Explicit expression for a first integral for some classes of polynomial differential systems

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1 Int. J. Adv. Appl. Math. and Mech ISSN: Journal homepage: International Journal of Advances in Applied Mathematics and Mechanics Eplicit epression for a first integral for some classes of polynomial differential systems Research Article Ahmed Bendjeddou a, Rachid Boukoucha b, a Departement de Mathematics, Faculty of Sciences, University of Setif, Setif, Algeria b Department of Mathematics, Faculty of Sciences Eactes and Informatics, University of Jijel, Jijel, Algeria Received 0 June 015; accepted in revised version 0 August 015 Abstract: In this paper we charecterize the integrability and introduce an eplicit epression of first integral then consequently the curves which are formed by the trajectories of the planar differentials systems of the form { = P n + Rm y = Q n + yrm, and { = P n y = y Q n MSC: where P n, Qn, Rm homogeneous polynomials of degree n, n, m respectively. Concrete eamples ehibiting the applicability of our result is introduced. 34C07 37C7 37K10 Keywords: Autonomous differential system kolmogorov system First integral, curves 015 The Authors. This is an open access article under the CC BY-NC-ND license 1. Introduction and statement of the main results The autonomous differential system on the plane given by = d t = F, y = d y d t = G. 1 where F and G are reals functions In the qualitative theory of planar dynamical systems see [1 7], one of the most important topics is related to the second part of the unsolved Hilbert 16th problem, There is a huge literature about limit cycles, most of them deal essentially with their detection, their number and their stability and rare are papers concerned by giving them eplicitly see [8 13]. There eist three main open problems in the qualitative theory of real planar differential systems, the distinction between a centre and a focus, the determination of the number of limit cycles and their distribution, and the determination of its integrability. The importance for searching first integrals of a given system was already noted by Poincaré in his discussion on a method to obtain polynomial or rational first integrals. One of the classical tools in the Corresponding author. adess: bendjeddou@univ-setif.dz Ahmed Bendjeddou, rachid_boukecha@yahoo.fr Rachid Boukoucha

2 Ahmed Bendjeddou, Rachid Boukoucha / Int. J. Adv. Appl. Math. and Mech classification of all trajectories of a dynamical system is to find first integrals, for or more details about first integral see for instance [14 3]. System 1 is integrable on an open set Ω of R if there eists a non constant C 1 function H : Ω R, called a first integral of the system on Ω, which is constant on the trajectories of the system 1 contained in Ω, i.e. if d H d t = H F + H y G 0 in the points of Ω. Moreover, H = h is the general solution of this equation, where h is an arbitrary constant. It is well known that for differential systems defined on the plane R the eistence of a first integral determines their phase portrait see [4]. In this paper we are interested in studying the integrability and the curves which are formed by the trajectories of the -dimensional polynomial systems of the form { = P n + Rm y = Q n + yrm, and { = P n y = y Q n 3 where P n, Qn, Rm homogeneous polynomials of degree n, n, m respectively. The autonomous differential system 3 on the plane known as Kolmogorov system see [5]. There are many natural phenomena which can be modeled the Kolmogorov systems such as mathematical ecology and population dynamics see [6] chemical reactions, plasma physics see [7], hyodynamics see [17], economics etc. We define the trigonometric functions f 1 = P n cos,sincos +Q n cos,sinsin, f = R m cos,sin, f 3 = Q n cos,sincos P n cos,sinsin, g 1 = P n cos,sincos +Q n cos,sinsin, g = Q n cos,sincos sin P n cos,sincos sin. Our main result on the integrability and the curves which are formed by the trajectories of the planar differentials systems and 3 is the following. Theorem 1.1. Consider polynomials systems and 3, then the following statements hold. a If f 3 0 and 0, then system has the first integral H = + y ep A ωdω ep A ωdω B wd w where A = f 1 f 3, B = f f 3, and = n m 1 Moreover, the curves which are formed by the trajectories of the differential system, written as + y = [ y A ωdω + ep ] A ωdω ep A ωdω B wd w b If f 3 0 and = 0, then system has the first integral H = + y ep A ω + B ωdω Moreover, the curves which are formed by the trajectories of the differential system, written as + y = A ω + B ωdω c If f 3 = 0 for all R, then system has the first integral H = y.

