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Non-normal numbers with respect to Markov partitions
Gevrey normal forms for nilpotent contact points of order two
1. | Hasselt University, Campus Diepenbeek, Agoralaan-Gebouw D, B-3590 Diepenbeek, Belgium |
References:
[1] |
P. Bonckaert and P. De Maesschalck, Gevrey and analytic local models for families of vector fields, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 377-400.
doi: 10.3934/dcdsb.2008.10.377. |
[2] |
Éric Benoît, Perturbation singulière en dimension trois: Canards en un point pseudo-singulier nœud, Bull. Soc. Math. France, 129 (2001), 91-113. |
[3] |
Bernard Candelpergher, Francine Diener and Marc Diener, Retard à la bifurcation: Du local au global, In "Bifurcations of Planar Vector Fields" (Luminy, 1989), Lecture Notes in Math., 1455, Springer, Berlin, (1990), 1-19.
doi: 10.1007/BFb0085388. |
[4] |
M. Canalis-Durand, J. P. Ramis, R. Schäfke and Y. Sibuya, Gevrey solutions of singularly perturbed differential equations, J. Reine Angew. Math., 518 (2000), 95-129.
doi: 10.1515/crll.2000.008. |
[5] |
Mireille Canalis-Durand and Reinhard Schäfke, Divergence and summability of normal forms of systems of differential equations with nilpotent linear part, Ann. Fac. Sci. Toulouse Math. (6), 13 (2004), 493-513.
doi: 10.5802/afst.1079. |
[6] |
P. De Maesschalck, F. Dumortier and R. Roussarie, Cyclicity of common slow-fast cycles, Indag. Math. (N.S.), 22 (2011), 165-206.
doi: 10.1016/j.indag.2011.09.008. |
[7] |
Peter De Maesschalck and Nikola Popović, Gevrey properties of the asymptotic critical wave speed in a family of scalar reaction-diffusion equations, J. Math. Anal. Appl., 386 (2012), 542-558.
doi: 10.1016/j.jmaa.2011.08.016. |
[8] |
Freddy Dumortier, Compactification and desingularization of spaces of polynomial Liénard equations, J. Differential Equations, 224 (2006), 296-313.
doi: 10.1016/j.jde.2005.08.011. |
[9] |
A. Fruchard and R. Schäfke, Overstability and resonance, Ann. Inst. Fourier (Grenoble), 53 (2003), 227-264.
doi: 10.5802/aif.1943. |
[10] |
Masaki Hibino, Borel summability of divergent solutions for singularly perturbed first-order ordinary differential equations, Tohoku Math. J. (2), 58 (2006), 237-258. |
[11] |
Gérard Iooss and Eric Lombardi, Normal forms with exponentially small remainder: application to homoclinic connections for the reversible $0^{2+}i\omega$ resonance, C. R. Math. Acad. Sci. Paris, 339 (2004), 831-838.
doi: 10.1016/j.crma.2004.10.002. |
[12] |
Frank Loray, Réduction formelle des singularités cuspidales de champs de vecteurs analytiques, J. Differential Equations, 158 (1999), 152-173.
doi: 10.1016/S0022-0396(99)80021-7. |
[13] |
Eric Lombardi and Laurent Stolovitch, Normal forms of analytic perturbations of quasihomogeneous vector fields: rigidity, invariant analytic sets and exponentially small approximation, Ann. Sci. Éc. Norm. Supér. (4), 43 (2010), 659-718. |
[14] |
Robert Roussarie, Putting a boundary to the space of Liénard equations, Discrete Contin. Dyn. Syst., 17 (2007), 441-448.
doi: 10.3934/dcds.2007.17.441. |
[15] |
Reinhard Schäfke, Gevrey asymptotics in singular perturbations of ODE, in "International Conference on Differential Equations, Vol. 1, 2" (Berlin, 1999), World Sci. Publ., River Edge, NJ, (2000), 118-123. |
[16] |
Yasutaka Sibuya, The Gevrey asymptotics in the case of singular perturbations, J. Differential Equations, 165 (2000), 255-314.
doi: 10.1006/jdeq.2000.3787. |
[17] |
Ewa Stróżyna and Henryk Żoładek, The analytic and formal normal form for the nilpotent singularity, J. Differential Equations, 179 (2002), 479-537.
doi: 10.1006/jdeq.2001.4043. |
[18] |
Ewa Stróżyna and Henryk Żoładek, Orbital formal normal forms for general Bogdanov-Takens singularity, J. Differential Equations, 193 (2003), 239-259.
