Chaos and control in dynamical systems

Computational Mathematics and Modeling

Volumn 12, Issue 4, 2001, Pages 314-352

  Loskutov, A Y ^a,b  

^a MARTIN LUTHER UNIVERSITY HALLE WITTENBERG   (Germany)

^b Gwangju Institute of Science and Technology (GIST)   (South Korea)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 3342899646     PISSN: 1046283X     EISSN: 1573837X     Source Type: Journal    
DOI: 10.1023/A:1012510914172     Document Type: Review

References (268)

1. H. Poincaré, Calcul des Probabilites, Gauthier-Villars, Paris (1912).
2. L. Boltzmann, "Über die mechanischen Analogien des zweiten Hauptsatzes der Thermodynamik," Journ. f. Mathem., 100, 201-212 (1887).
3. L. Boltzmann, Vorlesungen über Gastheorie, Leipzig (1896).
4. L. Boltzmann, Papers and Speeches [Russian translation], Nauka, Moscow (1970).
5. P. Ehrenfest and T. Ehrenfest, Enzyklopaedie d. Math., Vol. IV, part 32, Leipzig (1911).
6. P. Ehrenfest, Collected Papers [Russian translation], Nauka, Moscow (1972).
7. M. Kac, Probability and Related Issues in Physics [Russian translation], Mir, Moscow (1965).
8. E. G. Cohen and W. Thirring (eds.), The Boltzmann Equation: Theory and Application, Springer, Berlin (1973).
9. E. Fermi, J. Pasta, and S. Ulam, Studies of Nonlinear Problems, Los Alamos Scientific Report LA-1940 (1955).
10. J. Ford, "Equipartition of energy for nonlinear systems," J. Math. Phys., 2, No. 3, 387-393 (1961).
11. E. A. Jackson, "Nonlinear coupled oscillators. Perturbation theory: ergodic problem," J. Math. Phys., 4, No. 4, 551-558 (1963).
12. H. Poincaré, Collected Works [Russian translation], Vol. 1, Nauka, Moscow (1973).
13. G. D. Birkhoff, Dynamical Systems, AMS, New York (1927).
14. N. S. Krylov, Studies in Statistical Physics [in Russian], Izd. AN SSSR, Moscow-Leningrad (1950).
15. M. Born, quot;Is prediction possible in classical mechanics?quot; Usp. Fiz. Nauk, 62, No. 2, 173-187 (1959).
16. J. Ford, quot;Foreword to Symbolic Dynamics and Hyperbolic Dynamic Systems by V. M. Alekseev and M. V. Yakobson, quot; Phys. Rep., 75, No. 5, 288-289 (1981).
17. A. N. Kolmogorov, quot;A new metric invariant for transitive dynamical systems and Lebesgue space automorphisms, quot; Dokl. Akad. Nauk SSSR. 119, No. 5, 861-864 (1958).
18. A. N. Kolmogorov, quot;Entropy per unit time as a metric invariant of automorphisms, quot; Dokl. Akad. Nauk SSSR, 124, No. 4, 754-755 (1959).
19. Ya. G. Sinai, quot;The concept of entropy for a dynamical system, quot; Dokl. Akad. Nauk SSSR, 124, No. 4, 768-771 (1959).
20. S. Smale, quot;Diffeomorphisms with many periodic points, quot; in: S. S. Cairns, ed., Differential and Combinatorial Topology, Princeton Univ. Press (1965), pp. 63-80.
21. S. Smale, quot;Differentiable dynamical systems, quot; Usp. Mat. Nauk, 25, No. 1, 113-185 (1970).
22. D. V. Anosov, quot;Robustness of geodesic flows on compact Riemannian manifolds of negative curvature, quot; Dokl. Akad. Nauk SSSR, 145, No. 4, 707-709;
23. quot;Ergodic properties of geodesic flows on closed manifolds of negative curvature, quot; Dokl. Akad. Nauk SSSR, 151, No. 6, 1250-1252 (1963).
24. D. V. Anosov, Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature [in Russian], Nauka, Moscow (1967).
25. R. Bowen, Methods of Symbolic Dynamics [Russian translation], Mir, Moscow (1979).
26. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, Cambridge (1995).
27. Z. Nitecky, An Introduction to Differential Dynamics [Russian translation], Mir, Moscow (1975).
28. A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics, Springer, Berlin (1994).
29. Ya. G. Sinai, quot;Substantiation of the ergodic conjecture for one dynamical system in statistical mechanics, quot; Dokl. Akad. Nauk SSSR, 153, No. 6, 1261-1264 (1963).
30. Ya. G. Sinai, quot;A physical system with positive entropy, quot; Vestnik MGU, Mathem.-Mech., 5, 6-12 (1963).
31. L. A. Bunimovich and Ya. G. Sinai, quot;Statistical properties of Lorentz gas with periodic configuration of scatters, quot; Commun. Math. Phys., 78, No. 4, 479-497 (1981).
32. Dynamical Systems, Vol. 2, Series: Current Issues in Mathematics. Basic Directions [in Russian], VINITI (1985).
33. L. A. Bunimovich, quot;Conditions of stochasticity for two-dimensional billiards, quot; Chaos, 1, No. 2, 187-193 (1991).
34. A. A. Tabachnikov, Billiards, France Math. Soc. Press (1995).
35. Proceedings of the SPIE 1993 Annual Meeting on Chaos in Communications, San Diego, CA, July 11-16, 1993, Vol. 2038.
36. A. Yu. Loskutov and V. M. Tereshko, quot;Extraction of the prototypes encoded in a chaotic attractor, quot; in: I. Alexander and J. Taylor (eds.), Artificial Neural Networks, Elsevier, North Holland (1992), pp. 449-452.
