Dynamical systems, graphs, and algorithms

Lecture Notes in Mathematics

Volumn 1889, Issue , 2006, Pages 1-283

  Osipenko, George ^a  

^a SEVASTOPOL STATE UNIVERSITY   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33750565405     PISSN: 00758434     EISSN: None     Source Type: Book Series    
DOI: None     Document Type: Review

References (153)

1. A.V. Aho and J.E. Hopcroft and J.D. Ullman, Data Structures and Algorithms, Addison-Wesley, 1987
2. G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, 1983
3. V.M. Alekseev, Quasi-random oscillations and qualitative problems of celestial mechanics. In: 9th Mathematical school, Kiev, 1972 (in Russian)
4. V.M. Alekseev, Symbolic Dynamics, 11th Mathematical School, Kiev, 1976 (in Russian)
5. A.A. Andronov and L.S. Pontryagin, Rough Systems, Doklady Academy Nauk SSSR, v.14, no.5, 247-250 (1937) (in Russian)
6. D. V. Anosov, Geodesic flow on closed Riemannian manifold of negative curvature, Trudy Math. Steclov Institute, v.90, 1967, 210 p., (in Russian)
7. N. Ampilova, K. Mot'kina, An algorithm for construction of the homotopic paths on a symbolic image, The Fourth International Conference "Tools for Mathematical Modelling", St. Petersburg, 2003, book of abstracts, p. 160
8. J. Argyris, G. Faust, and M. Haase, An Exploration of Chaos: An Introduction for Natural Scientists and Engineers, Elsevier Science Ltd., 1994
9. L. Auslender and R. MacKenzie, Introduction to Differentiate Manifolds, N.Y., 1963
10. V. Avrutin, R. Lammert, M. Schanz, G. Wackenhut, and G. Osipenko, On the software package AnT 4.669 for the investigation of dynamical systems, The Fourth International Conference "Tools for Mathematical Modelling", Mathematical Reaseach, v.9 (2003), 24-35, St. Petersburg State Polytechnic University, Russia
11. V. Avrutin and M. Schanz, Border-collision period-doubling scenario, Phys. Rev. E, v.70 (2004), 026222/1-11
12. N.P. Bhatia, G.P. Szego, Stability theory of dynamical systems, New York, Springer, 1970
13. G.D. Birkhoff, Nouvelles recherches sur les systemes dynamique, Memoriae Pont. Acad. Novi Lincaei v.3, no. 1(1935), 85-216
14. R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lectures Note in Mathematics, 470, Springer-Verlagh, N.Y., 1982
15. R. Bowen, Symbolic Dynamics, Ann. Math. Soc. Providence, R.I., vol.8, 1982
16. M. Brin, On incusion of diffeomorphism into flow, Izvestiya VUZov, no. 8(123), 1972, 19-25 (in Russian)
17. H. Broer, C. Simo, Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena, Bul. Soc. Bras. Mat., v.29 (1998), 253-293
18. I.U. Bronshtein, Theorem on structural stability of smooth extension of cascade, in Algebraic invariants of dynamical systems, Mat. Issledovaniya, v.67, Kishinev, Shtinisa, 12-29 (1980) (in Russian)
19. I.U. Bronshtein, Nonautonomous dynamical system, Kishinev, Shtinisa, 1984 (in Russian)
20. B.F. Bylov, R.E. Vinograd, D.M. Grobman, V.V. Nemytskii, Theory of Lyapunov exponents, Moscow, Nauka, 1966 (in Russian)
21. M.L. Cartwright and J.E. Littlewood, On non-linear differential equations of the second order, III, IV, Acta Math., v.97 (1957), no. 3-4, 267-308;
22. M.L. Cartwright and J.E. Littlewood, On non-linear differential equations of the second order, III, IV, v.98 (1957), no 1-2, 1-110
23. T. Cormen, C. Leiserson, and R. Rivest, Introduction to algorithms, MIT Press, 2000
24. P. Cvitanović, Periodic orbits as the skeleton of classical and quantum chaos, Physica D, 51, 1991
25. P. Cvitanović. Focus issue on periodic orbit theory. Chaos, 2, 1992
26. F. Colonius and W. Kliemann, The Morse spectrum of linear flows on vector bundles, Trans. Amer. Math. Soc., 348, 4355-4388 (1996)
27. F. Colonius and W. Kliemann, The Lyapunov spectrum of families of time varying matrices, report 504, Inst. of Math. Augsburg Univ., 1994
28. F. Colonius and W. Kliemann, The Dynamics of Control, Burkhauser, 2000
29. C. Conley, Isolated Invariant set and the Morse Index, CBMS Regional Conference Series, v.38, Amer. Math. Soc., Providence, 1978
