# American Institute of Mathematical Sciences

December  2014, 34(12): 5135-5164. doi: 10.3934/dcds.2014.34.5135

## Measuring the total amount of chaos in some Hamiltonian systems

 1 Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona

Received  February 2014 Revised  April 2014 Published  June 2014

We consider some simple Hamiltonian systems, variants or generalizations of the Hénon-Heiles system, in two and three degrees of freedom, around a positive definite elliptic point, in resonant and non-resonant cases. After reviewing some theoretical background, we determine a measure of the domain of chaoticity by looking at the frequency of positive Lyapunov exponents in a sample of initial conditions. The question we study is how this measure depends on the energy and parameters and which are the dynamical objects responsible for the observed behaviour.
Citation: Carles Simó. Measuring the total amount of chaos in some Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5135-5164. doi: 10.3934/dcds.2014.34.5135
##### References:
 [1] V. I. Arnold and A. Avez, Problèmes Ergodiques de la Mécanique Classique, Gauthier-Villars, Paris, 1967. [2] H. Broer, R. Roussarie and C. Simó, Invariant circles in the Bogdanov-Takens bifurcation for diffeomorphism, Ergod. Th. & Dynam. Systems, 16 (1996), 1147-1172. doi: 10.1017/S0143385700009950. [3] H. Broer and C. Simó, Hill's equation with quasi-periodic forcing: Resonance tongues, instability pockets and global phenomena, Bul. Soc. Bras. Mat., 29 (1998), 253-293. doi: 10.1007/BF01237651. [4] P. M. Cincotta, C. M. Giordano and C. Simó, Phase space structure of multidimensional systems by means of the Mean Exponential Growth factor of Nearby Orbits (MEGNO), Physica D, 182 (2003), 151-178. doi: 10.1016/S0167-2789(03)00103-9. [5] A. Delshams, V. Gelfreich, À. Jorba and T. Martínez-Seara, Exponentially small splitting of separatrices under fast quasiperiodic forcing, Comm. Math. Phys., 189 (1997), 35-71. doi: 10.1007/s002200050190. [6] A. Delshams, M. Gonchenko and P. Gutiérrez, Exponentially small lower bounds for the splitting of separatrices to whiskered tori with frequencies of constant type,, Preprint, (): 14. [7] E. Fontich and C. Simó, Invariant manifolds for near identity differentiable maps and splitting of separatrices, Ergod. Th. & Dynam. Systems, 10 (1990), 319-346. doi: 10.1017/S0143385700005575. [8] E. Fontich and C. Simó, The splitting of separatrices for analytic diffeomorphisms, Ergod. Th. & Dynam. Systems, 10 (1990), 295-318. doi: 10.1017/S0143385700005563. [9] C. Froeschlé, Numerical study of a four-dimensional mapping, Astronom. and Astrophys., 16 (1972), 172-189. [10] V. Gelfreich and C. Simó, High-precision computations of divergent asymptotic series and homoclinic phenomena, Discrete and Continuous Dynamical Systems B, 10 (2008), 511-536. doi: 10.3934/dcdsb.2008.10.511. [11] V. Gelfreich, C. Simó and A. Vieiro, Dynamics of $4D$ symplectic maps near a double resonance, Physica D, 243 (2013), 92-110. doi: 10.1016/j.physd.2012.10.001. [12] A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and C. Simó, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem, Journal of Differential Equations, 77 (1989), 167-198. doi: 10.1016/0022-0396(89)90161-7. [13] G. Gómez, J. M. Mondelo and C. Simó, A collocation method for the numerical fourier analysis of quasi-periodic functions. I: Numerical tests and examples, Discrete and Continuous Dynamical Systems B, 14 (2010), 41-74. doi: 10.3934/dcdsb.2010.14.41. [14] G. Gómez, J. M. Mondelo and C. Simó, A Collocation