37. A. Yu. Loskutov and V. M. Tereshko, quot;Processing information encoded in chaotic sets of dynamical systems, quot; SPIE, 2038, 263-272 (1993).
38. S. Hayes, C. Grebogi, and E. Ott, quot;Communicating with chaos, quot; Phys. Rev. Lett., 70, No. 20, 3031-3034 (1993).
39. S. Hayes, C. Grebogi, E. Ott, and A. Mark, quot;Experimental control of chaos for communication, quot; Phys. Rev. Lett., 73, No. 13, 1781-1784 (1994).
40. H. D. I. Abarbanel and P. S. Linsay, quot;Secure communications and unstable periodic orbits of strange attractors, quot; IEEE Trans. Circuits Systems., 40, No. 10, 643-645 (1993).
41. Physica D, 84, Nos. 1, 2 (1995).
42. A. Yu. Loskutov and G. E. Thomas, quot;On a possible mechanism of self-organization in a two-dimensional network of coupled quadratic maps, quot; SPIE, 2037, 238-249 (1993).
43. A. Yu. Loskutov, V. M. Tereshko, and K. A. Vasiliev, quot;Predicted dynamics for cyclic cascades of chaotic deterministic automata, quot; Int. J. Neural Syst., 6, 175-182 (1995).
44. L. Glass, quot;Cardiac arrhythmias and circle maps: A classical problem, quot; Chaos, 1, No. 1, 13-19 (1991).
45. A. Garfinkel, M. L. Spano, and W. L. Ditto, quot;Controlling cardiac chaos, quot; Science, 257, 1230-1235 (1992).
46. A. Yu. Loskutov and S. D. Rybalko, quot;Circle map dynamics with a parametric perturbation, quot; Vestnik MGU, Phys.-Astron., 34, No. 4, 19-27 (1993).
47. A. Yu. Loskutov, quot;Nonlinear dynamics and cardiac arrhythmia, quot; Prikladnaya Nelineinaya Dinamika, 2, No. 3-4, 14-25 (1994).
48. L. Bresler, G. Metcalfe, J. M. Ottino, and T. Shinbrot, "Isolated mixing regions: origin, robustness, and control," Chem. Eng. Sci., 58, 1671-1679 (1996).
49. T. Shinbrot and J. M. Ottino, "A geometric method to create coherent structures in chaotic flows," Phys. Rev. Lett., 71, 843-847 (1993).
50. A. Yu. Loskutov and G. E. Thomas, "Chaos and destochastization in a two-dimensional lattice of coupled maps," Vestnik MGU, Phys.-Astron., 34, No. 5, 3-11 (1993).
51. R. Lima and M. Pettini, "Suppression of chaos by resonant parametric perturbations," Phys. Rev. A, 41, No. 2, 726-733 (1990).
52. J. Singer, Y.-Z. Wang, and H. H. Bau, "Controlling a chaotic system," Phys. Rev. Lett., 66, 1123-1125 (1991).
53. L. Fronzoni, M. Geocondo, and M. Pettini, "Experimental evidence of suppression of chaos by resonant parametric perturbations," Phys. Rev. A, 43, 6483-6487 (1991).
54. Y. Braiman and I. Goldhirsch, "Taming chaotic dynamics with weak periodic perturbations," Phys. Rev. Lett. 66, 2545-2548 (1991).
55. S. Rajasekar and M. Lakshmanan, "Algorithms for controlling chaotic motion: application for the BVP oscillator," Physica D, 67, No. 1-3, 282-300 (1993).
56. S. Bielawski, D. Derozier, and P. Glorieux, "Controlling unstable periodic orbits by a delayed continuous feedback," Phys. Rev. E, 49, No. 2, 971-974 (1994).
57. Ph. V. Bayly and L. N. Virgin, "Practical considerations in the control of chaos," Phys. Rev. E, 50, No. 1, 604-607 (1994).
58. D. Vassiliadis, "Parametric adaptive control and parametric identification of low-dimensional chaotic systems," Physica D, 71, No. 1-2, 319-341 (1994).
59. B. Hübinger, R. Doerner, and W. Martienssen, "Controlling chaos experimentally in systems exhibiting large effective Lyapunov exponents," Phys. Rev. E, 50, No. 2, 932-948 (1994).
60. R. Mettini and T. Kurz, "Optimized periodic control of chaotic systems," Phys. Lett. A, 206, No. 5-6, 331-339 (1995).
61. R. Chacon, "Suppression of chaos by selective resonant parametric perturbations," Phys. Rev. E, 51, No. 1, 761-764 (1995).
62. T. Shinbrot, "Chaos: Unpredictable yet controllable?" Nonlinear Science Today, 3, No. 2, 1-8 (1993).
63. T. Shinbrot, C. Grebogi, E. Ott, and J. A. Yorke, "Using small perturbations to control chaos," Nature, 363, 411-417 (1993).
64. V. V. Alekseev and A. Yu. Loskutov, "Destochastization of a system with a strange attractor by parametric perturbation," Vestnik MGU, Phys.-Astron., 26, No. 3, 40-14 (1985).
65. V. V. Alekseev and A. Yu. Loskutov, "Controlling a system with a strange attractor by periodic parametric perturbation," Dokl. Akad. Nauk SSSR, 293, No. 6, 1346-1348 (1987).
66. A. Yu. Loskutov and A. I. Shishmarev, "A property of the family of quadratic maps in the presence of parametric perturbation," Usp. Mat. Nauk, 48, No. 1, 160-170 (1993).