30. B. Coomes, H. Kocak, K. Palmer, Periodic shadowing, chaotic numerics, Contemporary Mathematics, 172, 1994
31. B. Coomes, H. Kocak, K. Palmer, Computation of long period orbits in chaotic dynamical systems, Aust. Math. Soc. Gaz., v.24, no.5, 183-190 (1997)
32. M. Dellnitz, A. Hohmann, The computation of unstable manifolds using subdivision and continuation, in Nonlinear Dynamical Systems and Chaos, by Broer, H.W.; Gils, S.A. van; Hoveijn, I.; Takens, F. (eds.) PNLDE 19, Birkhauser, 449-459, 1996
33. M. Dellnitz, A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numerische Mathematik 75, 293-317, 1997
34. M. Dellnitz, O. Junge, An adaptive subdivision technique for the approximation of attractors and invariant measures, Corn-put, Visual. Sci. 1, 63-68, (1998)
35. M. Dellnitz, O. Junge, Almost invariant sets in Chuas circuit, Int. J. Bif. and Chaos, 7(11), 2475-2485, 1997
36. M. Dellnitz, G. Froyland, 0. Junge, The algorithms behind GAI0 - Set oriented numerical methods for dynamical systems, in Ergodic Theories, Analysis, and Efficient Simulation of Dynamical Systems, Springer, 2001, 145-175
37. M. Dellnitz and O. Junge, Set Oriented Numerical Methods for Dynamical Systems, in Handbook of Dynamical Systems II: Towards Applications, World Scientific, 2002, 221-264
38. R.L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley pub. com., 1990
39. P. Diamond, P. Kloeden, A. Pokrovskii, Cycles of spatial discretizations of shadowing dynamical systems, Math. Nachr., v.171 (1995), 95-110
40. W. Dijkstra, A Note on two problems in connection with graphs, Numerische Math., v.1 (1959), 269-271
41. M.J. Feigenbaum, The Universal Metric Properties of Nonlinear Transformations, J. Stat. Phys., v.21, no. 6 (1979), 669-707
42. M.J. Feigenbaum, Quantitative Universality for a Class of Nonlinear Transformations, J. Stat. Phys., v.19, no. 1 (1978), 25-53
43. J. Franke and J. Selgrade, Hyperbolicity and chain recurrence, J. Differential Equations, 26 (1977), 27-36
44. G. Froyland, O. Junge, G. Ochs, Rigorous computation of topological entropy with respect to finite partition, Web-article, URL: http://www.math. unipaderborn.de/agdellnitz/papers/topen15.ps.gz
45. D. Fundinger, T. Lindström, G. Osipenko, On the appearance of multiple attractors in discrete food-chains, submitted in Applied Mathematics and Computation, 2006
46. N.K. Gavrilov and L.P. Silnikov, On three-dimansional dynamical systems close to systems with a structurally unstable homoclinic curve, Math. USSR Sb., v.88 (1972), 467-485;
47. N.K. Gavrilov and L.P. Silnikov, On three-dimansional dynamical systems close to systems with a structurally unstable homoclinic curve, v.90 (1973), 139-156
48. G. Golub, C. Van Loan, Matrix computations, Mir, Moscow, 1999
49. A. Gragnami, O. De Feo, and S. Rinaldi, Food chains in the chemostat: Relationships between mean yield and complex dynamics, Bulletin of Math. Biology, v.60, 703-719, 1998
50. J. Guckenheimer, P. Holmes, Nonlinear Oscilations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, N.Y., 1983
51. B. Hao and W. Zheng, Applied Symbolic Dynamics and Chaos, World Scientific, Singapore, 1998
52. J. Hadamard, Les surfaces a courbures opposees et leurs lignes geodesiques, Journal de Matematiques Pure et Applique, v.4 (1898), 27-73
53. P. Hartman, Ordinary Differential Equations, N.Y., 1964
54. M.E. Henderson, Computing invariant manifolds by integrating fat trajectories, IBM Research, Technical Report RC22944, 2003
55. M. Henon, A two-dimensional mapping with strange attractor. Copmm. Math. Phys., 50, 1976, 69-77
56. M. Hirsch and S. Smale, Differential, Equations, Dynamical, Systems and Linear Algebra, N.Y. 1970
57. D. Hobson, An efficient method for computing invariant manifolds, J. Comput. Phys., v.104 (1991), 14-22
58. S.L. Hruska, Constructing an expanding metric for dynamical systems in one complex variable, Nonlinearity, v.18 (2005), 81-100
