Explicit expression for a first integral for some classes of two-dimensional differential systems

R. Boukoucha

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 903-913.

Résumé

In this paper we are interested in studying the existence of first integrals and then the trajectories for classes of two-dimensional differential systems of the forms \begin{equation*} \left\{ \begin{array}{l} x^{\prime }=\frac{P\left( x,y\right) ^{\alpha }}{T\left( x,y\right) ^{\beta } }+x\frac{R\left( x,y\right) ^{\gamma }}{S\left( x,y\right) ^{\delta }}, \\ y^{\prime }=\frac{Q\left( x,y\right) ^{\alpha }}{K\left( x,y\right) ^{\beta } }+y\frac{R\left( x,y\right) ^{\gamma }}{S\left( x,y\right) ^{\delta }}, \end{array} \right. \end{equation*} and \begin{equation*} \left\{ \begin{array}{l} x^{\prime }=x\left( \frac{P\left( x,y\right) ^{\alpha }}{T\left( x,y\right) ^{\beta }}+\frac{R\left( x,y\right) ^{\gamma }}{S\left( x,y\right) ^{\delta } }\right) , \\ y^{\prime }=y\left( \frac{Q\left( x,y\right) ^{\alpha }}{K\left( x,y\right) ^{\beta }}+\frac{R\left( x,y\right) ^{\gamma }}{S\left( x,y\right) ^{\delta } }\right) , \end{array} \right. \end{equation*} where $a,$ $b,$ $n,$ $m$ are positive integers, $\alpha ,$ $\beta ,$ $\gamma ,$ $\delta \in \mathbb{Q} $ and $P\left( x,y\right) ,$ $Q\left( x,y\right) ,$ $R\left( x,y\right) ,$ $ T\left( x,y\right) ,$ $K\left( x,y\right) ,$ $S\left( x,y\right) $ are homogeneous polynomials of degree $n,$ $n,$ $m,$ $a,$ $a,$ $b$ respectively. Concrete examples exhibiting the applicability of our result are introduced.

Détail

Keywords: autonomous differential system, Kolmogorov system, first integral, trajectories, Hilbert 16th problem.

Bibliographie

[1] A. A. Andronov, E. A. Leontovich, I. I. Gordon, A. G. Maier, Qualitative Theory of Second-Order Dynamic Systems, John Wiley and Sons, New York, 1973 | MR | Zbl

[2] A. Bendjeddou, R. Boukoucha, “Explicit non-algebraic limit cycles of a class of polynomial systems”, FJAM Volume, 91:2 (2015), 133–142 | DOI | MR | Zbl

[3] R. Boukoucha, A. Bendjeddou, “On the dynamics of a class of rational Kolmogorov systems”, Journal of Nonlinear Mathematical Physics, 23:1 (2016), 21–27 | DOI | MR

[4] R. Boukoucha, “On the Dynamics of a Class of Kolmogorov Systems”, Siberian Electronic Mathematical Reports, 13 (2016), 734–739 | MR | Zbl

[5] F. H. Busse, Transition to turbulence via the statistical limit cycle route, Synergetics, Springer-Verlag, Berlin, 1978

[6] L. Cairó, J. Llibre, “Phase portraits of cubic polynomial vector fields of Lotka–Volterra type having a rational first integral of degree 2”, J. Phys. A, 40 (2007), 6329–6348 | DOI | MR | Zbl

[7] C. Christopher, C. Li, S. Yakovenko, Advanced Course On Limit Cycles of Differential Equations, Centre de Recerca Matemàtica Bellaterra, Spain, 2006

[8] F. Dumortier, J. Llibre, J. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer, Berlin, 2006 | MR | Zbl

[9] P. Gao, “Hamiltonian structure and first integrals for the Lotka–Volterra systems”, Phys. Lett. A, 273 (2000), 85–96 | DOI | MR | Zbl

[10] Bull. Amer. Math. Soc., 8 (1902), 437–479 | DOI | MR | Zbl | Zbl

[11] A. Goriely, Integrability and nonintegrability of dynamical systems, Advanced Series in Nonlinear Dynamics, 19, World Scientific Publishing Co., Inc., River Edge, NJ, 2001 | DOI | MR | Zbl

[12] N. A. Korol, The integral curves of a certain differential equation, Minsk. Gos. Ped. Inst., Minsk, 1973, 47–51 (In Russian) | MR | Zbl

[13] G. Lavel, R. Pellat, “Plasma Physics”, Proceedings of Summer School of Theoreal Physics, Gordon and Breach, New York, 1975

[14] J. Llibre, T. Salhi, “On the dynamics of class of Kolmogorov systems”, J. Appl. Math. Comput., 225 (2013), 242–245 | DOI | MR | Zbl

[15] J. Llibre, C. Valls, “Polynomial, rational and analytic first integrals for a family of 3-dimensional Lotka–Volterra systems”, Z. Angew. Math. Phys., 62 (2011), 761–777 | DOI | MR | Zbl

[16] N. G. Llyod, J. M. Pearson, E. Saez, I. Szanto, “Limit cycles of a Cubic Kolmogorov System”, Appl. Math. Lett., 9:1 (1996), 15–18 | DOI | MR | Zbl

[17] R. M. May, Stability and complexity in Model Ecosystems, Princeton, New Jersey, 1974

[18] L. Perko, Differential equations and dynamical systems, Texts in Applied Mathematics, 7, Third edition, Springer-Verlag, New York, 2001 | DOI | MR | Zbl

[19] H. Poincaré, “Sur l'intégration algébrique des équations différentielles du premier ordre et du premier degré I and II”, Rendiconti del circolo matematico di Palermo, 5 (1891), 161–191 ; 11 (1897), 193–239 | DOI | Zbl | DOI | Zbl

[20] M. J. Prelle, M. F. Singer, “Elementary first integrals of differential equations”, Trans. Amer. Math. Soc., 279:1 (1983), 215–229 | DOI | MR | Zbl

[21] M. F. Singer, “Liouvillian first integrals of differential equations”, Trans. Amer. Math. Soc., 333 (1992), 673–688 | DOI | MR | Zbl