More Compactification for Differential Systems


This article is a review and promotion of the study of solutions of differential equations in the neighborhood of infinity via a non traditional compactification. We define and compute critical points at infinity of polynomial autonomuos differential systems and develop an explicit formula for the leading asymptotic term of diverging solutions to critical points at infinity. Applications to problems of completeness and incompleteness (the existence and nonexistence respectively of global solutions) of dynamical systems are provided. In particular a quadratic competing species model and the Lorentz equations are being used as arenas where our technique is applied. The study is also relevant to the Painlevé property and to questions of integrability of dynamical systems.

Share and Cite:

H. Gingold and D. Solomon, "More Compactification for Differential Systems," Advances in Pure Mathematics, Vol. 3 No. 1A, 2013, pp. 190-203. doi: 10.4236/apm.2013.31A027.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] H. Poincare, “Mémoire sur les Courbes Définies par Une équation Différentielle,” Journal de Mathématiques Pures et Appliquées, Vol. 7, 1881, pp. 375-422.
[2] R. K. W. Roeder, “On Poincaré’s Fourth and Fifth Examples of Limit Cycles at Infinity,” Rocky Mountain Journal of Mathematics, Vol. 33, No. 3, 2003, pp. 1057-1082. doi:10.1216/rmjm/1181069943
[3] I. Bendixson, “Sur les Courbes Définies par des équations Différentielles,” Acta Mathematica, Vol. 24, No. 1, 1901, pp. 1-88. doi:10.1007/BF02403068
[4] A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. L. Maier, “Qualitative Theory of Second-Order Dynamic Systems,” Wiley, New York, 1973.
[5] L. A. Ahlfors, “Complex Analysis,” McGraw-Hill, New York, 1979.
[6] E. Hille, “Analytic Function Theory,” Chelsea Publishing Company, New York, 1982.
[7] D. Jordan and P. Smith, “Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers,” 4th Edition, In: Oxford Texts in Applied and Engineering Mathematics, 2007.
[8] S. Lefschetz, “Differential Equations: Geometric Theory,” Dover, New York, 1977.
[9] L. Perko, “Differential Equations and Dynamical Systems,” 3rd Edition, In: Texts in Applied Mathematics, Vol. 7, Springer-Verlag, Berlin, 2001.
[10] G. Sansone and R. Conti, “Non-Linear Differential Equations,” Pergamon Press, Oxford, 1964.
[11] C. Chicone and J. Sotomayor, “On a Class of Complete Polynomial Vector Fields in the Plane,” Journal of Differential Equations, Vol. 61, No. 3, 1986, pp. 398-418. doi:10.1016/0022-0396(86)90113-0
[12] A. Cima and J. Llibre, “Bounded Polynomial Vector Fields Bounded Polynomial Vector Fields,” Transactions of the American Mathematical Society, Vol. 318, No. 2, 1990, pp. 557-579. doi:10.1090/S0002-9947-1990-0998352-5
[13] D. Schlomiuk and N. Vulpe, “The Full Study of Planar Quadratic Differential Systems Possessing a Line of Singularities at Infinity,” Journal of Dynamics and Differential Equations, Vol. 20, No. 4, 2008, pp. 737-775. doi:10.1007/s10884-008-9117-2
[14] Y. I. Gingold and H. Gingold, “Geometric Properties of a Family of Compactifications,” Balkan Journal of Geometry and Its Applications, Vol. 12, No. 1, 2007, pp. 44-55.
[15] G. Q. Chen and Z. J. Liang, “Affine Classification for the Quadratic Vector Fields without the Critical Points at Infinity,” Journal of Mathematical Analysis and Applications, Vol. 172, No. 1, 1993, pp. 62-72. doi:10.1006/jmaa.1993.1007
[16] H. Gingold, “Compactification Applied to a Discrete Competing System Model,” International Journal of Pure and Applied Mathematics, Vol. 66, No. 3, 2011, pp. 297-320.
[17] H. Gingold, “Divergence of Solutions of Polynomials Finite Difference Equations,” Proceedings A of the Royal Society of Edinburgh, Vol. 142, No. 4, 2012, pp. 787-804. doi:10.1017/S0308210510000077
[18] H. Gingold, “Compactification and Divergence of Solutions of Polynomial Finite Difference Systems of Equations,” Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applied Algorithms, Vol. 18, No. 3, 2011, pp. 315-335.
[19] H. Gingold, “Approximation of Unbounded Functions via Compactification,” Journal of Approximation Theory, Vol. 131, No. 2, 2004, pp. 284-305. doi:10.1016/j.jat.2004.08.001
[20] S. Willard, “General Topology,” Addison-Wesley, Reading, 1970.