Method for the Numerical Fourier Analysis of Quasi-periodic Functions. II: Analytical error estimates, Discrete and Continuous Dynamical Systems B, 14 (2010), 75-109. doi: 10.3934/dcdsb.2010.14.75. [15] M. Hénon and C. Heiles, The applicability of the third integral of motion: Some numerical experiments, Astronom. J., 69 (1964), 73-79. doi: 10.1086/109234. [16] H. Ito, Non-integrability of the Hénon-Heiles system and a theorem of Ziglin, Koday Math. J., 8 (1985), 120-138. doi: 10.2996/kmj/1138037004. [17] J. Laskar, The chaotic motion of the solar system. A numerical estimate of the size of the chaotic zones, Icarus, 88 (1990), 266-291. doi: 10.1016/0019-1035(90)90084-M. [18] F. Ledrappier, M. Shub, C. Simó and A. Wilkinson, Random versus deterministic exponents in a rich family of diffeomorphisms, J. Statist. Phys., 113 (2003), 85-149. doi: 10.1023/A:1025770720803. [19] A. Luque and J. Villanueva, Quasi-Periodic Frequency Analysis Using Averaging-Extrapolation Methods, SIAM J. Appl. Dyn. Syst., 13 (2014), 1-46. doi: 10.1137/130920113. [20] R. Martínez and C. Simó, Non-integrability of Hamiltonian systems through high order variational equations: Summary of results and examples, Regular and Chaotic Dynamics, 14 (2009), 323-348. doi: 10.1134/S1560354709030010. [21] R. Martínez and C. Simó, Non-integrability of the degenerate cases of the Swinging Atwood's Machine using higher order variational equations, Discrete and Continuous Dynamical Systems A, 29 (2011), 1-24. doi: 10.3934/dcds.2011.29.1. [22] N. Miguel, C. Simó and A. Vieiro, From the Hénon conservative map to the Chirikov standard map for large parameter values, Regular and Chaotic Dynamics, 18 (2013), 469-489. doi: 10.1134/S1560354713050018. [23] J. J. Morales-Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Progress in Mathematics, 179, Birkhäuser, 1999. doi: 10.1007/978-3-0348-8718-2. [24] J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems I, Methods and Applications of Analysis, 8 (2001), 33-95. [25] J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems II, Methods and Applications of Analysis, 8 (2001), 97-112. [26] J. J. Morales, J. P. Ramis and C. Simó, Integrability of hamiltonian systems and differential galois groups of higher variational equations, Annales Sci. de l'ENS 4$^e$ série, 40 (2007), 845-884. doi: 10.1016/j.ansens.2007.09.002. [27] J. J. Morales and C. Simó, Non integrability criteria for Hamiltonians in the case of Lamé normal variational equations, J. Diff. Equations, 129 (1996), 111-135. doi: 10.1006/jdeq.1996.0113. [28] A. Morbidelli and A. Giorgilli, Superexponential stability of KAM tori, J. Statist. Phys, 78 (1995), 1607-1617. doi: 10.1007/BF02180145. [29] J. Moser, Stable and Random Motions in Dynamical Systems: With Special Emphasis on Celestial Mechanics, Princeton University Press, 1973. [30] A. I. Neishtadt, The separation of motions in systems with rapidly rotating phase, Prikladnaja Matematika i Mekhanika, 48 (1984), 133-139. doi: 10.1016/0021-8928(84)90078-9. [31] N. N. Nekhorosev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems, Russian Mathematical Surveys, 32 (6) (1977), 5-66. [32] J. B. Pesin, Characteristic exponents and smooth ergodic theory, Russian Math. Surveys, 32 (4) (1977), 55-112. [33] J. Sánchez, M. Net and C. Simó, Computation of invariant tori by Newton-Krylov methods in large-scale dissipative systems, Physica D, 239 (2010), 123-133. doi: 10.1016/j.physd.2009.10.012. [34] C. Siegel and J. Moser, Lectures on Celestial Mechanics, Springer, 1971. [35] C. Simó, On the analytical