67. A. Yu. Loskutov and A. I. Shishmarev, "Control of dynamical systems behavior by parametric perturbations: an analytic approach," Chaos, 4, No. 2, 351-355 (1994).
68. M. Pettini, "Controlling chaos through parametric excitations," in: R. Lima, L. Streit, and R. Vilela Mendes (eds.), Dynamics and Stochastic Processes, Springer, Berlin (1990), pp. 242-250.
69. Yu. S. Kivshar, B. Rödelsperger, and H. Benner, "Suppression of chaos by nonresonant parametric perturbations," Phys. Rev. E, 49, 319-324 (1994).
70. A. B. Corbet, "Suppression of chaos in ID maps," Phys. Lett. A, 130, No. 4-5, 267-270 (1988).
71. A. Hübler, R. Georgii, M. Kuckler, W. Stelzl, and E. Lüscher, "Resonant stimulation of nonlinear damped oscillators by Poincaré maps," Helv. Phys. Acta, 61, 897-900 (1988).
72. A. Hübler, "Adaptive control of chaotic systems," Helv. Phys. Acta, 62, 343-346 (1989).
73. E. Lüscher and A. Hübler, "Resonant stimulations of complex systems," Helv. Phys. Acta, 62, 544-551 (1989).
74. E. A. Jackson and A. Hübler, "Periodic entrainment of chaotic logistic map dynamics," Physica D, 44, 407-420 (1990).
75. B. A. Huberman and E. Lumer, "Dynamics of adaptive systems," IEEE Trans. Circ. Syst., 37, 547-550 (1990).
76. K. Pyragas, "Stabilization of unstable periodic and aperiodic obits of chaotic systems by self-controlling feedback," Z. Naturforsch. A, 48, 629-632 (1993).
77. G. I. Dykman, P. S. Landa, and Yu. I. Neimark, "Synchronization of the chaotic oscillations by external force," Chaos, Solitons, and Fractals, 1, No. 4, 339-353.
78. Yu. I. Neimark and P. S. Landa, Stochastic and Chaotic Oscillations [in Russian], Nauka, Moscow (1987).
79. E. Ott, C. Grebogi, and J. A. Yorke, "Controlling chaos," Phys. Rev. Lett., 64, 1196-1199 (1990).
80. F. J. Romeiras, E. Ott, C. Grebogi, and W. P. Dayawansa, "Controlling chaotic dynamical systems," Physica D, 58, 165-192 (1992).
81. T. Shinbrot, E. Ott, C. Grebogi, and J. A. Yorke, "Using chaos to direct trajectories to targets," Phys. Rev. Lett., 65, 3215-3218 (1990).
82. T. Shinbrot, C. Grebogi, E. Ott, and J. A. Yorke, "Using chaos to target stationary states of flows," Phys. Lett. A, 169, 349-354 (1992).
83. E. Kostelich, C. Grebogi, E. Ott, and J. A. Yorke, "Higher dimensional targeting," Phys. Rev. E, 47, 305-310 (1993).
84. R. Meucci, W. Gadomski, M. Ciofini, and F. T. Arecchi, "Experimental control of chaos by weak parametric perturbations," Phys. Rev. E, 49, No. 4, 2528-2531 (1994).
85. J. Milnor, "On the concept of attractor," Commun. Math. Phys., 99, No. 2, 177-196 (1985).
86. V. S. Afraimovich, "On attractors," in: A. V. Gaponov-Grekhov and M. I. Rabinovich (eds.), Nonlinear Waves. Dynamics and Evolution [in Russian], Nauka, Moscow (1989), pp. 16-29.
87. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 3rd printing, Springer, Berlin (1990).
88. V. I. Arnol'd, V. S. Afraimovich, Yu. S. Il'yashenko, and L. P. Shil'nikov, "Bifurcation theory," in: Modern Issues in Mathematics. Basic Directions [in Russian], Vol. 5, VINITI, Moscow (1986), pp. 5-218.
89. N. N. Bautin and E. A. Leontovich, Methods and Techniques for Qualitative Investigation of Dynamical Systems on a Plane [in Russian], Nauka, Moscow (1990).
90. J. Marsden and M. McCracken, Hopf Bifurcation and Its Application [Russian translation], Mir, Moscow (1980).
91. B. Hassard, N. Kazarinov, and I. Ven, Theory and Application of Hopf Bifurcation [Russian translation], Mir, Moscow (1985).
92. V. I. Arnol'd, Supplementary Chapters in the Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (1978).
93. A. Yu. Loskutov and A. S. Mikhailov, Introduction to Synergetics [in Russian], Nauka, Moscow (1990).
94. J. P. LaSalle and S. Lefschetz, Stability by Lyapunov's Direct Method, Academic Press, New York (1961).
95. L. G. Khazin and E. E. Shnol, Stability of Critical Equilibria [in Russian], Izd. AN SSSR, Pushchino (1958).
96. E. Jury, Innors and Dynamical System Stability [Russian translation], Nauka, Moscow (1979).
97. N. N. Bautin, Behavior of Dynamical Systems Near the Stability Boundary [in Russian], Nauka, Moscow (1984).
98. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Bifurcation Theory for Dynamical Systems in a Plane [in Russian], Nauka, Moscow (1967).
99. V. S. Afraimovich, "Internal bifurcations and attractor crises," in: A. V. Gaponov-Grekhov and M. I. Rabinovich (eds.), Nonlinear Waves. Structures and Bifurcations [in Russian], Nauka, Moscow (1987), pp. 189-213.