[21] U. Elias and H. Gingold, “Critical Points at Infinity and Blow Up of Solutions of Autonomous Polynomial Differential Systems via Compactification,” Journal of Mathematical Analysis and Applications, Vol. 318, No. 1, 2006, pp. 305-322.
[22] H. Gingold and D. Solomon, “On Completeness of Quadratic Systems,” Nonlinear Analysis, Vol. 74, No. 12, 2011, pp. 4234-4240.
[23] J. Guckenheimer and P. Holmes, “Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,” Springer-Verlag, New York, 1983.
[24] W. Hahn, “Stability of Motion,” Springer Verlag, Berlin, 1967.
[25] P. Hartman, “Ordinary Differential Equations,” 2nd Edition, Birkhauser, 1982.
[26] E. L. Ince, “Ordinary Differential Equations,” Dover, New York, 1956.
[27] A. C. J. Luo, “Global Transversality, Resonance, and Chaotic Dynamics,” World Scientific Publishing Co. Pte. Ltd., Hackensack, 2008.
[28] M. Tabor, “Chaos and Integrability in Nonlinear Dynamics,” John Wiley & Sons, New York, 1989.
[29] B. Coomes, “The Lorenz System Does Not Have a Polynomial Flow,” Journal of Differential Equations, Vol. 82, No. 2, 1989, pp. 386-407. doi:10.1016/0022-0396(89)90140-X
[30] G. Meisters, “Polynomial Flows on ,” Proceedings of the Semester on Dynamical Systems, Mokotowska, 1986, p. 25.
[31] M. J. Ablowitz and H. Segur, “Solitons and the Inverse Scattering Transform,” Society for Industrial and Applied Mathematics, Philadelphia, 1981. doi:10.1137/1.9781611970883
[32] T. C. Bountis, V. Papageorgiou and P. Winternitz, “On the Integrability of Systems of Nonlinear Ordinary Differential Equations with Superposition Principles,” Journal of Mathematical Physics, Vol. 27, No. 5, 1986, pp. 1215-1224. doi:10.1063/1.527128
[33] F. J. Bureau, “équations Différentielles du Second Ordre en Y et du Second Degré en ? dont l’Intégrale Générale Est á Points Critiques Fixes,” Annali di Matematica Pura ed Applicata, Vol. 91, No. 4, 1972, pp. 163-281.
[34] F. J. Bureau, “Differential Equations with Fixed Critical Points,” Annali di Matematica Pura ed Applicata, Vol. 66, No. 1, 1964, pp. 1-116. doi:10.1007/BF02412437
[35] J. Chazy, “Sur les équations Différentielles du Troisième Ordre et d’Ordre Supérieur dont l’Intégrale Générale a Ses Points Critiques Fixes,” Acta Mathematica, Vol. 34, No. 1, 1911, pp. 317-385.
[36] J. Chazy, “Sur les équations Différentielles dont l’Intégrale Générale Possede Un Coupure Essentielle Mobile,” Les Comptes Rendus de l’Académie des sciences (Paris), Vol. 150, 1910, pp. 456-458.
[37] B. Gambier, “Sur les équations Différentielles du Second Ordre et du Premier Degré dont l’Intégrale Générale Est a Points Critiques Fixes,” Acta Mathematica, Vol. 33, No. 1, 1910, pp. 1-55. doi:10.1007/BF02393211
[38] R. Garnier, “Sur des Systémes Différentiels du Second Ordre dont l’Intégrale Générale est Uniforme,” Annales Scientifiques de l’école Normale Supérieure, Vol. 77, 1960, pp. 123-144.
[39] E. Hille, “On a Class of Series Expansions in the Theory of Emden’s Equation,” Proceedings of the Royal Society of Edinburgh, Section A, Vol. 71, Part 2, 1973, pp. 95-110.
[40] E. Hille, “A Note on Quadratic Systems,” Proceedings of the Royal Society of Edinburgh, Section A, Vol. 72, No. 1, 1974, pp. 17-37.
[41] P.-F. Hsieh and Y. Sibuya, “Basic Theory of Ordinary Differential Equations,” Universitext, Springer-Verlag, New York, 1999.