and numerical approximation of invariant manifolds, Modern methods in celestial mechanics, D. Benest and C. Froeschlé, editors, 285-330, Editions Frontières, Paris, (1990). (Also available at http://www.maia.ub.es/dsg/2004/). [36] C. Simó, Averaging under fast quasiperiodic forcing, Integrable and chaotic behaviour in Hamiltonian Systems, I. Seimenis, editor,, Plenum Pub. Co., New York, 331 (1994), 13-34. [37] C. Simó, Invariant Curves of Perturbations of Non Twist Integrable Area Preserving Maps, Regular and Chaotic Dynamics, 3 (1998), 180-195. doi: 10.1070/rd1998v003n03ABEH000088. [38] C. Simó, Global Dynamics and Fast Indicators, Global Analysis of Dynamical Systems, H. W. Broer, B. Krauskopf and G. Vegter, editors, 373-389, IOP Publishing, Bristol, (2001). [39] C. Simó and T. Stuchi, Central Stable/Unstable Manifolds and the destruction of KAM tori in the planar Hill problem, Physica D, 140 (2000), 1-32. doi: 10.1016/S0167-2789(99)00211-0. [40] C. Simó, P. Sousa-Silva and M. Terra, Practical Stability Domains near $L_{4,5}$ in the Restricted Three-Body Problem: Some preliminary facts, Progress and Challenges in Dynamical Systems, Springer Proceedings in Mathematics & Statistics Series, 54, S. Ibáñez et al., editors, 367-382, Springer, (2013). [41] C. Simó, P. Sousa-Silva and M. Terra, Evidences of Diffusion Related to the Centre Manifold of $L_3$ in the 3D RTBP,, Work in progress., (). [42] C. Simó and D. Treschev, Stability islands in the vicinity of separatrices of near-integrable symplectic maps, Discrete and Continuous Dynamical Systems B, 10 (2008), 681-698. doi: 10.3934/dcdsb.2008.10.681. [43] C. Simó and A. Vieiro, Resonant zones, inner and outer splittings in generic and low order resonances of Area Preserving Maps, Nonlinearity, 22 (2009), 1191-1245. doi: 10.1088/0951-7715/22/5/012. [44] C. Simó and A. Vieiro, Dynamics in chaotic zones of area preserving maps: close to separatrix and global instability zones, Physica D, 240 (2011), 732-753. doi: 10.1016/j.physd.2010.12.005. [45] D. Treschev, Multidimensional symplectic separatrix maps, J. Nonlinear Sci., 12 (2002), 27-58. doi: 10.1007/s00332-001-0460-2. [46] S. L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics I, Funct. Anal. Appl., 16 (1982), 181-189.

show all references

##### References:
 [1] V. I. Arnold and A. Avez, Problèmes Ergodiques de la Mécanique Classique, Gauthier-Villars, Paris, 1967. [2] H. Broer, R. Roussarie and C. Simó, Invariant circles in the Bogdanov-Takens bifurcation for diffeomorphism, Ergod. Th. & Dynam. Systems, 16 (1996), 1147-1172. doi: 10.1017/S0143385700009950. [3] H. Broer and C. Simó, Hill's equation with quasi-periodic forcing: Resonance tongues, instability pockets and global phenomena, Bul. Soc. Bras. Mat., 29 (1998), 253-293. doi: 10.1007/BF01237651. [4] P. M. Cincotta, C. M. Giordano and C. Simó, Phase space structure of multidimensional systems by means of the Mean Exponential Growth factor of Nearby Orbits (MEGNO), Physica D, 182 (2003), 151-178. doi: 10.1016/S0167-2789(03)00103-9. [5] A. Delshams, V. Gelfreich, À. Jorba and T. Martínez-Seara, Exponentially small splitting of separatrices under fast quasiperiodic forcing, Comm. Math. Phys., 189 (1997), 35-71. doi: 10.1007/s002200050190. [6] A. Delshams, M. Gonchenko and P. Gutiérrez, Exponentially small lower bounds for the splitting of separatrices to whiskered tori with frequencies of constant type,, Preprint, (): 14. [7] E. Fontich and C. Simó, Invariant manifolds for near identity differentiable maps and splitting of separatrices, Ergod. Th. & Dynam. Systems, 