Diffusing orbits along chains of cylinders

Marian Gidea; Jean-Pierre Marco

doi:10.3934/dcds.2022121

Article Contents

2022, Volume 42, Issue 12: 5737-5782. Doi: 10.3934/dcds.2022121

This issue Previous Article Asymptotic lattices, good labellings, and the rotation number for quantum integrable systems Next Article Ground state solutions of magnetic Schrödinger equations with exponential growth

Diffusing orbits along chains of cylinders

Marian Gidea^1, , and
Jean-Pierre Marco^2,

1.
Department of Mathematical Sciences, Yeshiva University, 215 Lexington Ave, New York, NY 10016, USA
2.
Sorbonne Université (Paris 6) and Institut de Mathématiques de Jussieu-Paris Rive Gauche, 4 Place Jussieu, 75005 Paris cedex 05, France

^*Corresponding author: Marian Gidea
^*Corresponding author: Marian Gidea

Received: July 2021

Revised: June 2022

Early access: September 2022

Published: December 2022

The first author is supported by NSF grants DMS-0635607, DMS-1515851 and DMS-1814543, and by the Alfred P. Sloan Foundation grant G-2016-7320

Abstract / Introduction Full Text(HTML) Figure(6) Related Papers Cited by

Abstract

We develop a geometric mechanism to prove the existence of orbits that drift along a prescribed sequence of cylinders, under some general conditions on the dynamics. This mechanism can be used to prove the existence of Arnold diffusion for large families of perturbations of Tonelli Hamiltonians on $ {{\mathbb A}}^3 $. Our approach can also be applied to more general Hamiltonians that are not necessarily convex.

The main geometric objects in our framework are $ 3 $–dimensional invariant cylinders with boundary (not necessarily hyperbolic), which are assumed to admit center-stable and center-unstable manifolds. These enable us to define chains of cylinders, i.e., finite, ordered families of cylinders where each cylinder admits homoclinic connections, and any two consecutive cylinders in the chain admit heteroclinic connections.

Our main result is on the existence of diffusing orbits which drift along such chains of cylinders, under precise conditions on the dynamics on the cylinders – i.e., the existence of Poincaré sections with the return maps satisfying a tilt condition – and on the geometric properties of the intersections of the center-stable and center-unstable manifolds of the cylinders – i.e., certain compatibility conditions between the tilt map and the homoclinic maps associated to its essential invariant circles.

We give two proofs of our result, a very short and abstract one, and a more constructive one, aimed at possible applications to concrete systems.

Keywords:

Mathematics Subject Classification: Primary: 37J40, 70H08; Secondary: 37E40.

Citation:

Full Text(HTML)

Figure 1. An illustration of a cylinder chain, consisting of a sequence of tame cylinders with boundary, connected via homoclinic and heteroclinic orbits. This chain also contains a singular cylinder (shown in the center of the figure); see Section 3.3 for details

Download: Full-size image PowerPoint slide

Figure 2. Homoclinic correspondence on the cylinder and the induced homoclinic correspondence on the Poincaré section