100. M. J. Feigenbaum, "Quantitative universality for a class of nonlinear transformations," J. Stat. Phys., 19, 25-52 (1978).
101. M. J. Feigenbaum, "Universal metric properties of nonlinear transformations," J. Stat. Phys., 21, 669-706 (1979).
102. Ya. G. Sinai, Current Issues in Ergodic Theory [in Russian], Nauka, Moscow (1995).
103. E. B. Vul, Ya. G. Sinai, and K. M. Khanin, "Feigenbaum universality and thermodynamic formalism," Usp. Mat. Nauk, 39, No. 3(237), 3-37 (1984).
104. P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhauser, Boston (1980).
105. O. E. Lanford III, "A computer-assisted proof of the Feigenbaum conjecture," Bull. AMS, 6, 427-434 (1982).
106. F. Moon, Chaotic Oscillations [Russian translation], Mir, Moscow (1990).
107. M. S. El Naschie, Stress, Stability, and Chaos in Structural Engineering: An Energy Approach, McGraw-Hill, London (1990).
108. E. A. Jackson, Perspectives of Nonlinear Dynamics, Vols. I, II, Cambridge Univ. Press, Cambridge (1989, 1990).
109. Hao Bai-Lin (ed.), Chaos, World Sci. (1990).
110. P. Manneville, Dissipative Structures and Weak Turbulence, Academic Press, London (1990).
111. n," J. Stat. Phys., 25, 1-14(1980).
112. M. J. Feigenbaum, "The onset spectrum of turbulence," Phys. Lett. A, 74, 375-378.
113. G. Schuster, Deterministic Chaos: An Introduction [Russian translation], Mir, Moscow (1988).
114. A. Lichtenberg and M. Lieberman, Regular and Stochastic Dynamics [Russian translation], Mir, Moscow (1984).
115. M. Feigenbaum, "Universality and nonlinear system behavior," Usp. Fiz. Nauk, 141, No. 2, 343-374 (1983).
116. J. Crutchfield, V. Nauenberg, and J. Rudnick, "Scaling for external noise at the onset of chaos," Phys. Rev. Lett., 46, No. 14, 933-935 (1981).
117. M. J. Feigenbaum and B. Hasslacher, "Irrational decimations and path integrals for external noise," Phys. Rev. Lett., 49, No. 9, 605-609 (1982).
118. J.-P. Eckmann, "Roads to turbulence in dissipative dynamical systems," Rev. Mod. Phys., 53, No. 4, Part 1, 643-654 (1981).
119. Y. Pomeau and P. Manneville, "Intermittent transition to turbulence in dissipative dynamical systems," Commun. Math. Phys., 4, No. 47, 189-197 (1980).
120. P. Manneville and Y. Pomeau, "Different ways to turbulence in dissipative dynamical systems," Physica D, 1, No. 2, 219-226 (1980).
121. P. Berge, Y. Pomeau, and C. Vidal, Order in Chaos [Russian translation], Mir, Moscow (1991).
122. V. S. Afraimovich and L. P. Shil'nikov, "On some global bifurcations associated with the disappearance of a saddle-node fixed point," Dokl. Akad. Nauk SSSR, 219, No. 3, 1281-1285 (1974).
123. V. I. Luk'yanov and L. P. Shil'nikov, "On some bifurcations of dynamical systems with homoclinic structures," Dokl. Akad. Nauk SSSR, 243, No. 1, 26-29 (1978).
124. A. Yu. Loskutov and A. S. Mikhailov, Complex Patterns, Springer, Berlin (1991).
125. B. Hu, "Functional renormalization-group equations approach to the transition to chaos," in: Y. Kuramoto (ed.), Chaos and Statistical Methods, Springer, Berlin (1983), pp. 72-82.
126. B. Hu and J. Rudnick, "Exact solutions to the Feigenbaum renormalization-group equation for intermittency," Phys. Rev. Lett., 48, No. 24, 1645-1648 (1982).
127. B. Hu, "Introduction to real-space renormalization-group methods in critical and chaotic phenomena," Phys. Rep., 91, No. 5, 233-295 (1982).
128. P. Manneville, "Intermittency, self-similarity, and 1/f-spectrum in dissipative dynamical systems," J. de Phys., 41, No. 11, 1235-1243 (1980).
129. I. Procaccia and H. G. Schuster, "Functional renormalization group theory of universal 1/f-noise in dynamical systems," Phys. Rev. A, 28, No. 2, 1210-1212 (1983).
130. V. S. Afraimovich and L. P. Shil'nikov, "Invariant two-dimensional tori, their collapse and stochasticity," in: Methods of Qualitative Theory of Differential Equations [in Russian], Gor'kii (1983), pp. 3-26.
131. K. Kaneko, Collapse of Tori and Genesis of Chaos in Dissipative Systems, World Science, Singapore (1986).
132. D. G. Aronson, M. A. Chory, G. R. Hall, and R. P. McGehee, "A discrete dynamical system with subtly wild behavior," in: New Approach to Nonlinear Problems in Dynamics, SIAM, Philadelphia (1980), pp. 339-360.
133. 2," Lecture Notes in Mathematics, Vol. 470, Springer, Berlin (1978), pp. 48-66.
134. V. S. Anishchenko, Complex Oscillations in Simple Systems [in Russian], Nauka, Moscow (1990).
135. J. Belair and L. Glass, "Universality and self-similarity in the bifurcations of circle maps," Physica D, 16, 143-145 (1985).
136. S. Newhouse, J. Palis, and F. Takens, "Bifurcations and stability of families of diffeomorphisms," Publ. Math. IHES, No. 57, 5-72 (1983).