10 (1990), 319-346. doi: 10.1017/S0143385700005575. [8] E. Fontich and C. Simó, The splitting of separatrices for analytic diffeomorphisms, Ergod. Th. & Dynam. Systems, 10 (1990), 295-318. doi: 10.1017/S0143385700005563. [9] C. Froeschlé, Numerical study of a four-dimensional mapping, Astronom. and Astrophys., 16 (1972), 172-189. [10] V. Gelfreich and C. Simó, High-precision computations of divergent asymptotic series and homoclinic phenomena, Discrete and Continuous Dynamical Systems B, 10 (2008), 511-536. doi: 10.3934/dcdsb.2008.10.511. [11] V. Gelfreich, C. Simó and A. Vieiro, Dynamics of $4D$ symplectic maps near a double resonance, Physica D, 243 (2013), 92-110. doi: 10.1016/j.physd.2012.10.001. [12] A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and C. Simó, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem, Journal of Differential Equations, 77 (1989), 167-198. doi: 10.1016/0022-0396(89)90161-7. [13] G. Gómez, J. M. Mondelo and C. Simó, A collocation method for the numerical fourier analysis of quasi-periodic functions. I: Numerical tests and examples, Discrete and Continuous Dynamical Systems B, 14 (2010), 41-74. doi: 10.3934/dcdsb.2010.14.41. [14] G. Gómez, J. M. Mondelo and C. Simó, A Collocation Method for the Numerical Fourier Analysis of Quasi-periodic Functions. II: Analytical error estimates, Discrete and Continuous Dynamical Systems B, 14 (2010), 75-109. doi: 10.3934/dcdsb.2010.14.75. [15] M. Hénon and C. Heiles, The applicability of the third integral of motion: Some numerical experiments, Astronom. J., 69 (1964), 73-79. doi: 10.1086/109234. [16] H. Ito, Non-integrability of the Hénon-Heiles system and a theorem of Ziglin, Koday Math. J., 8 (1985), 120-138. doi: 10.2996/kmj/1138037004. [17] J. Laskar, The chaotic motion of the solar system. A numerical estimate of the size of the chaotic zones, Icarus, 88 (1990), 266-291. doi: 10.1016/0019-1035(90)90084-M. [18] F. Ledrappier, M. Shub, C. Simó and A. Wilkinson, Random versus deterministic exponents in a rich family of diffeomorphisms, J. Statist. Phys., 113 (2003), 85-149. doi: 10.1023/A:1025770720803. [19] A. Luque and J. Villanueva, Quasi-Periodic Frequency Analysis Using Averaging-Extrapolation Methods, SIAM J. Appl. Dyn. Syst., 13 (2014), 1-46. doi: 10.1137/130920113. [20] R. Martínez and C. Simó, Non-integrability of Hamiltonian systems through high order variational equations: Summary of results and examples, Regular and Chaotic Dynamics, 14 (2009), 323-348. doi: 10.1134/S1560354709030010. [21] R. Martínez and C. Simó, Non-integrability of the degenerate cases of the Swinging Atwood's Machine using higher order variational equations, Discrete and Continuous Dynamical Systems A, 29 (2011), 1-24. doi: 10.3934/dcds.2011.29.1. [22] N. Miguel, C. Simó and A. Vieiro, From the Hénon conservative map to the Chirikov standard map for large parameter values, Regular and Chaotic Dynamics, 18 (2013), 469-489. doi: 10.1134/S1560354713050018. [23] J. J. Morales-Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Progress in Mathematics, 179, Birkhäuser, 1999. doi: 10.1007/978-3-0348-8718-2. [24] J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems I, Methods and Applications of Analysis, 8 (2001), 33-95. [25] J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems II, Methods and Applications of Analysis, 8 (2001), 97-112. [26] J. J. Morales, J. P. Ramis and C. Simó, Integrability of hamiltonian systems and differential galois groups of higher variational equations, Annales Sci. de l'ENS 4$^e$ série, 40 (2007), 