The coexistence problem" for conservative dynamical systems: a review

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1. [1] V. M. Alekseev, Quasirandom oscillations and qualitative problems of celestial mechanics, in: Ninth Mathematical Summer School, Naukova Dumka, Kiev 1976, 212-341 (in Russian); English transl. in: Three Papers on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, vol. 116, Providence, R.I., 1981, 97-169. 
2. [2] D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Trudy Mat. Inst. Steklov. 90 (1967) (in Russian); English transl. Amer. Math. Soc. Providence, R.I., 1969. 
3. [3] H. Aref, Integrable, chaotic, and turbulent vortex motion in two-dimensional flows, in: Annual Review of Fluid Mechanics, Vol. 15, Annual Reviews, Palo Alto, Calif., 1983, 345-389. 
4. [4] H. Aref, Chaos in the dynamics of a few vortices-fundamentals and applications, in: Theoretical and Applied Mechanics (Lyngby, 1984), F. I. Niordson and N. Olhoff (eds.), North-Holland, Amsterdam-New York 1985, 43-68. 
5. [5] H. Aref, J. B. Kadtke, I. Zawadzki, L. J. Campbell and B. Eckhardt, Point vortex dynamics: recent results and open problems, Fluid Dynamics Research 3 (1988), 63-74. 
6. [6] H. Aref and N. Pomphrey, Integrable and chaotic motions of four vortices I. The case of identical vortices, Proc. Roy. Soc. London Ser. A 380 (1982), 359-387. Zbl0483.76031
7. [7] V. I. Arnold, Mathematical Methods of Classical Mechanics, Nauka, Moscow 1974 (in Russian); English transl.: Springer, 1978. 
8. [8] G. Benettin, L. Galgani and J.-M. Strelcyn, Kolmogorov entropy and numerical experiments, Phys. Rev. A 14 (6) (1976), 2338-2345. 
9. [9] G. Benettin and J.-M. Strelcyn, Numerical experiments on the free motion of a point mass moving in a plane convex region: stochastic transition and entropy, ibid. 17 (2) (1978), 773-785. 
10. [10] P. Bergé (ed.), Le chaos, théorie et expériences, Eyrolles, Paris 1988. 
11. [11] O. I. Bogoyavlensky, On perturbations of the periodic Toda lattice, Comm. Math. Phys. 51 (1976), 201-209. 
12. [12] O. I. Bogoyavlensky, Methods in the Qualitative Theory of Dynamical Systems in Astrophysics and Gas Dynamics, Nauka, Moscow 1980 (in Russian); English transl. in: Springer Series in Soviet Mathematics, Springer, 1985. 
13. [13] M. Braun, Mathematical remarks on the Van Allen radiation belt: A survey of old and new results, SIAM Rev. 23 (1981), 61-93. Zbl0479.76128
14. [14] S. Bullett, Invariant circles for the piecewise linear standard map, Comm. Math. Phys. 107 (1986), 241-262. Zbl0617.58005
15. [15] G. Casati and J. Ford, Stochastic transition in the unequal-mass Toda lattice, Phys. Rev. A 12 (3) (1975), 1702-1709. 
16. [16] A. Chenciner, La dynamique au voisinage d'un point fixe elliptique conservatif: de Poincaré et Birkhoff à Aubry et Mather, Séminaire Bourbaki, Février 1984, Astérisque 121-122 (1985), 147-170. 
17. [17] B. V. Chirikov, Research concerning the theory of non-linear resonance and stochasticity, 1969, CERN Transl. 71-40, Geneva, October 1971. 
18. [18] B. V. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep. 52 (1979), 265-379. 
19. [19] B. V. Chirikov and F. M. Izrailev, Some numerical experiments with a nonlinear mapping: stochastic components, in: Transformations ponctuelles et leurs applications, Colloq. Internat. CNRS 229 (Toulouse 1973), Paris 1976, 409-428. 
20. [20] R. C. Churchill, G. Pecelli and D. L. Rod, A survey of the Hénon-Heiles Hamiltonian with applications to related examples, in: Stochastic Behavior in Classical and Quantum Hamiltonian Systems (Volta Memorial Conf., Como, 1977), G. Casati and J. Ford (eds.), Lecture Notes in Phys. 93, Springer, 1979, 76-136. Zbl0426.70020
21. [21] Y. Colin de Verdière, Sur les longueurs des trajectoires périodiques d'un billard, in: Géométrie symplectique et de contact: autour du théorème de Poincaré-Birkhoff, Travaux en Cours, Hermann, Paris 1984, 122-139. 