[42] D. D. Hua, L. Cairó, M. R. Feix, K. S. Govinder and P. G. L. Leach, “Connection between the Existence of First Integrals and the Painlevé Property in Lotka-Volterra and Quadratic Systems,” Proceedings of the Royal Society A, Vol. 452, No. 1947, 1996, pp. 859-880. doi:10.1098/rspa.1996.0043
[43] S. Kowalevski, “Sur le Probléme de la Rotation d’Un Corps Solide Autour d’Un Point Fixe,” Acta Mathematica, Vol. 12, No. 1, 1889, pp. 177-232. doi:10.1007/BF02592182
[44] P. G. L. Leach and J. Miritzis, “Competing Species: Integrability and Stability,” Journal of Nonlinear Mathema- tical Physics, Vol. 11, No. 1, 2004, pp. 123-133. doi:10.2991/jnmp.2004.11.1.9
[45] P. G. L. Leach, G. P. Flessas and S. Cotsakis, “Symmetry, Singularities and Integrability in Complex Dynamics I: The Reduction Problem,” Journal of Nonlinear Mathematical Physics, Vol. 7, No. 4, 2000, pp. 445-479. doi:10.2991/jnmp.2000.7.4.4
[46] M. Tabor and J. Weiss, “Analytic Structure of the Lorenz System,” Physical Review A, Vol. 24, No. 4, 1981, pp. 2157-2167. doi:10.1103/PhysRevA.24.2157
[47] A. Ramani, B. Grammaticos and T. Bountis, “The Painlevé Property and Singularity Analysis of Integrable and Nonintegrable Systems,” Physical Report, Vol. 180, No. 3, 1989, pp. 159-245.
[48] W. A. Coppel, “A Survey of Quadratic Systems,” Journal of Differential Equations, Vol. 2, No. 3, 1966, pp. 293-304. doi:10.1016/0022-0396(66)90070-2
[49] S.-B. Hsu, S. Hubbell and P. Waltman, “A Contribution to the Theory of Competing Predators,” Ecological Monographs, Vol. 48, No. 3, 1978, pp. 337-349. doi:10.2307/2937235
[50] J. C. Artes, J. Llibre, N. Vulpe, “Singular Points of Quadratic Systems: A Complete Classification in the Coefficient Space,” International Journal of Bifurcation and Chaos, Vol. 18, No. 2, 2008, pp. 313-362. doi:10.1142/S021812740802032X
[51] J. C. Artes, R. E. Kooij and J. Llibre, “Structurally Stable Quadratic Vector Fields,” Vol. 134, Memoirs of the American Mathematical Society, 1998.
[52] F. Dumortier, J. Llibre and J. Artes, “Qualitative Theory of Planar Differential Systems,” Universitext, Springer-Verlag, Berlin, 2006.
[53] H. Gingold, “The Leading Asymptotic Term of Real Valued Solutions of a Competing Species Model,” pp. 1-23.
[54] E. N. Lorenz, “Deterministic Nonperiodic Flow,” Journal of the Atmospheric Sciences, Vol. 20, No. 2, 1963, pp. 130-141. doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
[55] X.-Y. Chen, “On Generalized Rotated Vector Fields,” Journal of Nanjing University of Posts and Telecommunications (Natural Science), Vol. 1, 1975, pp. 100-108.
[56] G. F. D. Duff, “Limit-Cycles and Rotated Vector Fields,” Annals of Mathematics, Vol. 57, No. 1, 1953, pp. 15-31. doi:10.2307/1969724
[57] L. Perko, “Rotated Vector Fields and the Global Behavior of Limit Cycles for a Class of Quadratic Systems in the Plane,” Journal of Differential Equations, Vol. 18, No. 1, 1975, pp. 63-86. doi:10.1016/0022-0396(75)90081-9
[58] A. Yu. Fishkin, “On the Number of Limit Cycles of Planar Quadratic Vector Fields,” Doklady Akademii Nauk (Russian), Vol. 428, No. 4, 2009, pp. 462-464.
[59] J. C. Artés, J. Llibre and J. C. Medrado, “Nonexistence of Limit Cycles for a Class of Structurally Stable Quadratic Vector Fields,” Discrete and Continuous Dynamical Systems, Vol. 17, No. 2, 2007, pp. 259-271.
[60] N. Bautin, “On Periodic Solutions of a System of Differential Equations,” Prikladnaya Matematika i Mekhanika, Vol. 18, 1954, p. 128.
[61] Yu. S. Ilyashenko, “Finiteness Theorems for Limit Cycles,” Translated from the Russian by H. H. McFaden. Translations of Mathematical Monographs, 94, American Mathematical Society, Providence, 1991
[62] Y. Yanqian, “Theory of Limit Cycles,” American Mathematical Society, Providence, 1986.
[63] C. Sparrow, “The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors,” Vol. 41, Applied Mathematical Sciences, Springer-Verlag, Berlin, 1982. doi:10.1007/978-1-4612-5767-7
[64] H. Gingold and D. Solomon, “The Lorenz System Has a Global Repeller at Infinity,” Journal of Nonlinear Mathematical Physics, Vol. 18, No. 2, 2011, pp. 183-189.
[65] M. Holland and I. Melbourne, “Central Limit Theorems and Invariance Principles for Lorenz Attractors,” Journal of the London Mathematical Society, Vol. 76, No. 2, 2007, pp. 345-364.
[66] B. Marlin, “An Upper Semi-Continuous Model for the Lorenz Attractor,” Topology Proceedings, Vol. 40, 2012, pp. 73-81.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.