# American Institute of Mathematical Sciences

December  2018, 38(12): 6047-6072. doi: 10.3934/dcds.2018261

## Arnold diffusion for a complete family of perturbations with two independent harmonics

 Departament de Matemàtiques and Lab of Geometry and Dynamical Systems, Universitat Politècnica de Catalunya, Av. Doctor Marañón, 44-50, Barcelona, 08028, Spain

To Rafael de la Llave on the occasion of his 60th birthday

Received  September 2017 Revised  January 2018 Published  September 2018

Fund Project: This work has been partially supported by the Spanish MINECO-FEDER grant MTM2015-65715 and the Catalan grant 2017SGR1049. AD has been also partially supported by the Russian Scientific Foundation grant 14-41-00044 at the Lobachevsky University of Nizhny Novgorod. RS has been also partially supported by CNPq, Conselho Nacional de Desenvolvimento Científico e Tecnológico - Brasil

We prove that for any non-trivial perturbation depending on any two independent harmonics of a pendulum and a rotor there is global instability. The proof is based on the geometrical method and relies on the concrete computation of several scattering maps. A complete description of the different kinds of scattering maps taking place as well as the existence of piecewise smooth global scattering maps is also provided.

Citation: Amadeu Delshams, Rodrigo G. Schaefer. Arnold diffusion for a complete family of perturbations with two independent harmonics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6047-6072. doi: 10.3934/dcds.2018261
##### References:
 [1] E. Canalias, A. Delshams, J.J. Masdemont and P. Roldan, The scattering map in the planar restricted three body problem, Celestial Mechanics and Dynamical Astronomy, 95 (2006), 155-171. doi: 10.1007/s10569-006-9010-4. Google Scholar [2] M. J. Capinski, M. Gidea and R. de la Llave, Arnold diffusion in the planar elliptic restricted three-body problem: mechanism and numerical verification, Nonlinearity, 30 (2016), 329-360. doi: 10.1088/1361-6544/30/1/329. Google Scholar [3] C. -Q. Cheng, Dynamics around the double resonance, Camb. J. Math., 5 (2017), 153-228. doi: 10.4310/CJM.2017.v5.n2.a1. Google Scholar [4] L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 1-144. Google Scholar [5] M. N. Davletshin and D. V. Treschev, Arnold diffusion in a neighborhood of strong resonances, Proc. Steklov Inst. Math., 295 (2016), 63-94. doi: 10.1134/S0371968516040051. Google Scholar [6] A. Delshams, M. Gidea and P. Roldán, Transition map and shadowing lemma for normally hyperbolic invariant manifolds, Discrete & Continuous Dynamical Systems - A, 33 (2013), 1089-1112. Google Scholar [7] A. Delshams, M. Gidea and P. Roldán, Arnold's mechanism of diffusion in the spatial circular restricted three-body problem: A semi-analytical argument, Physica D: Nonlinear Phenomena, 334 (2016), 29-48. doi: 10.1016/j.physd.2016.06.005. Google Scholar [8] A. Delshams and G. Huguet, Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems, Nonlinearity, 22 (2009), 1997-2077. doi: 10.1088/0951-7715/22/8/013. Google Scholar [9] A. Delshams and G. Huguet, A geometric mechanism of diffusion: Rigorous verification in a priori unstable Hamiltonian systems, J. Differential Equations, 250 (2011), 2601-2623. doi: 10.1016/j.jde.2010.12.023. Google Scholar [10] 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 ${\bf{T}}^2$, Comm. Math. Phys., 209 (2000), 353-392. doi: 10.1007/PL00020961. Google Scholar [11] A. Delshams, R. de la Llave and T. M. Seara, A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model, Mem. Amer. Math. Soc., 179 (2006), 1-141. doi: 10.1090/memo/0844. Google Scholar [12] 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. Google Scholar [13] A. Delshams, J. J. Masdemont and P. Roldán, Computing the scattering map in the spatial Hill's problem, Discrete & Continuous Dynamical Systems - B, 10 (2008), 455-483. doi: 10.3934/dcdsb.2008.10.455. Google Scholar [14] A. Delshams and T. M. Seara, Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom, Math. Phys. Electron. J., 3 (1997), Paper 4, 40 pp. Google Scholar [15] 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. Google Scholar [16] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, in Mathematics and its Applications (Soviet Series) Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9. Google Scholar [17] E. Fontich and P. Martín, Differentiable invariant manifolds for partially hyperbolic tori and a lambda lemma, Nonlinearity, 13 (2000), 1561-1593. doi: 10.1088/0951-7715/13/5/309. Google Scholar [18] E. Fontich and P. Martín, Hamiltonian systems with orbits covering densely submanifolds of small codimension, Nonlinear Anal., 52 (2003), 315-327. doi: 10.1016/S0362-546X(02)00115-3. Google Scholar [19] V. Gelfreich and D. Turaev, Arnold diffusion in a priori chaotic symplectic maps, Comm. Math. Phys., 353 (2017), 507-547. doi: 10.1007/s00220-017-2867-0. Google Scholar [20] M. Gidea, R. de la Llave and T. M. Seara, A general mechanism of diffusion in Hamiltonian systems: qualitative results, preprint, arXiv: 1405.0866.Google Scholar [21] M. Gidea and J. -P. Marco, Diffusion along chains of normally hyperbolic cylinders, preprint, arXiv: 1708.08314.Google Scholar [22] L. Lazzarini, J. -P. Marco and D. Sauzin, Measure and capacity of wandering domains in Gevrey near-integrable exact symplectic systems, preprint, to appear in Mem. Amer. Math. Soc., arXiv: 1507.02050Google Scholar [23] J. -P. Marco, Arnold diffusion for cusp-generic nearly integrable convex systems on $\mathbb A^3$, preprint, arXiv: 1602.02403.Google Scholar