doi: 10.1016/S0022-0396(03)00137-2. |
show all references
References:
[1] |
P. Bonckaert and P. De Maesschalck, Gevrey and analytic local models for families of vector fields, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 377-400.
doi: 10.3934/dcdsb.2008.10.377. |
[2] |
Éric Benoît, Perturbation singulière en dimension trois: Canards en un point pseudo-singulier nœud, Bull. Soc. Math. France, 129 (2001), 91-113. |
[3] |
Bernard Candelpergher, Francine Diener and Marc Diener, Retard à la bifurcation: Du local au global, In "Bifurcations of Planar Vector Fields" (Luminy, 1989), Lecture Notes in Math., 1455, Springer, Berlin, (1990), 1-19.
doi: 10.1007/BFb0085388. |
[4] |
M. Canalis-Durand, J. P. Ramis, R. Schäfke and Y. Sibuya, Gevrey solutions of singularly perturbed differential equations, J. Reine Angew. Math., 518 (2000), 95-129.
doi: 10.1515/crll.2000.008. |
[5] |
Mireille Canalis-Durand and Reinhard Schäfke, Divergence and summability of normal forms of systems of differential equations with nilpotent linear part, Ann. Fac. Sci. Toulouse Math. (6), 13 (2004), 493-513.
doi: 10.5802/afst.1079. |
[6] |
P. De Maesschalck, F. Dumortier and R. Roussarie, Cyclicity of common slow-fast cycles, Indag. Math. (N.S.), 22 (2011), 165-206.
doi: 10.1016/j.indag.2011.09.008. |
[7] |
Peter De Maesschalck and Nikola Popović, Gevrey properties of the asymptotic critical wave speed in a family of scalar reaction-diffusion equations, J. Math. Anal. Appl., 386 (2012), 542-558.
doi: 10.1016/j.jmaa.2011.08.016. |
[8] |
Freddy Dumortier, Compactification and desingularization of spaces of polynomial Liénard equations, J. Differential Equations, 224 (2006), 296-313.
doi: 10.1016/j.jde.2005.08.011. |
[9] |
A. Fruchard and R. Schäfke, Overstability and resonance, Ann. Inst. Fourier (Grenoble), 53 (2003), 227-264.
doi: 10.5802/aif.1943. |
[10] |
Masaki Hibino, Borel summability of divergent solutions for singularly perturbed first-order ordinary differential equations, Tohoku Math. J. (2), 58 (2006), 237-258. |
[11] |
Gérard Iooss and Eric Lombardi, Normal forms with exponentially small remainder: application to homoclinic connections for the reversible $0^{2+}i\omega$ resonance, C. R. Math. Acad. Sci. Paris, 339 (2004), 831-838.
doi: 10.1016/j.crma.2004.10.002. |
[12] |
Frank Loray, Réduction formelle des singularités cuspidales de champs de vecteurs analytiques, J. Differential Equations, 158 (1999), 152-173.
doi: 10.1016/S0022-0396(99)80021-7. |
[13] |
Eric Lombardi and Laurent Stolovitch, Normal forms of analytic perturbations of quasihomogeneous vector fields: rigidity, invariant analytic sets and exponentially small approximation, Ann. Sci. Éc. Norm. Supér. (4), 43 (2010), 659-718. |
[14] |
Robert Roussarie, Putting a boundary to the space of Liénard equations, Discrete Contin. Dyn. Syst., 17 (2007), 441-448.
doi: 10.3934/dcds.2007.17.441. |
[15] |
Reinhard Schäfke, Gevrey asymptotics in singular perturbations of ODE, in "International Conference on Differential Equations, Vol. 1, 2" (Berlin, 1999), World Sci. Publ., River Edge, NJ, (2000), 118-123. |
[16] |
Yasutaka Sibuya, The Gevrey asymptotics in the case of singular perturbations, J. Differential Equations, 165 (2000), 255-314.
doi: 10.1006/jdeq.2000.3787. |
[17] |
Ewa Stróżyna and Henryk Żoładek, The analytic and formal normal form for the nilpotent singularity, J. Differential Equations, 179 (2002), 479-537.
doi: 10.1006/jdeq.2001.4043. |
[18] |
Ewa Stróżyna and Henryk Żoładek, Orbital formal normal forms for general Bogdanov-Takens singularity, J. Differential Equations, 193 (2003), 239-259.
doi: 10.1016/S0022-0396(03)00137-2. |
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