial damping and a polynomial restoring force. We prove that Liénard differential systems are not Darboux integrable excluding subfamilies with certain restrictions on the degrees of the polynomials arising in the systems. We demonstrate that if the degree of a polynomial responsible for the restoring force is greater than the degree of a polynomial producing the damping, then a generic Liénard differential system is not Liouvillian integrable with the exception of linear Liénard systems. However, for any fixed degrees of the polynomials describing the damping and the restoring force we present subfamilies possessing Liouvillian first integrals. As a by-product of our results, we find a number of novel Liouvillian integrable subfamilies. In addition, we study the existence of nonautonomous Darboux first integrals and nonautonomous Jacobi last multipliers with a time-dependent exponential factor.     

Integrable nonholonomic deformation of modified Volterra lattice equation

The modified Volterra lattice equation with nonholonomic constrain has been considered in this paper. The integrability of the deformed model has been demonstrated by providing a Lax pair. Applying the gauge transformation to the Lax pair, we establish Darboux transformation theorem for the nonholonomic deformation equation. Some analytic solutions of the system are obtained via the one-fold and two-fold Darboux transformations. The deformation on explicit solutions exhibits different curvy profiles and propagation trajectories that were not found in modified Volterra lattice equation.     

Generalized cofactors and nonlinear superposition principles

It is known from Lie's works that the only ordinary differential equation of first order in which the knowledge of a certain number of particular solutions allows the construction of a fundamental set of solutions is, excepting changes of variables, the Riccati equation. For planar complex polynomial differential systems, the classical Darboux integrability theory exists based on the fact that a sufficient number of invariant algebraic curves permits the construction of a first integral or an inverse integrating factor. In this paper, we present a generalization of the Darboux integrability theory based on the definition of generalized cofactors.     

Varieties of local integrability of analytic differential systems and their applications

In this paper we provide a characterization of local integrability for analytic or formal differential systems in Rⁿ${\mathbb{R}}^n$ or Cⁿ${\mathbb{C}}^n$ via the integrability varieties. Our result generalizes the classical one of Poincaré and Lyapunov on local integrability of planar analytic differential systems to any finitely dimensional analytic differential systems. As an application of our theory we study the integrability of a family of four-dimensional quadratic Hamiltonian systems.     

Remarks on integrable systems

The problem of integrability conditions for systems of differential equations is discussed. Darboux’s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an n-dimensional space, which admit the algebra of symmetry fields of dimension ? n. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.     

Integrability,computation and applications

The study of integrable systems and the notion of integrability has been re-energized with the discovery that infinite-dimensional systems such as the Korteweg-de Vries equation are integrable. In this paper, the following novel aspects of integrability are described: (i) solutions of Darboux, Brioschi, Halphen-type systems and their relationships to monodromy problems and automorphic functions, (ii) computational chaos in integrable systems, (iii) we explain why we believe that homoclinic structures and homoclinic chaos associated with nonlinear integrable wave problems, will be observed in appropriate laboratory experiments.     

Darboux polynomials,balances and Painlevé property

For a given polynomial differential system we provide different necessary conditions for the existence of Darboux polynomials using the balances of the system and the Painlevé property. As far as we know, these are the first results which relate the Darboux theory of integrability, first, to the Painlevé property and, second, to the Kovalevskaya exponents. The relation of these last two notions to the general integrability has been intensively studied over these last years.     

Analytic integrability inside a family of degenerate centers

In this paper we study the analytic integrability around the origin inside a family of degenerate centers or perturbations of them. For this family analytic integrability does not imply formal orbital equivalence to a Hamiltonian system. It is shown how difficult is the integrability problem even inside this simple family of degenerate centers or perturbations of them.     

Integrability conditions for complex systems with homogeneous

In this paper we obtain conditions for the existence  of a local analytic first integral for four eight-parameter families of quintic complex system. We also discuss computational difficulties arising in the study of the problem of integrability for these systems.     