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##### References:
 [1] E. Canalias, A. Delshams, J.J. Masdemont and P. Roldan, The scattering map in the planar restricted three body problem, Celestial Mechanics and Dynamical Astronomy, 95 (2006), 155-171. doi: 10.1007/s10569-006-9010-4. Google Scholar [2] M. J. Capinski, M. Gidea and R. de la Llave, Arnold diffusion in the planar elliptic restricted three-body problem: mechanism and numerical verification, Nonlinearity, 30 (2016), 329-360. doi: 10.1088/1361-6544/30/1/329. Google Scholar [3] C. -Q. Cheng, Dynamics around the double resonance, Camb. J. Math., 5 (2017), 153-228. doi: 10.4310/CJM.2017.v5.n2.a1. Google Scholar [4] L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 1-144. Google Scholar [5] M. N. Davletshin and D. V. Treschev, Arnold diffusion in a neighborhood of strong resonances, Proc. Steklov Inst. Math., 295 (2016), 63-94. doi: 10.1134/S0371968516040051. Google Scholar [6] A. Delshams, M. Gidea and P. Roldán, Transition map and shadowing lemma for normally hyperbolic invariant manifolds, Discrete & Continuous Dynamical Systems - A, 33 (2013), 1089-1112. Google Scholar [7] A. Delshams, M. Gidea and P. Roldán, Arnold's mechanism of diffusion in the spatial circular restricted three-body problem: A semi-analytical argument, Physica D: Nonlinear Phenomena, 334 (2016), 29-48. doi: 10.1016/j.physd.2016.06.005. Google Scholar [8] A. Delshams and G. Huguet, Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems, Nonlinearity, 22 (2009), 1997-2077. doi: 10.1088/0951-7715/22/8/013. Google Scholar [9] A. Delshams and G. Huguet, A geometric mechanism of diffusion: Rigorous verification in a priori unstable Hamiltonian systems, J. Differential Equations, 250 (2011), 2601-2623. doi: 10.1016/j.jde.2010.12.023. Google Scholar [10] 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 ${\bf{T}}^2$, Comm. Math. Phys., 209 (2000), 353-392. doi: 10.1007/PL00020961. Google Scholar [11] A. Delshams, R. de la Llave and T. M. Seara, A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model, Mem. Amer. Math. Soc., 179 (2006), 1-141. doi: 10.1090/memo/0844. Google Scholar [12] 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. Google Scholar [13] A. Delshams, J. J. Masdemont and P. Roldán, Computing the scattering map in the spatial Hill's problem, Discrete & Continuous Dynamical Systems - B, 10 (2008), 455-483. doi: 10.3934/dcdsb.2008.10.455. Google Scholar [14] A. Delshams and T. M. Seara, Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom, Math. Phys. Electron. J., 3 (1997), Paper 4, 40 pp. Google Scholar [15] 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. Google Scholar [16] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, in