Abstract:We find the order of contact of the boundaries of the cusp for two-parameter families of vector fields on the real line or diffeomorphisms of the real line, for cusp bifurcations of codimensions 1 and 2. Moreover, we create a machinery that can be used for the same problem in higher codimensions and perhaps for other, similar problems.
- V. I. Arnold, Small denominators I, Mappings of the Circumference onto Itself, Amer. Math. Soc. Translations 46, 213–284 (1965).
- Henk W. Broer, Martin Golubitsky, and Gert Vegter, The geometry of resonance tongues: a singularity theory approach, Nonlinearity 16 (2003), no. 4, 1511–1538. MR 1986309, DOI 10.1088/0951-7715/16/4/319
- Henk Broer and Carles Simó, Resonance tongues in Hill’s equations: a geometric approach, J. Differential Equations 166 (2000), no. 2, 290–327. MR 1781258, DOI 10.1006/jdeq.2000.3804
- Henk Broer and Carles Simó, Hill’s equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena, Bol. Soc. Brasil. Mat. (N.S.) 29 (1998), no. 2, 253–293. MR 1654831, DOI 10.1007/BF01237651
- Henk Broer, Joaquim Puig, and Carles Simó, Resonance tongues and instability pockets in the quasi-periodic Hill-Schrödinger equation, Comm. Math. Phys. 241 (2003), no. 2-3, 467–503. MR 2013807, DOI 10.1007/s00220-003-0935-0
- Henk Broer, Carles Simó, and Joan Carles Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms, Nonlinearity 11 (1998), no. 3, 667–770. MR 1617974, DOI 10.1088/0951-7715/11/3/015
- Jack K. Hale and Hüseyin Koçak, Dynamics and bifurcations, Texts in Applied Mathematics, vol. 3, Springer-Verlag, New York, 1991. MR 1138981, DOI 10.1007/978-1-4612-4426-4
- MichałMisiurewicz and Ana Rodrigues, Double standard maps, Comm. Math. Phys. 273 (2007), no. 1, 37–65. MR 2308749, DOI 10.1007/s00220-007-0223-5
- MichałMisiurewicz and Ana Rodrigues, On the tip of the tongue, J. Fixed Point Theory Appl. 3 (2008), no. 1, 131–141. MR 2402913, DOI 10.1007/s11784-008-0052-y
- Joaquim Puig and Carles Simó, Analytic families of reducible linear quasi-periodic differential equations, Ergodic Theory Dynam. Systems 26 (2006), no. 2, 481–524. MR 2218772, DOI 10.1017/S0143385705000362
- Michał Misiurewicz
- Affiliation: Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216 – and – Institute of Mathematics of the Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland
- MR Author ID: 125475
- Email: firstname.lastname@example.org
- Ana Rodrigues
- Affiliation: Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216 – and – CMUP, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
- Address at time of publication: Matematiska Institutionen, KTH, SE-100 44 Stockholm, Sweden
- Email: email@example.com, firstname.lastname@example.org
- Received by editor(s): December 14, 2008
- Received by editor(s) in revised form: April 23, 2009
- Published electronically: February 18, 2011
- Additional Notes: The first author was partially supported by NSF grant DMS 0456526.
The second author was supported by FCT Grant BPD/36072/2007. Research of the second author was supported in part by Centro de Matemática da Universidade do Porto (CMUP) financed by FCT through the programmes POCTI and POSI, with Portuguese and European Community structural funds.
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
- Journal: Trans. Amer. Math. Soc. 363 (2011), 3553-3572
- MSC (2010): Primary 37G15, 37E99
- DOI: https://doi.org/10.1090/S0002-9947-2011-05114-7
- MathSciNet review: 2775818