Annotation of the results obtained in 2021
The third stage of the project is devoted to developing a new theory of integrability of two-dimensional polynomial differential systems. This theory is based on the properties of asymptotic Puiseux series satisfying a non-autonomous first order ordinary differential equation associated with the initial system. This theory, called the theory of Puiseux integrability, can be considered as a generalization of the theory of Darboux integrability. According to the new theory, algebraic and exponential invariants lose the polynomial dependence on one of the variables. Coefficients of polynomials describing invariants become Puiseux series in neighborhoods of a finite or an infinite point. Since the sets of Puiseux series are algebraically closed fields, it becomes possible to consider only the invariants of the first degree. The necessary and sufficient conditions of the Puiseux integrability are obtained for an arbitrary two-dimensional differential system. New theory can find two-dimensional systems with first integrals that are not Liouville functions. Certain examples of systems Liouville non-integrable, but Puiseux integrable are obtained. The Liénard differential systems and their generalizations, often called the Levinson–Smith or the Kukles systems, are studied in details. Upper bounds on the degrees of irreducible invariant algebraic curves for the Levinson-Smith systems quadratic and cubic with respect to one of the variables are obtained. It is proved that these systems cannot have more than three pairwise distinct invariant algebraic curves simultaneously. The problem of Liouville integrability is completely solved for non-resonant near infinity Liénard systems. The resonance in such systems appears only for certain restrictions on the degrees of the polynomials in the systems. The set of resonant systems has Lebesgue measure zero in the set of all polynomial Liénard systems. During the third stage of the project, it is proved that the generalized Sundman transformations and some other nonlocal transformations map autonomous invariant curves to autonomous invariant curves whenever the functions giving the transformations do not explicitly contain the independent variable. Explicit formulae connecting invariant curves and cofactors are constructed. The families of transformations permit to extend known classifications of invariant curves to the whole equivalence class of a given system. Thus, it becomes possible to perform a classification of invariant curves for non-polynomial differential systems. Equivalence classes for two planar differential systems connected with the equations II and VII of the Painlevé–Gambier classification are constructed. These classes describe cubic and quadratic with respect to the first derivative autonomous non-linear oscillators. An explicit form of the differential relations on the functional coefficients of the equations giving the equivalence criteria is obtained. Parametric representations of the general solutions are found. Examples of two-dimensional differential systems belonging to equivalence classes under consideration are considered. In particular, is shown that the set of equations appearing via the traveling wave reductions in a reaction-diffusion equation is equivalent to a differential system connected with the equation II of the Painlevé–Gambier classification. The method of finding algebraic invariants developed in the project is used for studying the properties and for constructing exact solutions of three-dimensional and four-dimensional polynomial differential systems important from an applied point of view. In particular, algebraically invariant solutions for the Rucklidge system and for traveling wave reductions of the non-integrable generalizations of the Kortweg–de Vries equation and the Kawahara equation are classified.

 

