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On Instability Leading to Chaos in Dynamical Systems

Published online by Cambridge University Press:  12 April 2016

L.M. Saha
L.M. Saha*
Affiliation:
Department of Mathematics, Zakir Husain College, Ajmeri Gate, New Delhi-110006, India

Abstract

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Instability of orbits in dynamical systems leading to chaos has been reviewed briefly. Stability criteria for some unimodal mapping which provide various periodic regimes during the period doubling bifurcations has been discussed in detail. Stability conditions are reviewed for standard map (or Chirikov-Taylor map), and results obtained for range of values of the non-linear parameter appearing in the map have been studied. Strange attractor has also been discussed.

Type
Part I Chaos
Information
International Astronomical Union Colloquium , Volume 132: Instability, Chaos and Predictability in Celestial Mechanics and Stellar Dynamics , 1993 , pp. 47 - 60
Copyright
Copyright © Nova Science Publishers 1993

References

[1] Ruelle, D. (1989), Chaotic evolution and strange attractors, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[2] Oono, Y. and Osikawa, M. (1980), Prog. Theor. Phys., 64, 54–67.CrossRefGoogle Scholar
[3] Goldberger, A.L., Rigney, D.R. and West, B.J. (1990), Sci. Amer., February, 35–41.Google Scholar
[4] McRobie, A. and Thompson, M. (1990), New Scientist, 9 June, 41.Google Scholar
[5] Lesurf, J. (1990), New Scientist, 30 June, 63.Google Scholar
[6] Guckenheimer, J. and Holmes, P. (1983), Nonlinear oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer Verlag, New York.CrossRefGoogle Scholar
[7] Hénon, M. (1976), Comm. Math. Phys., 50, 69–79.CrossRefGoogle Scholar
[8] Chirikov, B.V. (1979), Phys. Reports, 52, 263–379.CrossRefGoogle Scholar
[9] Helleman, R.H.G. (1980), Fundamental Problems of Statistical Mechanics, Ed. Cohen, E.G.D North-Holland Publ., Amsterdam, 5, 165–233.Google Scholar
[10] May, R.M. (1974), Science, 186, 645–647.CrossRefGoogle Scholar
[11] May, R.M. (1976), Nature, 261, 459.CrossRefGoogle Scholar
[12] Li, T. and Yorke, J.A. (1975), Amer. Math. Monthly, 82, 985–992.CrossRefGoogle Scholar
[13] Moser, J. (1986), SIAM Reviews, 28, 459.Google Scholar
[14] Tsonis, A.A. and Elsner, J.B. (1998), Nature, 333, 545–547.CrossRefGoogle Scholar
[15] Tsonis, A.A. (1989), Chaos and unpredictability of Weather.CrossRefGoogle Scholar
[16] Moon, F.C. (1987), Chaotic vibrations, J.Wiley and Sons, New York.Google Scholar
[17] Palmer, T. (1989), New Scientist, 11 Nov. 56–59.Google Scholar
[18] Scott, S. (1989), New Scientist, 2 December, 53–59.Google Scholar
[19] Moser, J. (1973), Stable and Random Motions in Dynamical systems, Princeton University Press, Princeton, N.J.Google Scholar
[20] Cook, P.A. (1986), Nonlinear Dynamical Systems, Prentice Hall International, New Delhi.Google Scholar
[21] Hénon, M. and Heiles, C. (1964), Astrn. J. 69, 73–79.CrossRefGoogle Scholar
[22] Hao, B.L. (1984), Chaos.World Scientific Publ., Singapore.Google Scholar
[23] Lorenz, E.N. (1963), J.Atmos. Sci., 20, 130–141.2.0.CO;2>CrossRefGoogle Scholar
[24] Feigenbaum, M.J. (1978), J. Stat. Phys., 19, 25–52 (1979) J.Stat. Phys., 21, 669-706.CrossRefGoogle Scholar
[25] Feigenbaum, M.J. (1960), Corran. Maths. Phys., 77, 65–86.CrossRefGoogle Scholar
[26] Feigenbaum, M.J. (1982), Lecture Notes in Physics, Ed. Carrido, L., 179, Proceedings, Stiges, 131–148.Google Scholar
[27] Feigenbaum, M.J. (1979), Phys., Lett. 74A, 375.CrossRefGoogle Scholar
[28] Feigenbaum, M.J., Kadanoff, L.P. and Shenker, S.J, (1982), Physica, 5D, 370.Google Scholar