845-884. doi: 10.1016/j.ansens.2007.09.002. [27] J. J. Morales and C. Simó, Non integrability criteria for Hamiltonians in the case of Lamé normal variational equations, J. Diff. Equations, 129 (1996), 111-135. doi: 10.1006/jdeq.1996.0113. [28] A. Morbidelli and A. Giorgilli, Superexponential stability of KAM tori, J. Statist. Phys, 78 (1995), 1607-1617. doi: 10.1007/BF02180145. [29] J. Moser, Stable and Random Motions in Dynamical Systems: With Special Emphasis on Celestial Mechanics, Princeton University Press, 1973. [30] A. I. Neishtadt, The separation of motions in systems with rapidly rotating phase, Prikladnaja Matematika i Mekhanika, 48 (1984), 133-139. doi: 10.1016/0021-8928(84)90078-9. [31] N. N. Nekhorosev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems, Russian Mathematical Surveys, 32 (6) (1977), 5-66. [32] J. B. Pesin, Characteristic exponents and smooth ergodic theory, Russian Math. Surveys, 32 (4) (1977), 55-112. [33] J. Sánchez, M. Net and C. Simó, Computation of invariant tori by Newton-Krylov methods in large-scale dissipative systems, Physica D, 239 (2010), 123-133. doi: 10.1016/j.physd.2009.10.012. [34] C. Siegel and J. Moser, Lectures on Celestial Mechanics, Springer, 1971. [35] C. Simó, On the analytical and numerical approximation of invariant manifolds, Modern methods in celestial mechanics, D. Benest and C. Froeschlé, editors, 285-330, Editions Frontières, Paris, (1990). (Also available at http://www.maia.ub.es/dsg/2004/). [36] C. Simó, Averaging under fast quasiperiodic forcing, Integrable and chaotic behaviour in Hamiltonian Systems, I. Seimenis, editor,, Plenum Pub. Co., New York, 331 (1994), 13-34. [37] C. Simó, Invariant Curves of Perturbations of Non Twist Integrable Area Preserving Maps, Regular and Chaotic Dynamics, 3 (1998), 180-195. doi: 10.1070/rd1998v003n03ABEH000088. [38] C. Simó, Global Dynamics and Fast Indicators, Global Analysis of Dynamical Systems, H. W. Broer, B. Krauskopf and G. Vegter, editors, 373-389, IOP Publishing, Bristol, (2001). [39] C. Simó and T. Stuchi, Central Stable/Unstable Manifolds and the destruction of KAM tori in the planar Hill problem, Physica D, 140 (2000), 1-32. doi: 10.1016/S0167-2789(99)00211-0. [40] C. Simó, P. Sousa-Silva and M. Terra, Practical Stability Domains near $L_{4,5}$ in the Restricted Three-Body Problem: Some preliminary facts, Progress and Challenges in Dynamical Systems, Springer Proceedings in Mathematics & Statistics Series, 54, S. Ibáñez et al., editors, 367-382, Springer, (2013). [41] C. Simó, P. Sousa-Silva and M. Terra, Evidences of Diffusion Related to the Centre Manifold of $L_3$ in the 3D RTBP,, Work in progress., (). [42] C. Simó and D. Treschev, Stability islands in the vicinity of separatrices of near-integrable symplectic maps, Discrete and Continuous Dynamical Systems B, 10 (2008), 681-698. doi: 10.3934/dcdsb.2008.10.681. [43] C. Simó and A. Vieiro, Resonant zones, inner and outer splittings in generic and low order resonances of Area Preserving Maps, Nonlinearity, 22 (2009), 1191-1245. doi: 10.1088/0951-7715/22/5/012. [44] C. Simó and A. Vieiro, Dynamics in chaotic zones of area preserving maps: close to separatrix and global instability zones, Physica D, 240 (2011), 732-753. doi: 10.1016/j.physd.2010.12.005. [45] D. Treschev, Multidimensional symplectic separatrix maps, J. Nonlinear Sci., 12 (2002), 27-58. doi: 10.1007/s00332-001-0460-2. [46] S. L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics I, Funct. Anal. Appl., 16 (1982), 181-189.
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