22. [22] E. Cornelis and M. Wojtkowski, A criterion of the positivity of the Lyapunov characteristic exponent, Ergodic Theory Dynamical Systems 4 (1984), 527-539. Zbl0578.22006
23. [23] R. Cushman, Examples of non-integrable analytic Hamiltonian vector fields with no small divisor, Trans. Amer. Math. Soc. 238 (1978), 45-55. Zbl0388.58008
24. [24] P. Cvitanović (ed.), Universality in Chaos. A selection of reprints, A. Hilger, Bristol, and Heyden, Philadelphia, Pa., 1984. 
25. [25] R. L. Devaney, A piecewise linear model for the zones of instability of an area preserving map, preprint. Zbl0588.58009
26. [26] T. Dombre, U. Frisch, J. M. Greene, M. Hénon, A. Mehr and A. M. Soward, Chaotic streamlines in the ABC flows, J. Fluid Mech. 167 (1986), 353-391. Zbl0622.76027
27. [27] M. M. Dvorin and V. F. Lazutkin, The existence of infinite number of elliptic and hyperbolic periodic trajectories for a convex billiard, Funktsional. Anal. Prilozhen. 7 (2) (1973), 20-27 (in Russian); English transl.: Functional Anal. Appl. 7 (2) (1973), 103-109. Zbl0298.58006
28. [28] B. Eckhardt, Irregular scattering of vortex pairs, Europhys. Lett. 5 (1988), 107- 111. 
29. [29] B. Eckhardt and H. Aref, Integrable and chaotic motion of four vortices II: collision dynamics of vortex pair, Philos. Trans. Roy. Soc. London Ser. A 326 (1988), 655-696. Zbl0661.76019
30. [30] M. Feingold, L. P. Kadanoff and O. Pirro, Passive scalars, three-dimensional volume preserving maps and chaos, J. Statist. Phys. 50 (1988), 529-565. Zbl0987.37055
31. [31] A. F. Filippov, Determination of characteristic exponents of linear systems with quasiperiodical coefficients, Mat. Zametki 44 (2) (1988), 231-243 (in Russian); English transl.: Math. Notes 44 (2) (1988), 609-619. Zbl0662.34008
32. [32] C. Froeschlé, Etude numérique de transformations ponctuelles planes conservant les aires, C. R. Acad. Sci. Paris Sér. A 266 (1968), 747-749 and 846-848. Zbl0159.26401
33. [33] L. Galgani, A. Giorgilli and J.-M. Strelcyn, Chaotic motions and transition to stochasticity in the classical problem of the heavy rigid body with a fixed point, Nuovo Cimento 61B (1) (1981), 1-20. 
34. [34] D. Goroff, Hyperbolic sets for twist maps, Ergodic Theory Dynamical Systems 5 (1985), 337-339. Zbl0551.58028
35. [35] Hao Bai-Lin (ed.), Chaos, World Scientific, 1984. 
36. [36] A. Hayli et Th. Dumont, Éxperiences numériques sur des billards $C^1$ formés de quatre arcs de cercles, Celestial Mech. 38 (1986), 23-66. Zbl0602.70022
37. [37] A. Hayli, Th. Dumont, J. Moulin-Ollagnier et J.-M. Strelcyn, Quelques résultats nouveaux sur les billards de Robnik, J. Phys. A 20 (1987), 3237-3249. Zbl0663.58021
38. [38] R. M. G. Helleman, Self-generated chaotic behavior in non-linear mechanics, in: Fundamental Problems in Statistical Mechanics, V (Proc. Fifth Internat. Summer School, Enschede, 1980), E. G. D. Cohen (ed.), North-Holland, Amsterdam-New York 1980, 165-233. 
39. [39] M. Hénon, Numerical study of quadratic area preserving mappings, Quart. Appl. Math. 26 (1969), 291-312. Zbl0191.45403
40. [40] M. Hénon and C. Heiles, The applicability of the third integral of motion: some numerical experiments, Astronom. J. 69 (1964), 73-79. 