[29] Feigenbaum, M.J., Jensen, M.H. and Procacci, A.I. (1980), Phys, Rev. Lett. 57, 1503.CrossRefGoogle Scholar
[30] Ruelle, D. (1990), Proc. Roy. Soc. Lond., A427, 241–248.Google Scholar
[31] Carrido, L. and Simo, C. (1982), Lectures Notes in Physics, Ed. Carrido, L., 179, Proceedings, Stiges, 1–27.Google Scholar
[32] Benettin, G., Froschle, C. and Scheidecker, J.P. (1979), Phys. Rev., A19, 2454.CrossRefGoogle Scholar
[33] Hadamard, J. (1898), J. Math. Pure Appl. 4, 23–73.Google Scholar
[34] Duhem, P. (1906), La theorie physique. Son object et sa structure, Paris: Chevalier et Riviere.Google Scholar
[35] Poincare, H. (1908), Sciences et Methode.Google Scholar
[36] Metropolis, N., Stein, M. and Stein, P.R. (1973), J. J. Combinatorial Theo., 15, 25–44.CrossRefGoogle Scholar
[37] Kolmogorov, A.N. (1954), Les Alamas Scientific Laboratory Translation, LA-TR-71-6 by Dahly, H. of Akad. Nauk. SSSR Doklady 93, 763, (1953), Doklady Akad. Nauk. 98, 527 (1954).Google Scholar
[38] Arnold, V.I. (1961), Izv. Akad. Nauk. SSSR, Ser .Mat., 25. 21.Google Scholar
[39] Arnold, V.I. (1963), Uspekhi Mat. Nauk., 18, 6, 91.Google Scholar
[40] Moser, J. (1962), Nach. Akad. Wiss. Gottingen Math. Phys., K1, 21.Google Scholar
[41] Moser, J. (1968), Lectures on Hamiltonian Systems. Mem. AMS 81.Google Scholar
[42] Arnold, V.I. and Avez, V. (1968), Ergodic Problems of Classical Mechanics, Benjamin, New York.Google Scholar
[43] Fermi, E., Pasta, J. and Ulam, S. (1965), Los Alamas Report LA-1940: The collected papers by Fermi, E., University of Chicago Press, 2, 978.Google Scholar
[44] Collet, P. and Eckmann, J.P. (1975), Iterated Maps on the interval as dynamical systems, Birkhauser.Google Scholar
[45] Hao, B.-L. (1983), Prog. Phys., 3. 329.Google Scholar
[46] Yi-Sui-Sun, (1985), Celes. Mech., 37, 171–181.CrossRefGoogle Scholar
[47] Slarkovski, A.N. (1964), Ukr. Math. Zurn., 16, 61–71.Google Scholar
[48] Ostlund, S., Rand, D., Sethna, J.P. and Siggia, E.D. (1983), Physica, 89, 303.Google Scholar
[49] Arnold, V.I. (1978), Mathematical Methods of Classical Mechanics, Springer, New York.CrossRefGoogle Scholar
[50] Ruelle, D. and Takens, F. (1971), Comm. Math. Phys., 20, 167–192.CrossRefGoogle Scholar
[51] Newhouse, S., Ruelle, D. and Takens, F. (1978), Comm. Math. Phys., 64, 35–40.CrossRefGoogle Scholar
[52] Takens, F. (1981), Lecture Notes in Mathematics, Springer, 898, 712–716.Google Scholar
[53] Vivaldi, F. (1989), New Scientist, 28 Oct. 46–49.Google Scholar
[54] Stewart, I. (1989), New Scientist, 4 Nov., 42–46.Google Scholar
[55] Kaplan, J.L. and Yorke, J.A. (1979), Ann. N.Y. Acad. Sci., Vol. 316.CrossRefGoogle Scholar
[56] Jacobi, C. (1834), Prog. Ann., 32, 229.Google Scholar
[57] Poincare, H. (1885), Acta Math., 7, 259–380.CrossRefGoogle Scholar
[58] Landau, L.D. (1944), Dokl. Akad. Nauk, 342. SSSR, 44, 339–342.Google Scholar
[59] Landau, L.D. and Lifshitz, E.M. (1959), Fluid Mechanics, Oxford, Pergamon.Google Scholar
[60] Hopf, E. (1948), Comm. Pure Appl. Math., 1, 303–322.CrossRefGoogle Scholar
[61] Szebehely, V. (1986), Particle Accelerators, 19, 43–47.Google Scholar
[62] Szebehely, V. (1986), New Non-deterministic Celestial Mechanics, D. Reidel Co. Dordrecht, Holland.Google Scholar
[63] Szebehely, V. (1988), Celest Mech., 43, 139–145.CrossRefGoogle Scholar
[64] Eichhorn, H. (1987), Celest, Mech., 43, 418.Google Scholar
[65] Marchal, C. (1987), Proc. 96th IAU Coll. Ed. Valtonen, M.J. Turku, Finland, 91–97.Google Scholar
[66] Richardson, L.F. (1948), Psychometrika, 13, 147–174.CrossRefGoogle Scholar
[67] Grebogi, C., Ott, E. and Yorke, J.A. (1982), Phys.Rev. Lett., 50, 935–938.CrossRefGoogle Scholar