Download: Full-size image PowerPoint slide

Figure 3. Splitting arcs

Download: Full-size image PowerPoint slide

Figure 4. A triangular domain associated to a right splitting arc

Download: Full-size image PowerPoint slide

Figure 5. The setting of Proposition 3.2

Download: Full-size image PowerPoint slide

Figure 6. Positively and negatively tilted arcs

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	V. I. Arnold, Instability of dynamical systems with several degrees of freedom, Collected Works, Volume 1, Springer-Verlag Berlin Heidelberg, 2009,423-427. doi: 10.1007/978-3-642-01742-1_26.
[2]	P. W. Bates, K. Lu and C. Zeng, Invariant foliations near normally hyperbolic invariant manifolds for semiflows, Transactions of the American Mathematical Society, 352 (2000), 4641-4676. doi: 10.1090/S0002-9947-00-02503-4.
[3]	P. Berger, Persistence of laminations, Bull. Braz. Math. Soc., 41 (2010), 259-319. doi: 10.1007/s00574-010-0013-0.
[4]	P. Berger and A. Bounemoura, A geometrical proof of the persistence of normally hyperbolic submanifolds, Dynamical Systems: An International Journal, 28 (2013), 567-581. doi: 10.1080/14689367.2013.835386.
[5]	P. Bernard, The dynamics of pseudographs in convex Hamiltonian systems, J. Amer. Math. Soc., 21 (2008), 615-669. doi: 10.1090/S0894-0347-08-00591-2.
[6]	P. Bernard, V. Kaloshin and K. Zhang, Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders, Acta Mathematica, 217 (2016), 1-79. doi: 10.1007/s11511-016-0141-5.
[7]	M. Berti, L. Biasco and P. Bolle, Drift in phase space: A new variational mechanism with optimal diffusion time, J. Math. Pures Appl., 82 (2003), 613-664. doi: 10.1016/S0021-7824(03)00032-1.
[8]	U. Bessi, An approach to Arnold's diffusion through the calculus of variations, Nonlinear Anal., 26 (1996), 1115-1135. doi: 10.1016/0362-546X(94)00270-R.
[9]	S. Bolotin and D. Treschev, Unbounded growth of energy in nonautonomous Hamiltonian systems, Nonlinearity, 12 (1999), 365-387. doi: 10.1088/0951-7715/12/2/013.
[10]	P. Boyland, Rotation sets and Morse decompositions in twist maps, Ergod. Th. & Dynam. Sys., 8 (1988), 33-61. doi: 10.1017/S0143385700009329.
[11]	M. J. Capiński and M. Gidea, Arnold diffusion, quantitative estimates and stochastic behavior in the three-body problem, Communications on Pure and Applied Mathematics. doi: 10.1002/cpa.22014.
[12]	M. J. Capiński, J. Gonzalez, J.-P. Marco and J. D. Mireles James, Computer assisted proof of drift orbits along normally hyperbolic manifolds, Communications in Nonlinear Science and Numerical Simulations, 106 (2022), 105970. doi: 10.1016/j.cnsns.2021.105970.
[13]	M. Chaperon, The Lipschitz core of some invariant manifold theorems, Ergod. Th. & Dynam. Sys., 28 (2008), 1419-1441. doi: 10.1017/S0143385707000910.
[14]	C.-Q. Cheng, The genericity of Arnold diffusion in nearly integerable Hamiltonian systems, Asian J. Math., 23 (2019), 401-438. doi: 10.4310/AJM.2019.v23.n3.a3.
[15]	C.-Q. Cheng and J. Yang, Arnold diffusion in Hamiltonian systems: Nearly integrable case, J. Diff. Geom., 82 (2009), 229-277. doi: 10.4310/jdg/1246888485.
[16]	L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 1-144.
[17]	A. Delshams, R. de la Llave and T. M. Seara, A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of $T^2$, Comm. Math. Phys., 209 (2000), 353-392. doi: 10.1007/PL00020961.
[18]	A. Delshams, R. de la Llave and T. M. Seara, A Geometric Mechanism for Diffusion in Hamiltonian Systems Overcoming the Large Gap Problem, Memoirs AMS, 179, American Mathematical Soc., 2006. doi: 10.1090/memo/0844.