41. [41] M. Hénon and J. Wisdom, The Benettin-Strelcyn oval billiard revisited, Physica 8D (1983), 157-169. Zbl0538.58032
42. [42] M. R. Herman, Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d'un théorème d'Arnold et de Moser sur le tore de dimension 2, Comment. Math. Helv. 58 (1983), 453-502. Zbl0554.58034
43. [43] M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau, Vol. 2, Astérisque 144 (1986). 
44. [44] K. Kaneko, Collaps of Tori and Genesis of Chaos in Dissipative Systems, World Scientific, 1986. Zbl0647.58032
45. [45] A. Katok, Some remarks on Birkhoff and Mather twist map theorem, Ergodic Theory Dynamical Systems 2 (1982), 185-194. Zbl0521.58048
46. [46] A. Katok, Periodic and quasi-periodic orbits for twist maps, in: Dynamical Systems and Chaos, L. Garrido (ed.), Lecture Notes in Phys. 179, Springer, 1983, 47-65. Zbl0517.58032
47. [47] A. Katok and J.-M. Strelcyn (with the collaboration of F. Ledrappier and F. Przytycki), Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities, Lecture Notes in Math. 1222, Springer, 1986. Zbl0658.58001
48. [48] Y. Kimura and H. Hasimoto, Motion of a pair of vortices in a semi-circular domain, J. Phys. Soc. Japan 55 (1986), 5-8. 
49. [49] V. V. Kozlov, Integrability and non-integrability in Hamiltonian mechanics, Uspekhi Mat. Nauk 38 (1) (1983), 1-67 (in Russian); English transl.: Russian Math. Surveys 38 (1) (1983), 1-76. Zbl0525.70023
50. [50] V. F. Lazutkin, The convex billiards and eigenfunctions of the Laplace operator, Izdat. Leningrad. Univ., Leningrad 1981 (in Russian). Zbl0532.58031
51. [51] V. F. Lazutkin, Splitting of separatrices of Chirikov's standard map, preprint, 1984 (in Russian). 
52. [52] V. F. Lazutkin, Splitting of complex separatrices, Funktsional Anal. i Prilozhen. 22 (2) (1988), 83-84 (in Russian); English transl.: Functional. Anal. Appl. 22 (2) (1988), 154-156. Zbl0658.58027
53. [53] V. F. Lazutkin, Analytic integrals of semi-standard maps and the splitting of separatrices, Algebra i Analiz 1 (2) (1989), 116-131 (in Russian). 
54. [54] V. F. Lazutkin, The width of the instability zone around separatrices of a standard mapping, Dokl. Akad. Nauk SSSR 313 (1990), 268-272 (in Russian). 
55. [55] V. F. Lazutkin, M. B. Tabanov and I. G. Shakhmanskiĭ, The splitting of separatrices for standard and semi-standard maps, preprint, 1985 (in Russian). 
56. [56] P. Le Calvez, Les ensembles d'Aubry-Mather d'un difféomorphisme conservatif de l'anneau déviant la verticale sont en général hyperboliques, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), 51-54. 
57. [57] M. Levi, Qualitative analysis of the periodically forced relaxation oscillations, Mem. Amer. Math. Soc. 244 (1981). Zbl0448.34032
58. [58] A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion, Springer, 1983. 
59. [59] R. S. MacKay, Transition to chaos for area preserving maps, in: Nonlinear Dynamics Aspects of Particle Accelerators, J. M. Jowett, M. Month and S. Turner (eds.), Lecture Notes in Phys. 247, Springer, 1986, 390-454. 
60. [60] R. S. MacKay and J. D. Meiss (eds.), Hamiltonian Dynamical Systems, A Reprint Selection, A. Hilger, 1987. 
61. [61] R. S. MacKay and I. C. Percival, Converse KAM: theory and practice, Comm. Math. Phys. 98 (1985), 469-512. Zbl0585.58032
62. [62] J. Moser, Stable and Random Motions in Dynamical Systems, Ann. of Math. Stud. 77, Princeton Univ. Press, Princeton, N.J., 1973. Zbl0271.70009
63. [63] S. A. Orszag and J. B. McLaughlin, Evidence that random behavior is generic for nonlinear differential equations, Physica 1D (1980), 68-79. Zbl1194.37060
64. [64] Ya. B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Uspekhi Mat. Nauk 32 (4) (1977), 55-112 (in Russian); English transl.: Russian Math. Surveys 32 (4) (1977), 55-114. 