Coupled systems of nonlinear wave equations and finite-dimensional lie algebras I

We study coupled systems of nonlinear wave equations from the point of view of their formal Darboux integrability. By making use of Vessiot's geometric theory of differential equations, it is possible to associate to each system of nonlinear wave equations a module of vector fields on the second-order jet bundle — the Vessiot distribution. By imposing certain conditions of the structure of the Vessiot distributions, we identify the so-called separable Vessiot distributions. By expressing the separable Vessiot distributions in a basis of singular vector fields, we show that there are, at most, 27 equivalence classes of such distributions. Of these, 14 classes are associated with Darboux integrable nonlinear systems. We take one of these Darboux integrable classes and show that it is in correspondence with the class of six-dimensional simply transitive Lie algebras. Finally, this later result is used to reduce the problem of constructing exact general solutions of the nonlinear wave equations understudy to the integration of Lie systems. These systems were first discovered by Sophus Lie as the most general class of ordinary differential equations which admit nonlinear superposition principles.     

Integrable maps from Galois differential algebras,Borel transforms and number sequences

A new class of integrable maps, obtained as lattice versions of polynomial dynamical systems is introduced. These systems are obtained by means of a discretization procedure that preserves several analytic and algebraic properties of a given differential equation, in particular symmetries and integrability (see Tempesta, 2010 [40]). Our approach is based on the properties of a suitable Galois differential algebra, that we shall call a Rota algebra. A formulation of the procedure in terms of category theory is proposed. In order to render the lattice dynamics confined, a Borel regularization is also adopted. As a byproduct of the theory, a connection between number sequences and integrability is discussed.     

The Center Conditions and Hopf Cyclicity for a 3D Lotka-Volterra System

The main objective of this paper is not only to find necessary and The main objective of this paper is not only to find necessary and sufficient conditions for the existence of a center on a local center manifold for a three dimensional Lotka-Volterra system, but also to determine the maximum number of limit cycles that can bifurcate from the positive equilibrium as a fine focus. Firstly, the singular point quantities are computed and simplified to obtain necessary conditions for local integrability, and Darboux method is applied to show the sufficiency. Then, the Hopf bifurcation on the center manifold is investigated, from this, the conclusion of at most five small limit cycles generated in the vicinity of the equilibrium is obtained. To the best of our knowledge, this is the first case with five possible limit cycles for the cyclicity of 3D Lotka-Volterra systems.     

Darboux System as Three-Dimensional Analog of Liouville Equation

We discuss the problems of the connections of the modern theory of integrability and the corresponding overdetermined linear systems with works of geometers of the late nineteenth century. One of these questions is the generalization of the theory of Darboux–Laplace transforms for second-order equations with two independent variables to the case of three-dimensional linear hyperbolic equations of the third order. In this paper we construct examples of such transformations. We consider applications to the problem of orthogonal curvilinear coordinate systems in ?³.     

Darboux integrability and algebraic limit cycles for a class of polynomial differential systems

This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems x=x-y+P n+1(x,y)+xF2n(x,y),y=x+y+Q n+1(x,y)+yF2n(x,y),where P i(x,y),Q i(x,y)and F i(x,y)are homogeneous polynomials of degree i.Within this class,we identify some new Darboux integrable systems having either a focus or a center at the origin.For such Darboux integrable systems having degrees 5and 9 we give the explicit expressions of their algebraic limit cycles.For the systems having degrees 3,5,7 and 9and restricted to a certain subclass we present necessary and sufficient conditions for being Darboux integrable.     

Reduction of integrable planar polynomial differential systems

The integrability problem consists in finding the class of functions, a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit a Darboux, elementary, Liouvillian or Weierstrass first integral. The reduction problem of an integrable planar system consists in finding the class of functions, a map that reduces the original system (transforms into a simple system or equation) must belong to. We identify the class of functions of this map for polynomial, rational, Darboux, elementary, Liouvillian and Weierstrass integrable systems.     

Integrability conditions of a resonant saddle in Liénard-like complex systems

We consider a complex differential system with a resonant saddle that remind the classical Liénard systems in the real plane. For such systems we determine the conditions of analytic integrability of the resonant saddle.