[19]	A. Delshams, R. de la Llave and T. M. Seara, Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows, Adv. Math., 202 (2006), 64-188. doi: 10.1016/j.aim.2005.03.005.
[20]	A. Delshams, R. de la Llave and T. M. Seara, Geometric properties of the scattering map of a normally hyperbolic invariant manifold, Adv. Math., 217 (2008), 1096-1153. doi: 10.1016/j.aim.2007.08.014.
[21]	A. Delshams, M. Gidea and P. Roldan, Transition map and shadowing lemma for normally hyperbolic invariant manifolds, Discrete Contin. Dyn. Syst., 33 (2013), 1089-1112. doi: 10.3934/dcds.2013.33.1089.
[22]	A. Delshams and P. Gutiérrez, Splitting Potential and the Poincaré-Melnikov Method for Whiskered Tori in Hamiltonian Systems, J. Nonlinear Sci., 10 (2000), 433-476. doi: 10.1007/s003329910016.
[23]	A. Delshams and R. G. Schaefer, Arnold diffusion for a complete family of perturbations, Regular and Chaotic Dynamics, 22 (2017), 78-108. doi: 10.1134/S1560354717010051.
[24]	N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1972), 193-226. doi: 10.1512/iumj.1972.21.21017.
[25]	E. Fontich and P. Martín, Hamiltonian systems with orbits covering densely submanifolds of small codimension, Nonlinear Analysis, 52 (2003), 315-327. doi: 10.1016/S0362-546X(02)00115-3.
[26]	V. Gelfreich and D. Turaev, Arnold diffusion in a priori chaotic symplectic maps, Communications in Mathematical Physics, 353 (2017), 507-547. doi: 10.1007/s00220-017-2867-0.
[27]	M. Gidea and R. de la Llave, Topological methods in the instability problem of Hamiltonian systems, Discrete Contin. Dyn. Syst., 14 (2006), 295-328. doi: 10.3934/dcds.2006.14.295.
[28]	M. Gidea, R. de la Llave and T. M-Seara, A general mechanism of diffusion in Hamiltonian systems: Qualitative results, Communications on Pure and Applied Mathematics, 73 (2020), 150-209. doi: 10.1002/cpa.21856.
[29]	M. Gidea and C. Robinson, Shadowing orbits for transition chains of invariant tori alternating with Birkhoff zones of instability, Nonlinearity, 20 (2007), 1115-1143. doi: 10.1088/0951-7715/20/5/004.
[30]	M. Gidea and C. Robinson, Obstruction argument for transition chains of tori interspersed with gaps, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 393-416. doi: 10.3934/dcdss.2009.2.393.
[31]	M. Gidea and C. Robinson, Diffusion along transition chains of invariant tori and Aubry-Mather sets, Ergodic Theory & Dynamical Systems, 33 (2013), 1401-1449. doi: 10.1017/S0143385712000363.
[32]	N. Gourmelon, Adapted metrics for dominated splitting, Erg. Th. and. Dyn. Syst., 27 (2007), 1839-1849. doi: 10.1017/S0143385707000272.
[33]	M. Guardia, T. M. Seara P. Martín and L. Sabbagh, Oscillatory orbits in the restricted elliptic planar three body problem, Discrete and Continuous Dynamical Systems, 37 (2017), 229-256. doi: 10.3934/dcds.2017009.
[34]	G. R. Hall, Birkhoff Orbits for Non-Monotone Twist Maps, Ph.D thesis, Wisconsin Univ-Madison Mathematics Research Center, 1984.
[35]	M.-R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau, Astérisque, 103-104 1983.
[36]	M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics 583, Springer-Verlag, 1977.
[37]	S. Hu, A variational principle associated to positive tilt maps, Commun. Math. Phys., 191 (1998), 627-639. doi: 10.1007/s002200050281.
[38]	V. Kaloshin and K. Zhang, Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom, (AMS-208), Princeton University Press, 2020.