137. K. Kaneko, "Supercritical behavior of disordered orbits of a circle map," Progr. Theor. Phys., 73, No. 6, 1089-1103 (1984).
138. P. L. Boyland, "Bifurcations of circle maps: Arnol'd tongues, bistability, and rotation intervals," Commun. Math. Phys., 106, 353-381 (1986).
139. M. J. Feigenbaum, L. P. Kadanoff, and S. J. Shenker, " Quasiperiodicity in dissipative systems: a renormalization group analysis," Physica D, 5, 370-386 (1982).
140. S. J. Shenker, "Scaling behavior in a map of a circle onto itself: empirical results," Physica D, 5, 405-411 (1982).
141. A. N. Sharkovskii, "On isomorphism of dynamical systems," in: Proceedings of the 5th Int. Conf. On Nonlinear Oscillations [in Russian], Vol. 2, Naukova Dumka, Kiev (1970), pp. 541-545.
142. L. Block, "Homoclinic points of mappings of the interval," Proc. AMS, 72, 576-580 (1978).
143. A. N. Sharkovskii, Yu. L. Maistrenko, and E. Yu. Romanenko, Difference Equations and Applications [in Russian], Naukova Dumka, Kiev (1986).
144. J. Palis and F. Takens, "Hyperbolicity and creation of homoclinic orbits," Ann. Math., 125, 337-374 (1987).
145. J. Palis and F. Takens, Hyperbolicity and Sensitive-Chaotic Dynamics at Homoclinic Bifurcations, Cambridge Univ. Press, Cambridge (1993).
146. L. Mora and M. Viana, "Abundance of strange attractors," Acta Math., 171, 1-71.
147. S. E. Newhouse, "The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms," Publ. Math. IHES, 50, 101-151 (1979).
148. S. E. Newhouse, "Lectures on dynamical systems," Progress in Mathematics, Birkhauser, Boston, No. 8, 1-114 (1978).
149. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, Berlin (1990).
150. P. J. Holmes and F. C. Moon, "Strange attractors and chaos in nonlinear mechanics," Trans. ASME, Ser. E, 50, No. 4, 1021-1032 (1983).
151. V. K. Mel'nikov, "Stability of a center for time-periodic perturbations," Trudy Mosk. Matem. Obshch., 12, 3-52 (1963).
152. J. A. Yorke and K. A. Alligopod, "Cascades of period doubling bifurcations: a prerequisite for horseshoes," Bull. AMS, 9, 319-322 (1983).
153. M. Viana, "Chaotic dynamical behavior," in: Proceedings of the 9th Int. Congress of Math. Phys. (Paris, 1993), International Press, Cambridge, MA (1995), pp. 1142-1154.
154. C. Robinson, "Bifurcation to infinitely many sinks," Commun. Math. Phys., 90, 433-459 (1983).
155. M. Viana, "Strange attractors in higher dimensions," Bull. Braz. Math. Soc., 24, 13-62 (1993).
156. N. Romero, Persistence of Homoclinic Tangencies in Higher Dimensions, Thesis, IMPA (1992).
157. J. Palis and M. Viana, "High dimension diffeomorphisms displaying infinitely many periodic attractors," Ann. Math., 140, 207-250 (1994).
158. L. P. Shil'nikov, "A case of existence of a countable set of periodic motions," Dokl. Akad. Nauk SSSR, 160, No. 3, 558-561 (1965).
159. L. Perko, Differential Equations and Dynamical Systems, Springer, Berlin (1996).
160. B. F. Bylov, R. E. Vinograd, D. M. Grobman, and V. V. Nemytskii, Lyapunov Exponent Theory and Its Applications to Stability Questions [in Russian], Nauka, Moscow (1966).
161. J.-P. Eckmann and D. Ruelle, "Ergodic theory of chaos and strange attractors," Rev. Mod. Phys., 57, No. 3, Part 1, 617-656 (1985).
162. T. S. Akhromeeva, S. P. Kurdyumov, G. G. Malinetskii, and A. A. Samarskii, Nonstationary Structures and Diffusion Chaos [in Russian], Nauka, Moscow (1992).
163. Lyapunov Exponents, Lecture Notes in Mathematics, Vol. 1186, Springer, Berlin (198).
164. N. Martin and J. England, Mathematical Entropy Theory [Russian translation], Mir, Moscow (1988).
165. V. S. Afraimovich and A. M. Reiman, "Dimension and entropy in multidimensional systems," in: A. V. Gaponov-Grekhov and I. M. Rabinovich (eds.), Nonlinear Waves. Dynamics and Evolution [in Russian], Nauka, Moscow (1989), pp. 238-262.
166. I. P. Kornfel'd, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory [in Russian], Nauka, Moscow (1980).
167. Ya. G. Sinai, "Stochasticity of dynamical systems," in: A. V. Gaponov-Grekhov (ed.), Nonlinear Waves [in Russian], Nauka, Moscow (1979), pp. 192-212.
168. D. Ornstein, Ergodic Theory, Randomness in Dynamical Systems [Russian translation], Mir, Moscow (1978).
169. P. Shields, The Theory of Bernoulli Shifts, Univ. Of Chicago Press, Chicago and London (1973).
170. Ya. G. Sinai, "Finite-dimensional randomness," Usp. Mat. Nauk, 46, No. 3, 147-159 (1991).
171. W. Feller, Introduction to Probability Theory and Its Applications [Russian translation], Vols. 1, 2, Mir, Moscow (1984).