3 11 Eplicit epression for a first integral for some classes of polynomial differential systems Moreover, the curves which are formed by the trajectories of the differential system, written as y = hhere h R d If g 0 and + 1 0, then system 3 has the first integral H = + y +1 ep + 1 C ωdω + 1 ep + 1 C ωdω D wd w where C = g 1 g, D = f g, and = n m 1 Moreover, the curves which are formed by the trajectories of the differential system 3, written as + y = + 1ep + 1 y C ωdω e If g 0 and + 1 = 0, then system 3 has the first integral H = + y ep C ω + D ωdω + 1 y C ωdω + ep + 1 w C ωdω D wd w +1 Moreover, the curves which are formed by the trajectories of the differential system 3, written as + y = C ω + D ωdω f If g = 0 for all R, then system 3 has the first integral H = y, Moreover, the curves which are formed by the trajectories of the differential system 3, written as y = hhere h R Proof. In order to prove our results a, b and c we write the polynomial differential system in polar coordinates r,, defined by = r cos, and y = r sin, then system becomes { r = f 1 r n + f r m+1 = f 3 r n 1 4 where the functionsf 1, f and f 3 are given in introduction, and r = d t, = d t. If f 3 0, and 0 We take as new independent variable the variable, then the differential system 4 becomes the differential equation 1 = A r + B r where the functions A and B are the ones defined in statement a of theorem 1, and = n m 1. We note that the differential equation 5 is a Bernoulli differential equation. By introducing the standard change of variables ρ = r the Bernoulli differential equation becomes the linear equation dρ = A ρ + B The general solution of linear equation 6 is [ w ] ρ = ep A ωdω α + ep A ωdω B wd w where α R, which has the first integral H = + y ep y A ωdω ep A ωdω B wd w 5 6 The curves H = h with h R, which are formed by trajectories of the differential system, can be written as + y = ep y A ωdω y A ωdω + ep w A ωdω B wd w

4 Ahmed Bendjeddou, Rachid Boukoucha / Int. J. Adv. Appl. Math. and Mech Hence statement a of Theorem 1.1 is proved. Suppose now that f 3 0, and = 0. Taking as new independent variable the coordinate, this differential system 4 writes = A + B r The general solution of equation 7 is r = αep A ω + B ωdω 7 where α R, which has the first integral H = + y ep A ω + B ωdω The curves H = h with h R, which are formed by trajectories of the differential system, can be written as + y = A ω + B ωdω Hence statement b of Theorem 1.1 is proved. Assume now that f 3 = 0 for all R, then, from 4 it follows that = 0. So the straight lines through the origin of coordinates of the differential system are invariant by the flow of this system. Hence, H = y is a first integral of the system, then since all straight lines through the origin are formed by trajectories. Then the curves H = h with h R, which are formed by trajectories of the differential system, can be written as y = hhere h R. This completes the proof of statement c of Theorem 1.1. In order to prove our results d, e and f, we write the polynomial differential system 3 in polar coordinates r,, defined by = r cos, and y = r sin, then system 3 becomes { r = g 1 r n+1 + f r m+1 = g r n 8 where f, g 1, g the trigonometric functions are given in introduction. If g 0, and Taking as new independent variable the coordinate, then the differential system 8 becomes the differential equation = C r + D r where the functions C and D are the ones defined in statement d of Theorem 1, and = n m 1 which is a Bernoulli equation. By introducing the standard change of variables ρ = r +1 we obtain the linear equation dρ = + 1 C ρ + D The general solution of linear Eq. 10 is [ w ] ρ = ep + 1 C ωdω α ep + 1 C ωdω D wd w where α R, which has the first integral H = + y +1 ep + 1 y C ωdω + 1 ep + 1 C ωdω D wd w 9 10 The curves H = h with h R, which are formed by trajectories of the differential system 3, can be written as + y = + 1ep + 1 y C ωdω Hence statement d of Theorem 1.1 is proved. + 1 y C ωdω + ep + 1 w C ωdω D wd w +1

5 114 Eplicit epression for a first integral for some classes of polynomial differential systems Suppose now that g 0, and + 1 = 0 Taking as new independent variable the coordinate, this differential system 8 writes = A + B r The general solution of equation 11 is r = αep C ω + D ωdω 11 where α R, which has the first integral H = + y ep C ω + D ωdω The curves H = h with h R, which are formed by trajectories of the differential system 3, can be written as + y = C ω + D ωdω Hence statement e of 1.1 is proved. Assume now that g = 0 for all R, then, from system 8 it follows that = 0. So the straight lines through the origin of coordinates of the differential system 3 are invariant by the flow of this system. Hence, H = y is a first integral of the system, then since all straight lines through the origin are formed by trajectories. Then the curves H = h with h R, which are formed by trajectories of the differential system 3, can be written as y = hhere h R. This completes the proof of statement f of Theorem Eamples The following eamples are given to illustrate our result. Eample.1. if we take P 3 = 3 y + y y 3, Q 3 = 3 + y + y + y 3, R 0 = 1, then system reads { = + 3 y + y y 3 y = y y + y + y 3 which has the first integral H = + y 1 ep y. The curves which are formed by the trajectories of the differential system written as +y 1 = y where h R. Eample.. if we take P 3 = 3 + y, Q 3 = y y + y, R = 3 + 3y, then system 3 reads { = 3 + y y y = y y y + y y which has the first integral H = y h R. + y + 3. The curves which are formed by the trajectories of the differential system written as y + y + 3 = h where Eample.3. if we take P 3 = 3 + y, Q 3 = 3 + y 3 + y + y, R = y, then system 3 reads { = 3 + y y y = y 3 + y 3 + y + y y which has the first integral H y + y + = + y +. y + y + The curves which are formed by the trajectories of the differential system written as + y + = h where h R.

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