Mathematics and its Applications (Soviet Series) Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9. Google Scholar [17] E. Fontich and P. Martín, Differentiable invariant manifolds for partially hyperbolic tori and a lambda lemma, Nonlinearity, 13 (2000), 1561-1593. doi: 10.1088/0951-7715/13/5/309. Google Scholar [18] E. Fontich and P. Martín, Hamiltonian systems with orbits covering densely submanifolds of small codimension, Nonlinear Anal., 52 (2003), 315-327. doi: 10.1016/S0362-546X(02)00115-3. Google Scholar [19] V. Gelfreich and D. Turaev, Arnold diffusion in a priori chaotic symplectic maps, Comm. Math. Phys., 353 (2017), 507-547. doi: 10.1007/s00220-017-2867-0. Google Scholar [20] M. Gidea, R. de la Llave and T. M. Seara, A general mechanism of diffusion in Hamiltonian systems: qualitative results, preprint, arXiv: 1405.0866.Google Scholar [21] M. Gidea and J. -P. Marco, Diffusion along chains of normally hyperbolic cylinders, preprint, arXiv: 1708.08314.Google Scholar [22] L. Lazzarini, J. -P. Marco and D. Sauzin, Measure and capacity of wandering domains in Gevrey near-integrable exact symplectic systems, preprint, to appear in Mem. Amer. Math. Soc., arXiv: 1507.02050Google Scholar [23] J. -P. Marco, Arnold diffusion for cusp-generic nearly integrable convex systems on $\mathbb A^3$, preprint, arXiv: 1602.02403.Google Scholar
Plane $\varphi \times I$ of inner dynamics for $\mu = 0.75$ and $\varepsilon = 0.01$
Finding $\tau^*(I,\theta)$ using the straight line $\sigma = \varphi$
Examples of piecewise smooth global scattering maps. The orbits of scattering maps are represented by the blue lines. In the red zones the values of $I$ on such orbits decrease, in the green one the values of $I$ increase
$\left|\alpha(I)\right|$ and $\left|\beta(I)\right|$ : Behavior of the crests and tangencies
Comparison between $\xi_{\text{M}}(I,\varphi)$ and $\eta_{\text{M}}(I,\sigma)$ for $\mu = 0.5$, $I = 0.68$ and $I = 0.72$ respectively
Different phase space of scattering maps $\mathcal{S}(I,\theta)$ associated to the same horizontal crest $C_{\text{M}}(I)$, for $\mu = 0.6$ and $\varepsilon = 0.01$. The orbits of scattering maps are represented by the blue lines which are, up to $\mathcal{O}(\varepsilon^2)$, level sets of the reduced Poincaré function $\mathcal{L}^*(I,\theta)$. In the red zones the values of $I$ on such orbits decrease, in the green one the values of $I$ increase. The white regions are regions where $\left|\mu\alpha(I)\sin\varphi\right|>1$ is satisfied
A piecewise smooth global scattering map divided into 3 regions. The vertical black lines are the boundaries of the domains of smooth scattering maps
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