172. G. M. Zaslavskii, Stochasticity of Dynamical Systems [in Russian], Nauka, Moscow (1984).
173. V. I. Oseledets, "Multiplicative ergodic theorem. Characteristic Lyapunov exponents of dynamical systems," Trudy Mosk. Matem. Obshch., 19, 179-210 (1968).
174. V. M. Millionshchikov, "Stability criterion for the probability spectrum of linear systems of differential equations with recursive coefficients and criterion of almost reducibility of systems with almost periodic coefficients," Matem. Sb., 78, No. 2, 179-201 (1969).
175. Ya. B. Pesin, "Characteristic Lyapunov exponents and smooth ergodic theory," Usp. Mat. Nauk, 32, No. 4, 55-111 (1977).
176. I. P. Kornfel'd and Ya. G. Sinai, "Basic concepts and examples in ergodic theory," in: Dynamical Systems [in Russian], Vol. 2, VINITI, Moscow (1985), pp. 7-35.
177. V. M. Alekseev and M. V. Yakobson, "Symbolic dynamics and hyperbolic dynamic systems," Phys. Rep., 75, No. 5, 287-325 (1981).
178. A. N. Kolmogorov and V. M. Tikhomirov, "ε-Entropy and ε-capacity of sets in functional spaces," Usp. Mat. Nauk, 14, No. 2, 3-86 (1959).
179. P. Billingsley, Ergodic Theory and Information [Russian translation], Mir, Moscow (1969).
180. P. Grassberger and I. Procaccia, "Measuring the strangeness of strange attractors," Physica D, 9, No. 1-2, 189-208 (1983).
181. J. Farmer, E. Ott, and J. A. Yorke, "The dimension of chaotic attractors," Physica D, 7, No. 1-3, 153-180 (1983).
182. H. G. E. Hentschel and I. Procaccia, "The infinite number of generalized dimensions of fractals and strange attractors," Physica D, 8, No. 3, 435-444 (1983).
183. G. Paladin and A. Vulpiani, "Anomalous scaling laws in multifractal objects," Phys. Rep., 156, No. 4, 147-225 (1987).
184. L.-S. Young, "Capacity of attractors," Ergod. Theory Dyn. Sys., 1, No. 3, 381-388 (1981).
185. F. Ledrappier, "Some relations between dimension and Lyapunov exponents," Commun. Math. Phys., 81, No. 2, 229-238 (1981).
186. L.-S. Young, "Dimension, entropy, and Lyapunov exponents," Ergod. Theory Dyn. Sys., 2, No. 1, 109-124 (1982).
187. Ya. B. Pesin, "On the relation of the dimension with respect to a dynamical system," Ergod. Theory Dyn. Sys., 4, No. 3, 405-420 (1984).
188. B. B. Mandelbrot, Fractals: Form, Chance, and Dimension, Freeman, San Francisco (1977).
189. L. Pietronero, E. Tosatti, Ya. G. Sinai, and I. M. Khalatnikov (eds.), Fractals in Physics [Russian translation], Mir, Moscow (1988).
190. F. Takens, "Detecting strange attractors in turbulence," Lecture Notes in Mathematics, Vol. 898, Springer, Berlin (1980), pp. 336-382.
191. F. Takens, "Distinguishing deterministic and random systems," in: G. I. Barenblatt, G. Iooss, and D. D. Joseph (eds.), Nonlinear Dynamics and Turbulence, Pitman, New York (1983), pp. 314-333.
192. Ya. G. Sinai, "Stochasticity of smooth dynamical systems. Elements of KAM theory," in: Dynamical Systems [in Russian], Vol. 2, VINITI, Moscow (1985), pp. 115-122.
193. E. A. Sataev, "Invariant measures for hyperbolic maps with singularities," Usp. Mat. Nauk, 47, No. 1, 147-202 (1992).
194. R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer, Berlin (1987).
195. M. L. Blank, "Small perturbations of chaotic dynamical systems," Usp. Mat. Nauk, 44, No. 6, 3-28 (1989).
196. Strange Attractors [Russian translations], Mir, Moscow (1981).
197. R. V. Plykin, "On the geometry of hyperbolic attractors of smooth cascades," Usp. Mat. Nauk, 39, No. 6, 75-113 (1984).
198. R. Lozi, "Un attracteur etrange du type attracteur de Henon," J. de Phys., 39, No. C5, 9-11 (1978).
199. M. Misiurewicz, "Strange attractors for the Lozi mappings," in: R. G. Helleman (ed.), Nonlinear Dynamics, Vol. 357, New York Acad. Sci., New York (1980), pp. 348-358.
200. P. Collet and Y. Levi, "Ergodic properties of the Lozi mappings," Commun. Math. Phys., 93, No. 4, 461-482 (1984).
201. R. V. Plykin, "Sources and sinks of surface A-diffeomorphisms," Matem. Sb., 94, No. 6, 243-264 (1974).
202. V. P. Belykh, "Models of discrete phase-synchronization systems," in: V. V. Shakhgil'dyan and L. N. Belyustina (eds.), Phase Synchronization Systems [in Russian], Radio i Svyaz', Moscow (1982), pp. 161-176.
203. L. A. Bunimovich, "Hyperbolic systems with singularities," in: Dynamical Systems [in Russian], Vol. 2, VINITI, Moscow (1985), pp. 173-204.
204. L. A. Bunimovich and Ya. G. Sinai, "Stochasticity of the attractor in the Lorentz model," in: A. V. Gaponov-Grekhov (ed.), Nonlinear Waves [in Russian], Nauka, Moscow (1979), pp. 212-226.