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Annotation of the results obtained in 2019
The first stage of the project is devoted to developing algebraic and analytic methods and approaches designed for investigating ordinary differential equations and dynamical systems. Detailed studies of physically relevant second-order autonomous and non-autonomous ordinary differential equations are implemented with the help of these methods. The problem of finding invariant algebraic curves for polynomial dynamical systems in the plane is considered. Necessary and sufficient conditions of their existence are obtained. These conditions are used to develop a method enabling one to perform a classification of irreducible invariant algebraic curves even if the curves contain dicritical singular points of the system under consideration. The main idea of the method is to use a factorization over the field of Puiseux series for bivariate polynomials related to algebraic curves. The field of Puiseux series is algebraically closed. This fact yields linear factors with respect to one of the variables. The case of Puiseux series with positive rational Fuchs indices arising in the factorization of polynomials producing algebraic curves is considered in detail. The novel method is used to perform the classification of irreducible invariant algebraic curves for polynomial Liénard differential systems and their generalizations under certain restrictions on the degrees of polynomial coefficients. Some families of the differential equations under study are interesting from an applied point of view. Let us name a cubic-quintic Duffing oscillator and nonlinear oscillators with quadratic dumping. As a result, the Poincaré problem of finding an upper bound on the degrees of irreducible invariant algebraic curves is solved for differential equations under consideration. The knowledge of the complete set of irreducible invariants (algebraic and exponential) for a given polynomial dynamical system enables one to solve the problem of Darboux and Liouvillian integrability. Necessary and sufficient conditions of Liouvillian integrability for some families of polynomial Liénard differential systems and their generalizations containing a quadratic term with respect to the first derivative are obtained. Liouvillian first integrals are found in explicit form. Some of the differential equations under consideration possess first integrals expressible via the hypergeometric function. These first integrals are neither analytic nor meromorphic. Necessary and sufficient conditions for a polynomial dynamical system in the plane to have time dependent Jacobi last multiplies of a special form are obtained. It is proved that these Jacobi last multiplies can be constructed via algebraic and exponential invariants. A classification of the Jacobi last multiplies is performed for cubic Liénard differential equations with quadratic dumping in the Liouvillian nonintegrable cases. Let us note that the famous Duffing – Van der Pol oscillator belongs to the family of differential equations under considerations. In addition, for this family of differential equations and in particular for the Duffing – Van der Pol oscillator it is established that all the limit cycles are non-algebraic. The equivalence problem for nonlinear oscillators with a quadratic with respect to the first derivative term and an external force and the Laguerre normal form of second order linear differential equation is considered. As equivalence transformations, we use nonlocal transformations, that generalize Sundman transformations. Coefficients of the considered equations are assumed to be arbitrary sufficiently smooth functions. A correlation on the coefficients of the considered family of equations that provide the necessary and sufficient conditions for linearizability is found in the explicit form. Let us remark that there are no restrictions on the function that describes external force. It is shown that each linearizable equation possesses a first integral and an explicit representation of this first integral via the functions that define the family of equations under consideration is found. We demonstrate that the previously known integrable case of the Duffing – Van der Pol oscillator with an external force belongs to the constructed class of linearized equations. With the help of the Darboux integrability theory the question of the existence of an additional independent first integral for the equations from the considered family is investigated. It is shown that the first integrals which are Darboux functions can be found by means of invariant surfaces. Decompositions in the Puiseux functional series in the vicinity of infinity are constructed to investigate the existence of invariant surfaces. In particular, for the equation corresponding to the degrees of 3 and 4 of polynomial coefficients, it is proved that there are no additional Darboux first integrals. Furthermore, we study the equivalence problem for nonlinear non-autonomous second-order differential equations with linear and square terms with respect to the first derivative and damped harmonic oscillator. It is assumed that the equivalence transformations are given by the generalized Sundman transformations, and the coefficients of the considered family of equations are sufficiently smooth functions. We explicitly find the correlations on the coefficients of the considered family of equations that provide the necessary and sufficient conditions for linearizability. As a consequence, we show that each of the linearized equations has a first integral and construct its explicit expression in terms of coefficients of the equation and functions determining nonlocal transformations. It is proved that this first integral is autonomous if and only if the functions that define nonlocal transformations do not explicitly depend on the independent variable. The corresponding family of Liénard equations that possesses an autonomous first integral is found. In addition, a restriction on the parameters of the damped linear harmonic oscillator is established, which corresponds to the rational first integrals of the linearizable equations from the considered family. Several examples of linearizable equations are constructed. In particular, the generalized non-autonomous dissipative Duffing equation, a non-autonomous generalization of the cubic oscillator with linear dissipation and a non-autonomous generalization of the Duffing – Van der Pol oscillator. For each of these equations, the general solution in the parametric form and a first integral are constructed. It is shown that these first integrals allow one to find periodic solutions including limit cycles of the corresponding equations. In particular, solutions in the form of limit cycles are found for non-autonomous generalization of the Duffing – Van der Pol oscillator. We also consider a family of cubic Liénard oscillators with linear damping. The question of linearizability via non-local transformations for members of this family is studied. Restrictions on parameters corresponding to the necessary and sufficient conditions for linearizability are found. We consider cases of autonomous and non-autonomous non-local transformations separately. As a result two subfamilies of the considered family of equations are constructed. The first of them has an autonomous Darboux first integral and is therefore integrable. The first integral of the second family of equations is non-autonomous. Nevertheless, for this case the general solution in the parametrical form is obtained by inverting nonlocal transformations. Finally, we prove that the considered family of differential equations can be linearized using non-local transformations if and only if it possesses a Darboux first integral, which can be found with the help of the corresponding invariant algebraic curves.

