Abstract
In this paper we introduce an explicit expression of first integral, then we prove the nonexistence of periodic orbits, then consequently the non-existence of limit cycles of two-dimensional Kolmogorov system, where R(x, y), S (x, y), P (x, y), Q(x, y),M (x, y), N (x, y) are homogeneous polynomials of degrees m, a, n, n, b, b, respectively. We introduce concrete example exhibiting the applicability of our result.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Gao, P. “Hamiltonian Structure and First Integrals for the Lotka–Volterra Systems”, Phys. Lett. A 273, No. 1–2, 85–96 (2000).
Li, C., Llibre, J. “The Cyclicity of Period Annulus of a Quadratic Reversible Lotka–Volterra System”, Nonlinearity 22, No. 12, 2971–2979 (2009).
Llibre, J., Valls, C. “Polynomial, Rational and Analytic First Integrals for a Family of 3-Dimensional Lotka–Volterra Systems”, Z. Angew.Math. Phys. 62, No. 5, 761–777 (2011).
Huang, X. “Limit in a Kolmogorov-Type Model”, Internat. J. Math. and Math Sci. 13, No. 3, 555–566 (1990).
Llibre, J., Salhi, T. “On the Dynamics of a Class of Kolmogorov Systems”, Appl.Math. Comput. 225, 242–245 (2013).
Llyod, N.G., Pearson, J.M., Sáez, E., Szánto, I. “Limit Cycles of a Cubic Kolmogorov System”, Appl.Math. Lett. 9, No. 1, 15–18 (1996).
May, R. M. Stability and Complexity in Model Ecosystems (Princeton, New Jersey, 1974).
Lavel, G., Pellat, R. “Plasma physics”, in: Proceedings of Summer School of Theoretical Physics (Gordon and Breach, New York, 1975).
Busse, F. H. “Transition to Turbulence via the Statistical Limit Cycle Route”, in: Synergetics (Springer-Verlag, Berlin, 1978), P. 39.
Boukoucha, R. “On the Dynamics of a Class of Kolmogorov Systems”, Sib. Elektron. Mat. Izv. 13, 734–739 (2016).
Boukoucha, R., Bendjeddou, A. “On theDynamics of a Class of Rational Kolmogorov Systems”, J.Nonlinear Math. Phys. 23, No. 1, 21–27 (2016).
Chavarriga, J., García, I. A. “Existence of Limit Cycles for Real Quadratic Differential Systems With an Invariant Cubic”, Pacific J.Math. 223, No. 2, 201–218 (2006).
Al-Dosary Khalil, I. T. “Non-Algebraic Limit Cycles for Parameterized Planar Polynomial Systems”, Int. J. Math. 18, No. 2, 179–189 (2007).
Dumortier, F., Llibre, J., Artés, J. Qualitative Theory of Planar Differential Systems (Springer, Universitex, Berlin, 2006).
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 Appl. Sci. Engrg. 24, No. 5, Article ID 1450065 (2014).
Bendjeddou, A., Boukoucha, R. “Explicit Non-Algebraic Limit Cycles of a Class of Polynomial Systems”, FJAM 91, No. 2, 133–142 (2015).
Bendjeddou, A., Boukoucha, R. “Explicit Limit Cycles of a Cubic Polynomial Differential Systems”, Stud. Univ. Babes-BolyaiMath. 61, No. 1, 77–85 (2016).
Gasull, A., Giacomini, H., Torregrosa, J. “Explicit Non-Algebraic Limit Cycles for Polynomial Systems”, J. Comput. Appl.Math. 200, No. 1, 448–457 (2007).
Cairó, 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 40, No. 24, 6329–6348 (2007).
Author information
Authors and Affiliations
Corresponding author
Additional information
The text was submitted by the author in English
About this article
Cite this article
Boukoucha, R. On the Non-Existence of Periodic Orbits for a Class of Two-Dimensional Kolmogorov Systems. Russ Math. 62, 1–6 (2018). https://doi.org/10.3103/S1066369X18010012
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X18010012