205. L. A. Bunimovich, "Statistical properties of Lorentz attractors," in: G. I. Barenblatt, G. Iooss, and D. D. Joseph, Nonlinear Dynamics and Turbulence, Pitman, New York (1983), pp. 71-92.
206. V. S. Afraimovich and L. P. Shilnikov, "On strange attractors and quasiattractors," in: G. I. Barenblatt, G. Iooss, and D. D. Joseph, Nonlinear Dynamics and Turbulence, Pitman, New York (1983), pp. 1-34.
207. R. Carrido and L. Simó, "Some ideas about strange attractors," Lecture Notes in Physics, Vol. 179, Springer, Berlin (1983), pp. 1-28.
208. V. S. Afraimovich, V. V. Bykov, and L. P. Shil'nikov, "On the existence of stable periodic motions in the Lorentz model," Usp. Mat. Nauk, 35, No. 5, 164-165 (1980).
209. V. V. Bykov, "On bifurcations of dynamical systems close to systems with a separatrix loop containing a saddle-focus," in: Methods of Qualitative Theory of Differential Equations [in Russian], GGU, Gor'kii (1980), pp. 44-72.
210. R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed., Addison-Wesley, New York (1993).
211. D. Singer, "Stable orbits and bifurcations of maps of the interval," SIAM J. Appl. Math., 35, No. 2, 260-267 (1978).
212. A. I. Ognev, "Metric properties of a class of maps of the interval into itself," Mat. Zametki, 30, No. 5, 723-736 (1981).
213. M. Misiurewicz, "Absolutely continuous measures for certain maps of an interval," Publ. Math. IHES, 53, 17-51 (1981).
214. W. De Melo and S. Van Strien, One-Dimensional Dynamics, Springer, Berlin (1993).
215. M. Tsudjii, "A proof of Benedicks-Carleson-Jakobson theorem for the quadratic family," Preprint, Kyoto Univ. (1992);
216. "Positive Lyapunov exponents in families of one-dimensional dynamical systems," Preprint, Kyoto Univ. (1992).
217. 2 on (-1, 1)," Annals. Math., 122, 1-25 (1985).
218. M. V. Jakobson, "Absolutely continuous invariant measures for one-parameter families of one-dimensional maps," Commun. Math. Phys., 81, No. 1, 39-88 (1981).
219. G. Świa̧tek, "Hyperbolicity is dense in the real quadratic family," Preprint, Stony Brook (1992).
220. Yu. I. Neimark, Dynamical Systems and Controllable Processes [in Russian], Nauka, Moscow (1978).
221. A. Yu. Loskutov, "Dynamics control of chaotic systems by parametric destochastization," J. Phys. A, 26, No. 18, 4581-4594 (1993).
222. A. Yu. Loskutov, "Non-feedback controlling complex behavior: an analytic approach," in: J. Awreicewicz (ed.), Nonlinear Dynamics: New Theoretical and Applied Results, Springer, Berlin (1995), pp. 125-150.
223. A. N. Deryugin, A. Yu. Loskutov, and V. M. Tereshko, "Inducing stable periodic behavior in a class of dynamical systems by parametric perturbations," Chaos, Solitons, and Fractals, 7, No. 10, 1555-1567 (1996).
224. A. Yu. Loskutov and S. D. Rybalko, "Parametric perturbations and suppression of chaos in n-dimensional maps," Preprint ICTP IC/94/347, Trieste, Italy (1994).
225. M. Roso, Nonlinear Oscillations and Stability Theory [Russian translation], Nauka, Moscow (1971).
226. E. N. Dudnik, Yu. I. Kuznetsov, I. I. Minakova, and Yu. M. Romanovskii, "Synchronization in systems with a strange attractor," Vestnik MGU, Phys.-Astron., 38, No. 4, 84-87 (1983).
227. Yu. I. Kuznetsov, V. V. Milyulin, I. I. Minakova, and B. A. Sil'nov, "Synchronization of chaotic modes," Dokl. Akad. Nauk SSSR, 275, Nos. 4-6, 1388-1391 (1984).
228. Yu. I. Kuznetsov, P. S. Landa, A. F. Ol'khovoi, and S. M. Perminov, "Relationship between synchronization amplitude threshold and entropy in stochastic oscillatory systems," Dokl. Akad. Nauk SSSR, 281, No. 2, 291-294 (1985).
229. V. V. Alekseev and A. Yu. Loskutov, "Controlling a system with a strange attractor," in: Topics in Ecological Monitoring and Ecosystem Simulation [in Russian], Vol. 8, Gidrometeoizdat, Leningrad (1985), pp. 175-189.
230. J. F. Linder and W. L. Ditto, "Removal, suppression, and control of chaos by nonlinear design," Appl. Mech. Rev., 48, No. 12, 795-807 (1995).
231. E. Ott and M. L. Spano, "Controlling chaos," Physics Today, 48, No. 5, 34-40 (1995).
232. P. So and E. Ott, "Controlling chaos using time delay coordinates via stabilization of periodic orbits," Phys. Rev. E, 51, No. 4, 2955-2962 (1995).
233. G. A. Johnson, M. Löcher, and E. R. Hunt, "Stabilized spatiotemporal waves in a convectively unstable open flow system: coupled diode resonators," Phys. Rev. E, 51, 1625-1628 (1995).
234. R. V. Solé and L. Menéndez de la Prida, "Controlling chaos in discrete neural networks," Phys. Lett. A, 199, No. 1-2, 65-69 (1995).
235. D. Auerbach, "Controlling extended systems of chaotic elements," Phys. Rev. Lett., 72, No. 8, 1184-1187 (1994).
236. M. Ding, E. Ott, and C. Grebogi, "Controlling chaos in a temporally irregular environment," Physica D, 74, No. 1-2, 386-394 (1994).