65. [65] F. Przytycki, Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behavior, Ergodic Theory Dynamical Systems 2 (1982), 439-463. Zbl0544.58012
66. [66] Ch. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc. 312 (1989), 1-54. Zbl0684.58008
67. [67] P. H. Richter and H. J. Scholz, Chaos in classical mechanics: the double pendulum, in: Stochastic Phenomena and Chaotic Behavior in Complex Systems, P. Schuster (ed.), Springer, 1984, 86-97. 
68. [68] M. Robnik, Classical dynamics of a family of billiards with analytic boundaries, J. Phys. A 16 (1983), 3971-3986. Zbl0622.70011
69. [69] D. L. Rod and R. C. Churchill, A guide to the Hénon-Heiles Hamiltonian, in: Singularities and Dynamical Systems (Iraklion, 1983) S. N. Pnevmatikos (ed.), North-Holland Math. Stud. 103, Elsevier, North-Holland, Amsterdam-New York 1985, 385-395. 
70. [70] R. Z. Sagdeev, D. A. Usikov and G. M. Zaslavsky, Nonlinear Series Physics. From the Pendulum to Turbulence and Chaos, Contemp. Concepts Phys. Series 4, Gordon and Breach, and Harwood, 1988. 
71. [71] N. Saitō, H. Hirooka, J. Ford, F. Vivaldi and G. H. Walker, Numerical study of billiard motion in an annulus bounded by non-concentric circles, Physica 5D (1982), 273-286. Zbl1194.65125
72. [72] K. Shiraiwa, Bibliography for dynamical systems, Preprint Series 1 (1985), Dept. of Math., College of General Educat., Furocho, Chikusa-Ku, Nagoya 464, Japan. Zbl0549.58001
73. [73] Yu. B. Suris, Integrable mappings of standard type, Funktsional. Anal. i Prilozhen. 23 (1) (1989), 84-85 (in Russian); English transl.: Functional Anal. Appl. 23 (1) (1989), 74-76. 
74. [74] M. Toda, Theory of Nonlinear Lattices, 2nd ed., Springer, 1989. 
75. [75] D. K. Umberger and J. D. Farmer, Fat fractals on the energy surfaces, Phys. Rev. Lett. 55 (1985), 661-664. 
76. [76] S. Wiggins, Global Bifurcations and Chaos, Analytical Methods, Appl. Math. Sci. 73, Springer, 1988. Zbl0661.58001
77. [77] M. Wojtkowski, A model problem with the coexistence of stochastic and integrable behaviour, Comm. Math. Phys. 80 (1981), 453-464. Zbl0473.28006
78. [78] M. Wojtkowski, On the ergodic properties of piecewise linear perturbations of the twist map, Ergodic Theory Dynamical Systems 2 (1982), 525-542. Zbl0528.58024
79. [79] M. Wojtkowski, Invariant families of cones and Lyapunov exponents, ibid. 5 (1985), 145-161. Zbl0578.58033
80. [80] M. Wojtkowski, Principle of the design of billiards with nonvanishing Lyapunov exponents, Comm. Math. Phys. 105 (1986), 391-414. Zbl0602.58029
81. [81] G. M. Zaslavsky, Stochasticity of Dynamical Systems, Nauka, Moscow 1984 (in Russian); English transl.: Chaos in Dynamical Systems, Harwood, 1985. 
82. [82] G. M. Zaslavsky and R. Z. Sagdeev, Introduction to Nonlinear Physics, From the Pendulum to Turbulence and Chaos, Nauka, Moscow 1988 (in Russian). Zbl0709.58003
83. [83] E. Zehnder, Homoclinic points near elliptic fixed points, Comm. Pure Appl. Math. 26 (1973), 131-182. Zbl0261.58002
84. [84] Zhou Jianying, Chaotic behavior in the Taylor mapping, Sci. Sinica Ser. A 28 (1) (1985), 47-60. Zbl0606.58034
85. [85] V. J. Donnay, Geodesic flow on the two-sphere, part I: Positive measure entropy, Ergodic Theory Dynamical Systems 8 (1988), 531-553. Zbl0645.58030
86. [86] V. Donnay and C. Liverani, Potentials on the two-torus for which the Hamiltonian flow is ergodicg, Comm. Math. Phys. 135 (1991), 267-302. Zbl0719.58022
87. [87] Hao Bai-Lin (ed.), Chaos IIg, World Scientific, 1990. 

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