59. C.S. Hsu, Cell-to-Cell Mapping. Springer-Verlag, N.Y., 1987
60. F. Hunt, Unique ergodicity and the approximation of attractors and their invariant measures using Ulam's method, Nonlinearity, v.11, no. 2, 307-317 (1998)
61. M. Hurley, Chain recurrence and attraction in non-compact spaces, Ergodic Theory and Dynamical Systems, 11 (1991), 709-729
62. K. Ikeda, Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system, Opt. Commun., 30, 257-261, 1979
63. A. Jorba, and C. Simo, On quasiperiodic perturbation of elliptic equilibrium points, SIAM J. of Math. Anal., 27 (1996), 1704-1737
64. O. Junge, Rigorous discretization of subdivision techniques, in Proceedings of Equadiff '99, Berlin, 2000
65. T. Kaczynsky, K. Mishaikow, M. Mrozek, Computation homology, Applied mathematic sciences, 157, Springer, N-Y, 2004
66. K. Kaneko (editor), Theory and Applications of Coupled Map Lattices, Wiley, New York, 1993
67. S. Khryashchev, On local controllability along trajectories, VINITI, 868-B 97, 1997, 1-14
68. S.Yu. Kobjakov, D.Yu. Matiassevich, G.S. Osipenko, Location of the invariant sets. Proceedings of the Fourth International Conference Tools for Mathematical Modelling, Saint-Petersburg State Polytechnic University, 2003, 300-3007
69. A.N. Kolmogorov, S.W. Fomin, An elements of the theory of functions and the functional analysis, Science, Moscow, 1968 (in Russian)
70. A.N. Kolmogorov, New mesure invariant for transitive dynamical systems and endomorphisms of Lebesgue spaces, Doklady AN USSR, 115, no. 5 (1958), 861-864
71. B. Krauskopf, H.M. Osinga, Globalizing two-dimensional unstable manifolds of maps, Int. J. Bifurcation and Chaos, v.8, no. 3 (1998), 483-503
72. B. Krauskopf, H.M. Osinga, E.J. Doedel, M.E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz, O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Int. J. Bifurcation and Chaos, v.15, no. 3 (2005)
73. B.Yu. Levit and V.N. Livshits, Nonlinear Network Transport Problems Moscow, Transport, 1972 (in Russian)
74. D. Lind, B. Marcus, An introduction to symbolic dynamics and coding, New York, 1995
75. T. Lindström, On the dynamics of discrete food chains: Low- and high-frequency behavior and optimality of chaos, Journal of Mathematical Biology, v.45 (2002), 396-418
76. E.N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20:130-141, 1963
77. A.M. Lyapunov, Probleme General de la Stabilite an Mouvement, Kharkov, 1892
78. G.G. Malinetskii, A.V. Potapov, A.I. Rakhmatov, E.B. Rodichev, Limination of delay reconstruction for chaotic systems with board spectrum, Phys. Lett. A., v.179, 15, 1993
79. G.G. Malinetskii, A.V. Potapov, A.I. Rakhmatov, Limination of delay reconstruction for chaotic systems with board spectrum, Phys. Rev. E., v.48, 904-912, 1993
80. G.G. Malinetskii, A.V. Potapov, Modern problems in nonlinear dynamics, Moscow, 2000 (in Russian)
81. R. Mane, Characterizations of AS diffeomorphisms, Lect. Notes Math., v.597, 389-394 (1977)
82. 1 stability conjecture, Publ. Math., Inst. Hautes Etud. Sci., 66, 161-210 (1988)
83. D.Yu. Matiyasevich, Localization of Invariant Sets of Dynamical Systems, Journal of Mathematical Sciences, v.124 (2004), no. 3, 4990-5000
84. D.Yu. Matiyasevich, E.I. Petrenko, Algorithms for the construction of isolated invariant subsets of the symbolic image, Proceedimgs of XXXVI conference "Control Processes and Stability", St.Petersburg, 2005, 341-347
85. J.R. Miller and J.A. Yorke, Finding all periodic orbits of maps using newton methods: Sizes of Basins, Physica D, 135: 195-211, 2000
86. K. Mischaikow, Topological techniques for efficient rigorous computations in dynamics, Acta Numerica, 2002
87. D.A. Mizin, Algorithms based on applied symbolic dynamics, PhD dissertation, St. Petersburg State Polytechnic University, 2002 (in Russian)
88. D.A. Mizin, G. Osipenko, S. Kobyakov, The estimats for the topological entropy of the dynamic system, Proceedings of the third international conference "Tools for mathematical modelling", 2001, 85-105
