Abstract
The singularity analysis of ordinary differential equations is based on the concept of the balance of the dominant terms. The equivalence of these terms is reflected by the possession of the same similarity symmetry. Different possible singular behaviours are reflected by different similarity symmeties. Hierarchal structures are observed. The generalised Chazy equation provides a simple example of this structure. The singularity analysis is based on the recently introduced ‘g function’ method.
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Ablowitz, M.J., Ramani, A., Segur, H. (1978): Nonlinear evolution equations and ordinary differential equations of Painlevé type. Lett. Nuovo Cimento 23, 333–338
Ablowitz, M.J., Ramani, A., Segur, H. (1980): A connection between nonlinear evolution equations and ordinary differential equations of P-type. I. J. Math. Phys. 21, 715–721
Ablowitz, M.J., Ramani, A., Segur, H. (1980): A connection between nonlinear evolution equations and ordinary differential equations of P-type. II. J. Math. Phys. 21, 1006–1015
Abraham-Shrauner, B. (1993): Hidden symmetries and linearization of the modified Painlevé-Ince equation. J. Math. Phys. 34, 4809–4816
Bureau, F.J. (1964): Differential equations with fixed critical points. Annali di Matematica pura ed applicata LXIV, 229–364
Chazy, Jean (1911): Sur les équations différentielles du troisiéme ordre et d'ordre supérieur dont l'intégrale générale a ses points critiques fixes. Acta Math. 34, 317–385
Conte, R. (1994): Singularities of differential equations and integrability in Introduction to methods of complex analysis and geometry for classical mechanics and nonlinear waves Benest, D. and Froeschlé, C. eds. (Éditions Frontières, Gif-sur-Yvette), 49–143
Feix, M.R., Geronimi, C., Cairó, L., Leach, P.G.L., Lemmer, R.L., Bouquet, S.É. (1997): On the singularity analysis of ordinary differential equations invariant under time translation and rescaling. J. Phys. A (to appear)
Feix, M.R., Geronimo, C., Pillay, T., Leach, P.G.L. (1997): The behaviour of the solution of an ordinary differential equation about its singular solutions and the connection with the Painlevé series. (preprint MAPMO, Université d'Orléans, UFR Sciences, BP 6759, F-45067 Orléans Cedex 2, France)
Feix, M.R., Geronimi, C., Leach, P.G.L. (1997): Periodic solutions and associated limit cycle for the generalised Chazy equation. (preprint MAPMO, Université d'Orléans, UFR Sciences, BP 6759, F-45067 Orléans Cedex 2, France)
Fordy, A., Pickering, A. (1991): Analysing negative resonances in the Painlevé test. Phys. Lett. A 160, 347–354
Gambier, B. (1909): Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est a points critiques fixes. Acta Math. 33, 1–55
Garnier, R. (1912): Sur des équations différentielles du troisième ordre dont l'intégrale générale est uniforme et sur une classe d'équations nouvelles d'ordre supérieur dont l'intégrale générale a ses points critiques fixes. Annales Scientifiques de l'École Normale Supérieur XXIX, 1–126
Govinder, K.S., Leach, P.G.L. (1995): On the determination of nonlocal symmetries. J. Phys. A 28, 5349–5359
Govinder, K.S., Leach, P.G.L. (1997): A group theoretic approach to a class of second order ordinary differential equations not possessing Lie point symmetries. J. Phys. A 30, 2055–2068
Hua, D.D., Cairo, L., Feix, M.R., Govinder, K.S., Leach, P.G.L. (1996): Connection between the existence of first integrals and the Painlevé property in two-dimensional Lotka-Volterra and Quadratic Systems. Proc. Roy. Soc. Lond. A 452, 859–880
Kowalevski, Sophie (1889): Sur les problème de la rotation d'un corps solide autour d'un point fixe. Acta Math. 12, 177–232
Kowalevski, Sophie (1889): Sur une propriété du système d'équations différentielles qui définit la rotation d'un corps solide autour d'un point fixe. Acta Math. 14, 81–93
Lemmer, R.L., Leach, P.G.L. (1993): The Painlevé test, hidden symmetries and the equation y″+yy′+ky 3=0. J. Phys. A 26, 5017–5024
Lie, Sophus (1967): Differentialgleichungen (Chelsea, New York)
Mahomed, F.M., Leach, P.G.L. (1990): Symmetry Lie algebras of nth order ordinary differential equations. J. Math. Anal. Appln. 151, 80–107
Painlevé, P. (1900): Mémoire sur les équations différentielles dont l'intégrale générale est uniforme. Bull. Math. Soc. France 28, 201–261
Painlevé, P. (1902): Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme. Acta Math. 25, 1–85
Ramani, A., Grammaticos, B., Bountis, T. (1989): The Painlevé property and singularity analysis of integrable and non-integrable systems. Phys. Rep. 180, 159–245
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Leach, P.G.L. (1999). Hierarchies of similarity symmetries and singularity analysis. In: Leach, P.G.L., Bouquet, S.E., Rouet, JL., Fijalkow, E. (eds) Dynamical Systems, Plasmas and Gravitation. Lecture Notes in Physics, vol 518. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0105935
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DOI: https://doi.org/10.1007/BFb0105935
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