An asymptotic universal focal decomposition for non-isochronous potentials

de Carvalho, C.; Peixoto, M.; Pinheiro, D.; Pinto, A.

doi:10.1090/S0002-9947-2013-05995-8

An asymptotic universal focal decomposition for non-isochronous potentials

Authors: C. A. A. de Carvalho, M. M. Peixoto, D. Pinheiro and A. A. Pinto
Journal: Trans. Amer. Math. Soc. 366 (2014), 2227-2263
MSC (2010): Primary 37E20, 34B15, 70H03, 70H09
DOI: https://doi.org/10.1090/S0002-9947-2013-05995-8
Published electronically: November 25, 2013
MathSciNet review: 3152729
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Abstract: Galileo, in the seventeenth century, observed that the small oscillations of a pendulum seem to have constant period. In fact, the Taylor expansion of the period map of the pendulum is constant up to second order in the initial angular velocity around the stable equilibrium. It is well known that, for small oscillations of the pendulum and small intervals of time, the dynamics of the pendulum can be approximated by the dynamics of the harmonic oscillator. We study the dynamics of a family of mechanical systems that includes the pendulum at small neighbourhoods of the equilibrium but after long intervals of time so that the second order term of the period map can no longer be neglected. We analyze such dynamical behaviour through a renormalization scheme acting on the dynamics of this family of mechanical systems. The main theorem states that the asymptotic limit of this renormalization scheme is universal: it is the same for all the elements in the considered class of mechanical systems. As a consequence, we obtain a universal asymptotic focal decomposition for this family of mechanical systems. This paper is intended to be the first in a series of articles aiming at a semiclassical quantization of systems of the pendulum type as a natural application of the focal decomposition associated to the two-point boundary value problem.

[1] Ralph Abraham and Jerrold E. Marsden, Foundations of mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. Second edition, revised and enlarged; With the assistance of Tudor Raţiu and Richard Cushman. MR 515141
[2] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover Publications (New York), 1965.
[3] Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, https://doi.org/10.1016/0022-1236(73)90051-7
[4] V. I. Arnol′d, Mathematical methods of classical mechanics, 2nd ed., Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989. Translated from the Russian by K. Vogtmann and A. Weinstein. MR 997295
[5] M. V. Berry, Tsunami asymptotics, New J. Phys. 7 (2005), 129.
[6] M. V. Berry, Focused tsunami waves, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007), no. 2087, 3055–3071. MR 2360189, https://doi.org/10.1098/rspa.2007.0051
[7] M. V. Berry and C. Upstill, Catastrophe optics: morphologies of caustics and their diffraction patterns, Prog. Optics 18 (1980), 257--346.
[8] Z. Bishnani and R. S. Mackay, Safety criteria for aperiodically forced systems, Dyn. Syst. 18 (2003), no. 2, 107–129. MR 1994783, https://doi.org/10.1080/14689360303087
[9] S. V. Bolotin and R. S. MacKay, Isochronous potentials, Localization and energy transfer in nonlinear systems (L. Vazquez, R. S. MacKay, and M. P. Zorzano, eds.), World Sci, 2003, pp. 217-224.
[10] F. Bowman, Introduction to elliptic functions with applications, English Universities Press, Ltd., London, 1953. MR 0058760
[11] C. G. Callan, Broken scale invariance in scalar field theory, Phys. Rev. D 2 (1970), 1541-1547.
[12] Francesco Calogero, Isochronous systems, Oxford University Press, Oxford, 2008. MR 2383111
[13] C. A. A. de Carvalho, R. M. Cavalcanti, E. S. Fraga, and S. E. Jorás, Semiclassical series at finite temperature, Ann. Physics 273 (1999), no. 1, 146–170. MR 1690487, https://doi.org/10.1006/aphy.1998.5900
[14] C. A. A. de Carvalho, R. M. Cavalcanti, E. S. Fraga, and S. E. Jorás, Improved semiclassical density matrix: Taming caustics, Physical Review E 65 (2002), no. 5, 56112-56221.