[39]	A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 54, 1997. doi: 10.1017/CBO9780511809187.
[40]	L. Lazzarini and J.-P. Marco, From the Birkhoff theory of twist maps to Arnold diffusion in the a priori unstable case, preprint, 2016.
[41]	L. Lazzarini, J.-P. Marco and D. Sauzin, Measure and Capacity of Wandering Domains in Gevrey Near-Integrable Exact Symplectic Systems, Mem. Amer. Math. Soc. 1235, 2019. doi: 10.1090/memo/1235.
[42]	P. Le Calvez, Propriétés dynamiques des difféomorphismes de l'anneau et du tore, Astérisque, 204 (1991), 131 pp.
[43]	P. Le Calvez, Existence d'orbites quasi-périodiques dans les attracteurs de Birkhoff, Comm. Math. Phys., 106 (1986), 383-394. doi: 10.1007/BF01207253.
[44]	P. Le Calvez, Propriétés dynamiques des régions d'instabilité, Ann. Scient. Éc. Norm. Sup., 20 (1987), 443-464. doi: 10.24033/asens.1539.
[45]	P. Le Calvez, Drift orbits for families of twist maps of the annulus, Ergodic Theory and Dynamical Systems, 27 (2007), 869-879. doi: 10.1017/S0143385706000903.
[46]	J.-P. Marco, Modèles pour les applications fibrées et les polysystèmes, C. R. Math. Acad. Sci. Paris, 346 (2008), 203-208. doi: 10.1016/j.crma.2007.11.017.
[47]	J.-P. Marco, Chains of compact cylinders for cusp-residual nearly integrable systems on ${{\mathbb A}}^3$, preprint, 2016, arXiv: 1602.02399.
[48]	J.-P. Marco, Arnold diffusion in cusp-residual nearly integrable systems on ${{\mathbb A}}^3$, preprint, 2016, arXiv: 1602.02403.
[49]	J.-P. Marco, Twist maps and Arnold diffusion for diffeomorphisms, in New Trends in Calculus of Variations’, Series: Radon Series on Computational and Applied Mathematics, Academic Press, 18, de Gruyter, 2017, 473-495.
[50]	J.-P. Marco, A symplectic approach to Arnold diffusion problems, to appear, Proccedings of HST.
[51]	J. N. Mather, Non-existence of invariant circles, Ergodic Theory and Dynamical Ssystems, 4 (1984), 301-309. doi: 10.1017/S0143385700002455.
[52]	J. N. Mather, Arnold diffusion, I: Announcement of Results, J. Math. Sci., 24 (2004), 5275-5289. doi: 10.1023/B:JOTH.0000047353.78307.09.
[53]	J. N. Mather, Order structure on action minimizing orbits, in Symplectic Topology and Measure Preserving Dynamical Systems, Contemp. Math., volume 512, Amer. Math. Soc., Providence, RI, (2010), 41-125. doi: 10.1090/conm/512/10060.
[54]	J. N. Mather, Arnold diffusion by variational methods, in Essays in Mathematics and its Applications, Springer, Heidelberg, 2012,271-285. doi: 10.1007/978-3-642-28821-0_11.
[55]	R. Moeckel, Generic drift on Cantor sets of annuli, in Celestial Mechanics (eds. A. Chenciner R. Cushman, C. Robinson, Z.J. Xia), Contemp. Math., 292, Amer. Math. Soc., Providence, RI, 2002,163-171 doi: 10.1090/conm/292/04922.
[56]	M. Nassiri and E. R. Pujals, Robust transitivity in Hamiltonian dynamics, Annales Scientifiques de l'École Normale Supérieure, 45 (2012), 191-239. doi: 10.24033/asens.2164.
[57]	A.-N. Nekhoroshev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems, Russian Mathematical Surveys, 32 (1977), 5-66. doi: 10.1070/RM1977v032n06ABEH003859.
[58]	L. Sabbagh, An inclination lemmas for normally hyperbolic invariant manifolds with applications to diffusion, Ergodic Theory and Dynamical Systems, 35 (2015), 2269-2291. doi: 10.1017/etds.2014.30.
[59]	D. Treschev, Evolution of slow variables in a priori unstable Hamiltonian systems, Nonlinearity, 17 (2004), 1803-1841. doi: 10.1088/0951-7715/17/5/014.
[60]	J-C. Yoccoz, Travaux de Herman sur les tores invariants, Astérisque, 206 (1992).