Integrability and invariant algebraic curves for a class of Kolmogorov systems

R. Boukoucha

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 2, pp. 311-318

Résumé

There are many natural phenomena which can be modeled by Kolmogorov systems such as mathematical ecology, population dynamics, etc.. One of the more classical problems in the qualitative theory of planar differential systems is to characterize the existence or not of first integrals. For a two dimensional system the existence of a first integral completely determines its phase portrait. The question to determine the invariant algebraic curves of a given planar vector field, or to decide that no such curves exist, is part of a problem set forth by Poincare. There are strong relationships between the integrability of a system, and its number of invariant algebraic curves. It is shown that the existence of a certain number of algebraic curves for a system implies its Darboux integrability, that is the first integral is the product of the algebraic solutions with complex exponents. The study of the number and location of limit cycles is one of the most important topics which is related to the second part of the unsolved Hilbert 16th problem. In this paper we introduce an explicit expression of invariant algebraic curves, then we prove that these systems are Darboux integrable and introduce an explicit expression of a Liouvillian first integral. Then we discuss the possibility of existence and non-existence of limit cycles of the two dimensional Kolmogorov systems of the form $$ \left\{ \begin{array}{l} x^{\prime }=x\left( \displaystyle\frac{P\left( x,y\right) }{Q\left( x,y\right) }+\ln \Big\vert \displaystyle\frac{N\left( x,y\right) }{M\left( x,y\right) }\Big\vert \right) , \\ y^{\prime }=y\left( \displaystyle\frac{R\left( x,y\right) }{S\left( x,y\right) }+\ln \Big\vert \displaystyle\frac{N\left( x,y\right) }{M\left( x,y\right) }\Big\vert \right) , \end{array} \right. $$ where $P\left( x,y\right) ,Q\left( x,y\right) ,R\left( x,y\right) ,S\left( x,y\right) ,N\left( x,y\right) $ and $M\left( x,y\right) $ are homogeneous polynomials of degree $n,m,n,m,a$ and $a,$ respectively. The elementary method used in this paper seems to be fruitful to investigate more general planar differential Kolmogorov systems of ODEs in order to obtain explicit expression of invariant algebraic curves and for first integrals in order to characterize their trajectories. Finally, we discuss the possibility of existence and non-existence of limit cycles.

Détail

Keywords: Kolmogorov system, first integral, invariant algebraic curves, limit cycle, Hilbert 16th problem.

Bibliographie

[1] Bendjeddou A., Llibre J., Salhi T., “Dynamics of the differential systems with homogenous nonlinearity and a star node”, J. Differential Equations, 254 (2013), 3530–3537 | DOI | MR | Zbl

[2] Boukoucha R., “On the dynamics of a class of Kolmogorov systems”, J. Sib. Fed. Univ. Math. Phys., 9:1 (2016), 11–16 | DOI | MR

[3] Boukoucha R., “On the dynamics of a class of Kolmogorov systems”, Sib. Elektron. Mat. Izv., 13 (2016), 734–739 | DOI | MR

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

[5] Busse F.H., “Transition to turbulence via the statistical limit cycle route”, Ser. SSSYN, 11, Springer-Verlag, Berlin, 1978, 36–44 | DOI

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

[7] Chavarriga J., Grau M., “A family of non-Darboux-integrable quadratic polynomial differential systems with algebraic solutions of arbitrarily high degree”, Appl. Math. Lett., 16:6 (2003), 833–837 | DOI | MR | Zbl

[8] Christopher C. , Llibre J., “Integrability via invariant algebraic curves for planar polynomial differential systems”, Ann. Differential Equations, 16:1 (2000), 5–19 | MR | Zbl

[9] Darboux G., “Memoire sur les equations diffe rentielles algebriques du premier ordre et du premier degre”, Bull. Sci. Math. (2), 2:1 (1878), 60–96; 123-144; 151–200 | Zbl

[10] Dumortier F., Llibre J., Artes J.,, Qualitative Theory of Planar Differential Systems, ser. Qualitative, Springer, Berlin, 2006, 298 pp. | MR | Zbl

[11] Gao P., “Hamiltonian structure and first integrals for the Lotka-Volterra systems”, Phys. Lett. A, 273:1–2 (2000), 85–96 | DOI | MR | Zbl

[12] Gasull A., Giacomini H., Torregrosa J., “Explicit non-algebraic limit cycles for polynomial systems”, J. Comput. Appl. Math., 200:1 (2007), 448–457 | DOI | MR | Zbl

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

[14] Huang X., “Limit in a Kolmogorov-type model”, Internat J. of Mathematics and Mathematical Sciences, 13:3 (1990), 555–566 | DOI | MR | Zbl

[15] Laval G., Pellat R., Plasma Physics, Proc. of Summer School of Theoretical Physics, Gordon and Breach, New York, 1975

[16] Li C., Llibre J., “The cyclicity of period annulus of a quadratic reversible Lotka-Volterra system”, Nonlinearity, 22:12 (2009), 2971–2979 | DOI | MR | Zbl

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

[18] Llibre J., Valls C., “Polynomial, rational and analytic first integrals for a family of 3-dimensional Lotka-Volterra systems”, Z. Angew. Math. Phys., 62:5 (2011), 761–777 | DOI | MR | Zbl

[19] Llibre J., Yu J., Zhang X., “On the limit cycle of the polynomial differential systems with a linear node and Homogeneous nonlinearities”, Internat. J. Bifur. Chaos, 24:5, ser. Appl. Sci. Engrg. (2014), 1–7 | DOI | MR

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

[21] May R.M., Stability and complexity in model ecosystems, Princeton, New Jersey, 1974, 304 pp.

[22] Singer M.F., “Liouvillian first integrals of differential equations”, Trans. Amer. Math. Soc., 333 (1990), 673–688 | DOI | MR

[23] Poincare H., “Sur lintegration des equations differentielles du premier ordre et du premier degre I and II”, Rendiconti del Circolo Matematico di Palermo, 5 (1891), 161–191 | DOI | Zbl