237. J. E. S. Socolar, D. W. Sukow, and D. J. Gauthier, "Stabilizing unstable periodic orbits in fast dynamical systems," Phys. Rev. E, 50, No. 4, 3245-3248(1994).
238. A. Kittel, K. Pyragas, and R. Richter, "Prerecorded history of a system as an experimental tool to control chaos," Phys. Rev. E, 50, No. 1, 262-268 (1994).
239. M. A. Matias and J. Guemez, "Stabilization of chaos by proportional pulses in the system variables," Phys. Rev. Lett., 72, No. 1, 1455-1458 (1994).
240. G. Reiser, A. Hudie;bler, and E. Lüscher, "Algorithm for the determination of the resonances of anharmonic damped oscillators," Z. Naturforsch. A, 42, 803-807 (1987).
241. E. A. Jackson, "Control of dynamic flows with attractors," Phys. Rev. A, 44, 4839-4853 (1991).
242. J. D. Farmer and J. J. Sidorovich, "Optimal shadowing and noise reduction," Preprint, Los Alamos National Lab., LA-UR-90-653.
243. T. Shinbrot, E. Ott, C. Grebogi and J. A. Yorke, "Using chaos to direct orbits to targets in systems describable by a one-dimensional map," Phys. Rev. A, 45, No. 6, 4165-4168 (1992).
244. I. M. Starobinets and A. S. Pikovsky, "Multistep controlling chaos," Phys. Lett. A, 181, 149-152.
245. B. Hübinger, R. Doerner, and W. Martienssen, "Local control of chaotic motion," Z. f. Phys. B, 90, 103-106 (1993).
246. S. J. Schiff, K. Jerger, D. H. Duong, T. Chang, M. L. Spano, and W. L. Ditto, "Controlling chaos in the brain," Nature, 370, 615-620 (1994).
247. Y. Liu, N. Kikuchi, and J. Ohtsubo, "Controlling dynamical behavior of a semiconductor laser with external optimal feedback," Phys. Rev. E, 51, No. 4, 2697-2700 (1995).
248. V. Petrov, M. J. Crowle, and K. Showalter, "Tracking unstable periodic orbits in the Belousov-Zhabotinsky reaction," Phys. Rev. Lett., 72, No. 18, 2955-2958 (1994).
249. V. In, W. L. Ditto, and M. L. Spano, "Adaptive control and tracking of chaos in a magnetoelastic ribbon," Phys. Rev. E, 51, No. 4, 2689-2692 (1995).
250. B. Blazejczyk, T. Kapitaniak, J. Woewoda, and J. Brindley, "Controlling chaos in mechanical systems," Appl. Mech. Rev., 46, No. 7, 385-391 (1993).
251. N. L. Komarova and A. Yu. Loskutov, "Stabilization of chaotic oscillations in dynamical systems: rigorous results," SPIE, 2037, 71-81 (1993).
252. N. L. Komarova and A. Yu. Loskutov, "Stabilization of chaotic behavior of an oscillatory chemical reaction," Matem. Modelirovanie, 7, No. 10, 133-143 (1995).
253. A. Yu. Loskutov, S. D. Rybalko, U. Feudel, and J. Kurths, "Suppression of chaos by cyclic parametric excitation in two-dimensional maps," J. Phys. A, 29, No. 18, 5759-5773 (1996).
254. A. Yu. Loskutov, Yu. V. Mishenko, and S. D. Rybalko, "Stabilization of unstable behavior of dynamical systems and the information processing problem," Fizicheskaya Mysl' Rossii, 2/3, 53-66 (1997).
255. A. Yu. Loskutov, V. M. Tereshko, and K. A. Vasiliev, "Stabilization of chaotic dynamics of one-dimensional maps by a cyclic parametric transformation," Int. J. Bif. Chaos, 6, No. 4, 725-735 (1996).
256. A. Yu. Loskutov and A. K. Prokhorov, "Parametric perturbations and stabilization of chaotic behavior of dynamical systems," Fizicheskaya Mysl' Rossii, 2/3, 36-52 (1997).
257. A. N. Deryugin, A. Yu. Loskutov, and V. M. Tereshko, "Creation of stable periodic behavior in parametrically excited dynamically systems," TMF, 104, No. 3, 507-512 (1995).
258. Z. Galias, "New method for stabilization of unstable periodic orbits in chaotic systems," Int. J. Bif. Chaos, 5, No. 1, 281-295 (1995).
259. G. Duffing, Erzwungene Schwingungen bei Veranderlicher Eigenfrequenz, F. Vieweg, Braunschweig (1918).
260. P. L. Kapitsa, "Dynamical stability of a pendulum with an oscillating pivot," ZhETF, 21, No. 5, 588-597 (1951).
261. L. D. Landau and E. M. Lifshits, Mechanics [in Russian], Nauka, Moscow (1988).
262. W. Parry and M. Pollicott, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Soc. Math. de France (1990).
263. D. Ruelle, Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval, AMS (1994).
264. K. Shiraiva, Bibliography of Dynamical Systems, Preprint No. 1, Nagoya Univ. (1985).
265. Z. Shu-yu, Bibliography on Chaos, World Science (1991).
266. S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, Berlin (1982).
267. V. I. Arnol'd, "Catastrophe theory," in: Dynamical Systems [in Russian], Vol. 5, VINITI, Moscow, pp. 219-277.
268. R. Gilmore, Applied Catastrophe Theory [Russian translation], Vols. 1, 2, Mir, Moscow (1984).