[15] Elise E. Cawley, The Teichmüller space of an Anosov diffeomorphism of 𝑇², Invent. Math. 112 (1993), no. 2, 351–376. MR 1213107, https://doi.org/10.1007/BF01232439
[16] Lothar Collatz, Differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1986. An introduction with applications; Translated from the German by E. R. Dawson. MR 858128
[17] P. Coullet and C. Tresser, Itérations d'endomorphismes et groupe de renormalisation, C. R. Acad. Sci. Paris Sér. A - B 287 (1978), no. 7, A577-A580.
[18] C. A. A. de Carvalho and R. M. Cavalcanti, Tunneling catastrophes of the partition function, Brazilian Journal of Physics 27 (1997), 373-378.
[19] C. A. A. de Carvalho, M. Peixoto, D. Pinheiro, and A. A. Pinto, Focal decomposition, renormalization and semiclassical physics, J. Difference Equ. Appl. 17 (2011), no. 7, 1019–1029. MR 2822433, https://doi.org/10.1080/10236191003685916
[20] Carlos A. A. de Carvalho, Mauricio M. Peixoto, Diogo Pinheiro, and Alberto A. Pinto, Renormalization and focal decomposition, Dynamics, games and science. II, Springer Proc. Math., vol. 2, Springer, Heidelberg, 2011, pp. 25–40. MR 2883266, https://doi.org/10.1007/978-3-642-14788-3_2
[21] Edson de Faria and Welington de Melo, Rigidity of critical circle mappings. I, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 4, 339–392. MR 1728375, https://doi.org/10.1007/s100970050011
[22] Edson de Faria and Welington de Melo, Rigidity of critical circle mappings. II, J. Amer. Math. Soc. 13 (2000), no. 2, 343–370. MR 1711394, https://doi.org/10.1090/S0894-0347-99-00324-0
[23] Edson de Faria, Welington de Melo, and Alberto Pinto, Global hyperbolicity of renormalization for 𝐶^{𝑟} unimodal mappings, Ann. of Math. (2) 164 (2006), no. 3, 731–824. MR 2259245, https://doi.org/10.4007/annals.2006.164.731
[24] W. de Melo, Rigidity and renormalization in one-dimensional dynamical systems, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 765–778. MR 1648124
[25] W. de Melo and A. A. Pinto, Rigidity of 𝐶² infinitely renormalizable unimodal maps, Comm. Math. Phys. 208 (1999), no. 1, 91–105. MR 1729879, https://doi.org/10.1007/s002200050749
[26] Cecile DeWitt-Morette, Pierre Cartier, and Antoine Folacci (eds.), Functional integration, NATO Advanced Science Institutes Series B: Physics, vol. 361, Plenum Press, New York, 1997. Basics and applications; Papers from the NATO Advanced Study Institute held in Cargèse, September 1–14, 1996. MR 1477451
[27] Jürgen Ehlers and Ezra T. Newman, The theory of caustics and wave front singularities with physical applications, J. Math. Phys. 41 (2000), no. 6, 3344–3378. MR 1768638, https://doi.org/10.1063/1.533316
[28] G. F. R. Ellis, B. A. C. C. Bassett, and P. K. S. Dunsby, Lensing and caustic effects on cosmological distances, Classical Quantum Gravity 15 (1998), no. 8, 2345–2361. MR 1645577, https://doi.org/10.1088/0264-9381/15/8/015
[29] Mitchell J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978), no. 1, 25–52. MR 501179, https://doi.org/10.1007/BF01020332
[30] Mitchell J. Feigenbaum, The universal metric properties of nonlinear transformations, J. Statist. Phys. 21 (1979), no. 6, 669–706. MR 555919, https://doi.org/10.1007/BF01107909
[31] R. P. Feynman and A. R. Hibbs, Quantum mechanics and path integrals, McGraw-Hill, New York, 1965.