89. A.A. Moiseev, Symbolic image od dynamical system and algorithms for investigation, Differential Equations and applications, abstracts of the International Conference, St. Petersburg, 1996, 152-153
90. R. Montgomery, The N-body problem, the briad group, and action-minimizing periodic solutions, Nonlinearity, v.11 (1998), 363-376
91. M. Morse, A one-to-one representation of geodesics on a, surface of negative curvature, Amer. J. Math., v.43 (1921), no. 1, p.33-51
92. M. Morse, G.A. Hedlund, Symbolic dynamics I, II Amer: J. Math., v.60 (1938), no. 4, p.815-866;
93. M. Morse, G.A. Hedlund, Symbolic dynamics I, II v.62 (1940), no 1, p. 1-42
94. S.E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, v.13, 1974, 9-18
95. Z. Nitecki, Differentiate Dynamics. London, 1971
96. Z. Nitecki and M. Shub, Filtrations, decompositions, and explosions, Amer. J. of Math., v.97 (1975), no.4, 1029-1047
97. H.E. Nusse, J.A. Yorke, Dynamics: Numerical Explorations, Springer-Verlag, 1997
98. G.S. Osipenko, On a symbolic image of dynamical system, in Boundary value problems, Perm, 1983, 101-105 (in Russian)
99. G.S. Osipenko, Verification of the transversality condition by the symbolic-dynamical methods, Differential Equations, v.26, N9, 1126-1132;
100. translated from Differentsial'nye Uravneniya, v.26, N9, 1528-1536, 1990
101. G.S. Osipenko, The periodic points and symbolic dynamics, in Seminar on Dynamical Systems. Euler International Mathematical Institute, St. Petersburg, Russia, October and November, 1991, Birkhauser Verlag, Basel, Prog. Nonlinear Differ. Equ. Appl. 12, 261-267 (1993)
102. G.S. Osipenko, Localization of the chain recurrent set by symbolic dynamics methods, Proceedings of Dynamics Systems and Applications, v.1, (1994), 227-282, Dynamic Publishers Inc
103. G. Osipenko and I. Komarchev, Applied symbolic dynamics: construction of periodic trajectories, WSSIA A, 4(1995), 573-587
104. G. Osipenko and I.Il'in, Methods of Applied Symbolic Dynamics, Proceedings of Dynamics Systems and Applications, v.2 (1996), 451-160
105. G. Osipenko, Indestructibility of invariant locally non-unique manifolds, Discrete and Continuous Dynamical Systems, v.2, no.2 (1996), 203-220
106. G. Osipenko, E. Ershov and J. Kim, Lectures on Invariant Manifolds of Perturbed Differential Equations and Linearization, St. Petersburg Polytech. University, 1996
107. G. Osipenko and Eu. Ershov, Perturbation of invariant manifolds of ordinary differential equations, in Six lectures on dynamical systems, editors B. Aulbach and F. Colonius, Wold Scientific, Singapore, 1996
108. G. Osipenko, Morse Spectrum of Dynamical Systems and Symbolic Dynamics, Proceedings of 15th IMACS World Congress, v.1 (1997), 25-30
109. G. Osipenko, Linearization near a locally non-unique invariant manifold, Discrete and Continuous Dynamical Systems, v.3, no.2 (1997), 189-205
110. G. Osipenko and S. Campbell, Applied Symbolic Dynamics: Attractors and Filtrations, Discrete and Continuous Dynamical Systems, v.5, no.1&2, 43-60 (1999)
111. G. Osipenko, Spectrum of a Dynamical System and Applied Symbolic Dynamics, Journal of Mathematical Analysis and Applications, v.252, no. 2, 587-616 (2000)
112. G.S. Osipenko, J.V. Romanovsky, N.B. Ampilova, E.I. Petrenko, Computation of the Morse Spectrum, Journal of Mathematical Sciences, v.120, no. 2 (2004), 1155-1166
113. G.S. Osipenko, Lectures on symbolic analysis of dynamical systems, St. Petersburg State Polytechnic University, 2004
114. J. Palis, J. Melo, Geometric Theory of Dynamical Systems, Springer-Verlag, New York (1982)
115. P. Pilarczyk, Computer assisted method for proving existence of periodic orbits. TMNA, 13(2): 365-377, 1999
116. P. Pilarczyk, Homology software, in Computational homology program, http:www.gatech.edu/chom
117. 0- Topology, Springer-Verlag, 1994, Lee. Notes in Math., 1571, 180p