 

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Annotation of the results obtained in 2020
The second stage of the project is devoted to developing a method of constructing and classifying algebraic invariants for autonomous ordinary differential equations of arbitrary orders. Algebraic invariants are polynomials in two variables that define first order ordinary differential equations compatible with the original equation. The theory of compatible equations has some applications in various fields of mathematics. A universal method of finding algebraic invariants is developed. The method is applicable in both two-dimensional and higher-dimensional cases. The algorithm of the method allows a computer implementation in systems of symbolic computations. The method is used to construct and classify algebraically invariant solutions for differential equations that are important from an applied point of view. Transcendental meromorphic functions that are elliptic or rational with an exponential argument are commonly called W-meromorphic functions in honor of Karl Weierstrass. It is well known that W-meromorphic solutions of ordinary differential equations are algebraically invariant solutions. This fact allows us to use the method developed in the project to find W-meromorphic solutions. All the steps of the method in difficult cases are worked out in details. More specifically, autonomous ordinary differential equations that have several dominant differential monomials and (or) an infinite number of local solutions given by Laurent series in a neighborhood of movable poles are considered. The method of constructing algebraic invariants is also applicable in the case of aforementioned equations. This method generalizes a number of other methods such as ansatz methods and sub-equations methods. The Darboux integrability problem for quadratic–quintic Duffing–van der Pol equations belonging to degenerate families of polynomial Liénard differential equations is considered. Degenerate Liénard equations are difficult to study by analytical methods. This is due to the dependence on the parameters of Fuchs indices related to asymptotic series, in particular, Puiseux series. Necessary and sufficient conditions for the existence of autonomous and non-autonomous Darboux first integrals are obtained for all values of the parameter with the exception of a discrete set. These first integrals are found explicitly. It is proved that the existence of such first integrals is the necessary and sufficient condition of the linearizability by means of nonlocal transformations generalizing Zundman transformations. The connection between the Darboux integrability theory and the method of nonlocal transformations is probably demonstrated for the first time in the framework of the present project. During the second stage of the project equivalence problems via generalized nonlocal transformations are also considered. In particular, the linearization problem via these transformations for second order differential equations that are cubic with respect to the first derivative and a damped harmonic oscillator is studied. Correlations on the coefficients of this family of equations that provide the necessary and sufficient conditions of linearization are found in an explicit form. It is shown that any linearizable equation possesses a first integral. An explicit expression of this first integral involving the coefficients of the corresponding equation is found. In addition, we construct a Jacobi last multiplier or an integrating factor for linearizable equations. We demonstrate that these equations do not have limit cycles. Furthermore, equivalence problems for type I Painleve–Gambier equations and second order ordinary differential equations that are cubic with respect to the first derivative are considered. Equivalence transformations are given by generalized nonlocal transformations. Equivalence criteria are found in an explicit form. A classification of second order ordinary differential equations that are quadratic with respect to the first derivative and admit a certain integrating factor is carried out. Necessary and sufficient conditions for the existence of this integrating factor are obtained. Several examples including the cubic Liénard equations with linear damping and the fifth order Duffing–van der Pol equations are considered in details. New integrable families of Liénard differential equations that cannot be linearized via nonlocal transformations are derived.

 

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