[32] H. Friedrich and J. M. Stewart, Characteristic initial data and wavefront singularities in general relativity, Proc. Roy. Soc. London Ser. A 385 (1983), no. 1789, 345–371. MR 692204, https://doi.org/10.1098/rspa.1983.0018
[33] M. Gell-Mann and F. E. Low, Quantum electrodynamics at small distances, Phys. Rev. (2) 95 (1954), 1300–1312. MR 64652
[34] Jack K. Hale, Ordinary differential equations, 2nd ed., Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980. MR 587488
[35] Wolfgang Hasse, Marcus Kriele, and Volker Perlick, Caustics of wavefronts in general relativity, Classical Quantum Gravity 13 (1996), no. 5, 1161–1182. MR 1390106, https://doi.org/10.1088/0264-9381/13/5/027
[36] Yun Ping Jiang, Asymptotic differentiable structure on Cantor set, Comm. Math. Phys. 155 (1993), no. 3, 503–509. MR 1231640
[37] Yunping Jiang, Renormalization and geometry in one-dimensional and complex dynamics, Advanced Series in Nonlinear Dynamics, vol. 10, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1442953
[38] Yunping Jiang, Smooth classification of geometrically finite one-dimensional maps, Trans. Amer. Math. Soc. 348 (1996), no. 6, 2391–2412. MR 1321579, https://doi.org/10.1090/S0002-9947-96-01487-0
[39] D. W. Jordan and P. Smith, Nonlinear ordinary differential equations, 3rd ed., Oxford Texts in Applied and Engineering Mathematics, vol. 2, Oxford University Press, Oxford, 1999. An introduction to dynamical systems. MR 1743361
[40] L. P. Kadanoff, Scaling laws for Ising models near , Physics 2 (1966), 263-272.
[41] Ivan A. K. Kupka and M. M. Peixoto, On the enumerative geometry of geodesics, From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990) Springer, New York, 1993, pp. 243–253. MR 1246123
[42] Ivan Kupka, Mauricio Peixoto, and Charles Pugh, Focal stability of Riemann metrics, J. Reine Angew. Math. 593 (2006), 31–72. MR 2227139, https://doi.org/10.1515/CRELLE.2006.029
[43] Oscar E. Lanford III, Renormalization group methods for critical circle mappings with general rotation number, VIIIth international congress on mathematical physics (Marseille, 1986) World Sci. Publishing, Singapore, 1987, pp. 532–536. MR 915597
[44] Oscar E. Lanford III, Renormalization group methods for critical circle mappings, Nonlinear evolution and chaotic phenomena (New York) (G. Gallavotti and P. F. Zweifel, eds.), NATO Adv. Sci. Inst. Ser. B: Phys., vol. 176, Springer, 1988, pp. 25-36.
[45] Mikhail Lyubich, Feigenbaum-Coullet-Tresser universality and Milnor’s hairiness conjecture, Ann. of Math. (2) 149 (1999), no. 2, 319–420. MR 1689333, https://doi.org/10.2307/120968
[46] R. S. MacKay, A renormalisation approach to invariant circles in area-preserving maps, Phys. D 7 (1983), no. 1-3, 283–300. Order in chaos (Los Alamos, N.M., 1982). MR 719057, https://doi.org/10.1016/0167-2789(83)90131-8
[47] R. S. MacKay, Renormalisation in area-preserving maps, Advanced Series in Nonlinear Dynamics, vol. 6, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. MR 1336593
[48] R. S. MacKay and J. D. Meiss (eds.), Hamiltonian dynamical systems, Adam Hilger, Ltd., Bristol, 1987. MR 1103556
[49] M. MacNeish, On determination of a catenary with given directrix and passing through two given points, Ann. of Math. 7 (1906), 65-80.