118. S. Pissanetzky Sparse Matrix Technology, New York, Academic Press, 1984
119. V.A. Pliss, Integral Sets of Periodic System of Differential Equations, Moscow, 1977 (in Russian)
120. H. Poincare, Sur le probleme des trois corps et les equations de la, dynamique, Acta Mathematica v.13 (1890), 1-271
121. H. Poincare, Les methodes nouvelles de la mecanique celeste, I-III (1892-1899), Gauthiers-Villars
122. J. Robbin, A structural stability theorem, Annals of Math., v.94, no.3, 447-493 (1971)
123. 0-diffeomorphism, J. Diff. Equat., v.22, no.1, 28-73 (1976)
124. C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos, 1995
125. Yu.V. Romanovsky, Optimization and stationary control of discrete deterministic, process in dynamic programming Kibernetika 2, 66-78 (in Russian);
126. Engl. transl. Cybernetics 3, 1967
127. M.L. Rosenzweig, Exploitation in three throphic levels, The American Naturalist, 107 (954), 275-294, 1973
128. R. Sacker and G. Sell, Existence of dichotomies and invariant splitting for linear differential systems I-III, J. Diff. Eg. v.15, no3 (1974), 429-458,
129. 22, no.2 (1976) 476-522
130. R. Sacker and G. Sell, A spectral theory for liner differential systems, J. Diff. Eg., v.27, no. 3, (1978), 320-358
131. D. Salamon and E. Zehnder, Flows on vector bundles and hyperbolic sets, Trans. Amer. Math. Soc., v.306, no. 2 (1988), 623-649
132. J. Selgrade, Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc., v.203 (1975), 359-390
133. G. Sell, Nonautonomous differential equations and topological dynamics, Trans. AMS, 127 (1967), 241-283
134. R. Sedgewick, Algorithms in Modula 3, Addison-Wesley, Massachusetts, 1993
135. A.N. Sharkovsky, Structure theory of differentiable dynamical systems and weak nonwandering points, Abh. Akad. Wiss. DDR. Abt. Math. Naturwiss. Techn., v.4 (1977), 193-200
136. A.N. Sharkovsky, Yu.L. Maistrenko, E.Yu. Romanenko, Diffenrence equations and applications, Naukova dumka, Kyev, 1986 (in Russian)
137. M. Shub, Stabilite globale de systems denamiques, Asterisque, v.56 (1978), 1-21
138. L.P. Silnikov, A case of the existence of a denumerable set of periodic motions, Sov. Math. Dokl., v.6 (1965), 163-166
139. L.P. Silnikov, A contridution to the problem of the structure of an extended neighborhood of a, rough state of saddle-focus type, Math. USSR Sb., v.10 (1970), 91-102
140. C. Simo, Effective computations in hamiltonian dynamics, in Cent ans apes les Methodes Nouvelles de H. Poincare, 1-23, 1996
141. C. Simo, Invariant curves of analytic perturbed nontwist area preserving maps, Regular and Chaotic Dynamics, v.3 (1998), no. 3, 180-195
142. S. Smale, Diffeomorphisms with many periodic points, in Differential and Combinatorial Topology, Princeton Univ., 1965, 63-80
143. S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73, 1967
144. S. Smale, The Ω- stability theorem, in Global Analysis, Proc. Symp. in Pure Math., v.XlV, Amer. Math. Soc. 1970
145. C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer, N.Y., 1973
146. H.J. Sussmann, Some properties of vector fields systems witch are not altered by small perturbations, J. Differential Equations, v.20 (1976), no. 2, 292-315
147. R. Tarjan, Depth-First Search and Linear Graph Algorithms, SIAM J. Comput, v.1 (1972), 146-160
148. R. Tarjan, Data structures and network algorithms, Philadelphia, Pa., 1991
149. H. Whitney, Diferentiable Manifolds, Ann. Math., 37 (1936), pp. 645-680
150. S. Wiggins, Introduction to Applied non linear Dynamical Systems and Chaos, 1990
151. A. Wolf, J. Swift, H. Swinney and J. Vastano, Determining Lyapunov exponents from a time series, Physica D. 16 (1985), 285-317
152. Z. You, E.J. Kostelich, J.A.Yorke, Calculating stable and unstable manifolds, Internat. J. Bifur. Chaos Appl. Sci. Engrg., v.1, no. 3, (1991), 605-623
153. Zh.T. Zhusubaliyev and E. Mosekilde, Bifurcations and chaos in piecewise-smooth dynamical systems, v.44, Nonlinear Science A, New Jersey, 2003