[50] Jerrold E. Marsden and Tudor S. Ratiu, Introduction to mechanics and symmetry, 2nd ed., Texts in Applied Mathematics, vol. 17, Springer-Verlag, New York, 1999. A basic exposition of classical mechanical systems. MR 1723696
[51] Marco Martens, The periodic points of renormalization, Ann. of Math. (2) 147 (1998), no. 3, 543–584. MR 1637651, https://doi.org/10.2307/120959
[52] Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
[53] Stellan Östlund, David Rand, James Sethna, and Eric Siggia, Universal properties of the transition from quasiperiodicity to chaos in dissipative systems, Phys. D 8 (1983), no. 3, 303–342. MR 719630, https://doi.org/10.1016/0167-2789(83)90229-4
[54] M. M. Peixoto, On a generic theory of end point boundary value problems, An. Acad. Brasil. Ci. 41 (1969), 1–6. MR 252747
[55] M. M. Peixoto, On end-point boundary value problems, J. Differential Equations 44 (1982), no. 2, 273–280. Special issue dedicated to J. P. LaSalle. MR 657782, https://doi.org/10.1016/0022-0396(82)90017-1
[56] M. M. Peixoto, Sigma décomposition et arithmétique de quelqes formes quadratiques définies positives, R. Thom Festschift volume: Passion des Formes (M. Porte, ed.), ENS Editions (Paris), 1994, pp. 455-479.
[57] Boris N. Apanasov, Steven B. Bradlow, Waldyr A. Rodrigues Jr., and Karen K. Uhlenbeck (eds.), Geometry, topology and physics, Walter de Gruyter & Co., Berlin, 1997. MR 1605264
[58] M. M. Peixoto and A. R. da Silva, Focal decomposition and some results of S. Bernstein on the 2-point boundary value problem, J. London Math. Soc. (2) 60 (1999), no. 2, 517–547. MR 1718712, https://doi.org/10.1112/S0024610799007929
[59] M. M. Peixoto and R. Thom, Le point de vue énumératif dans les problèmes aux limites pour les équations différentielles ordinaires, C. R. Acad. Sci., Paris, Sér. I 303 (1986), 629-633; erratum, 307 (1988) 197-198; II, 303 (1986) 693-698.
[60] Alberto A. Pinto, David A. Rand, and Flávio Ferreira, Fine structures of hyperbolic diffeomorphisms, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2009. With a preface by Jacob Palis and Enrique R. Pujals. MR 2464147
[61] E. C. G. Stueckelberg and A. Petermann, La normalisation des constantes dans la theorie des quanta, Helv. Phys. Acta 26 (1953), 499–520 (French). MR 87504
[62] Dennis Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988) Amer. Math. Soc., Providence, RI, 1992, pp. 417–466. MR 1184622
[63] Dennis Sullivan, Linking the universalities of Milnor-Thurston, Feigenbaum and Ahlfors-Bers, Topological methods in modern mathematics (Stony Brook, NY, 1991) (Houston, Texas) (L. R. Goldberg and A. V. Philips, eds.), Publish or Perish, 1993, pp. 543-563.
[64] K. Symanzik, Small distance behaviour in field theory and power counting, Comm. Math. Phys. 18 (1970), no. 3, 227–246. MR 1552571, https://doi.org/10.1007/BF01649434
[65] I. Todhunter, A history of the calculus of variations, Chelsea Publishing (New York), 1861 (reprint).
[66] J. J. P. Veerman, Mauricio M. Peixoto, André C. Rocha, and Scott Sutherland, On Brillouin zones, Comm. Math. Phys. 212 (2000), no. 3, 725–744. MR 1779166, https://doi.org/10.1007/PL00020959
[67] S. Weinberg, New approach to the renormalization group, Phys. Rev. D 8 (1973), 3497-3509.
[68] E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469
[69] Kenneth G. Wilson, The renormalization group: critical phenomena and the Kondo problem, Rev. Modern Phys. 47 (1975), no. 4, 773–840. MR 0438986, https://doi.org/10.1103/RevModPhys.47.773
[70] Michael Yampolsky, Complex bounds for renormalization of critical circle maps, Ergodic Theory Dynam. Systems 19 (1999), no. 1, 227–257. MR 1677153, https://doi.org/10.1017/S0143385799120947
[71] Michael Yampolsky, Renormalization horseshoe for critical circle maps, Comm. Math. Phys. 240 (2003), no. 1-2, 75–96. MR 2004980, https://doi.org/10.1007/s00220-003-0891-8