Bifurcation of the periodic orbits of Hamiltonian systems: An analysis using normal form theory

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 54, Issue 2, 1996, Pages 2033-2070

  Sadovskii D A ^a,b,c   Delos, J B ^a,b  

^a Dept of Economics and Thomas Jefferson Program in Public Policy at the College of William and Mary   (United States)

^b UNIVERSITY OF COLORADO   (United States)

^c UNIVERSITÉ DU LITTORAL CÔTE D OPALE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001597114     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/physreve.54.2033     Document Type: Article

References (84)

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17. K. R. Meyer, Trans. Am. Math. Soc. 149, 95 (1970); 154, 273 (1971). See also K. R. Meyer, in Multiparameter Bifurcation Theory, Contemporary Mathematics Series Vol. 56, edited by M. Golubitsky and J. Guckenheimer (American Mathematical Society, Providence, RI, 1986), p. 373.
18. K. R. Meyer, Trans. Am. Math. Soc. 149, 95 (1970); 154, 273 (1971). See also K. R. Meyer, in Multiparameter Bifurcation Theory, Contemporary Mathematics Series Vol. 56, edited by M. Golubitsky and J. Guckenheimer (American Mathematical Society, Providence, RI, 1986), p. 373.
19. A brief account of this work is given by D. A. Sadovskií, J. A. Shaw, and J. B. Delos, Phys. Rev. Lett. 75, 2120 (1995).
20. 2-symmetric bifurcation of the perpendicular orbit is organized in the fashion similar to the k = 4 and 6 cases [6].
21. An extension of the generic theory [16] to symmetric systems was given by M. A. M. de Aguiar, C. P. Malta, M. Baranger, and K. T. R. Davies, Ann. Phys. (N.Y.) 180, 167 (1987); M. A. M. de Aguiar and C. P. Malta, Physica D 30, 413 (1988).
22. An extension of the generic theory [16] to symmetric systems was given by M. A. M. de Aguiar, C. P. Malta, M. Baranger, and K. T. R. Davies, Ann. Phys. (N.Y.) 180, 167 (1987); M. A. M. de Aguiar and C. P. Malta, Physica D 30, 413 (1988).
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31. In mathematics the term "generic" has a precise meaning. A generic subset of an appropriately defined set (of functions, mappings, etc.) has two important properties: it must be open and everywhere dense. These properties are closely related to structural stability; see Refs. [17], Chap. 3, [16], Chaps. VI-IIA, p. 202, and [54]. For example, integrable systems are structurally unstable in the set of all dynamical systems [1,14], because an infinitely small perturbation typically destroys integrability; these systems are also not dense, and hence they cannot be used to approximate all possible dynamic regimes.
32. The name comes from the pitchforklike bifurcation diagram in the parameter space.
33. K. Meyer told us about a number of instances where similar phenomena were observed. He stressed that these phenomena are not generic in the one-parameter theory. See also Table I, footnote (c) and remarks of de Aguiar et al. (Ref. [12]), pp. 188 and 200.
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35. B. I. Ẑhilinskií and I. M. Pavliĉhenkov, Zh. Eksp. Teor. Fiz. 92, 387 (1987) [Sov. Phys. JETP 65, 221 (1987)]; Ann. Phys. (N.Y.) 184, 1 (1988). For a general concise discussion see B. I. Ẑhilinskií, Teoriŷa Sloẑhnyk̂h Molekulŷarnyk̂h Spektrov (Moscow University Press, Moscow, 1989), Appendix 2 (in Russian) (English title: Theory of Complex Molecular Spectra).
36. B. I. Ẑhilinskií and I. M. Pavliĉhenkov, Zh. Eksp. Teor. Fiz. 92, 387 (1987) [Sov. Phys. JETP 65, 221 (1987)]; Ann. Phys. (N.Y.) 184, 1 (1988). For a general concise discussion see B. I. Ẑhilinskií, Teoriŷa Sloẑhnyk̂h Molekulŷarnyk̂h Spektrov (Moscow University Press, Moscow, 1989), Appendix 2 (in Russian) (English title: Theory of Complex Molecular Spectra).
37. B. I. Ẑhilinskií and I. M. Pavliĉhenkov, Zh. Eksp. Teor. Fiz. 92, 387 (1987) [Sov. Phys. JETP 65, 221 (1987)]; Ann. Phys. (N.Y.) 184, 1 (1988). For a general concise discussion see B. I. Ẑhilinskií, Teoriŷa Sloẑhnyk̂h Molekulŷarnyk̂h Spektrov (Moscow University Press, Moscow, 1989), Appendix 2 (in Russian) (English title: Theory of Complex Molecular Spectra).
38. k are not symplectic and thus should not be considered in the Hamiltonian case (cf. Appendix A).
39. crit only one transversal mode with eigenvalues λ can be in such a resonance. It follows that λ's are purely imaginary: λ=exp(±i2πn/k). Therefore, in this paper we only consider situations (7a) and (7b).
40. A classical example of such generic parametrization is the Mathieu-Hill equation, where any of the two roots of ω(ε) = 0 are separated by finite intervals (Lyapunov's oscillation theorem (Ref. [37], Sec. 2.1)).
41. ε of matrices, whose multipliers are of type (7a) [24], can have at most one degenerate pair (7b) ([17], Chap. 6, Sec. 30E). However, in a generic one-parameter family of N×N Hamiltonian matrices, another case can arise: there may be irreducible 4×4 blocks. We consider only the 2×2 case [24].
42. 1,2 would have to be used.
43. The idea of Ẑhilinskií and Pavliĉhenkov [22] is to classify bifurcations of the fixed points (equilibria) of generic one-degree-of-freedom Hamiltonians with possible a priori symmetries, and to study quantum and classical manifestations of these bifurcations. Their initial work [Ann. Phys. (N.Y.) 184, 1 (1988)] from where we cite our Table I deals with molecular rotation separated from vibration and electronic motion due to the Born-Oppenheimer principle; separation of vibrational modes of a molecule can be approximately achieved near the equilibrium configuration where the perturbation technique is valid [55]. In both cases symmetry enters as the a priori symmetry of the (equilibrium configuration of the) molecule. More generally this subject is discussed in [56]. As follows from our paper the range of applications of this theory is significantly broader.
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47. This "trick" has been suggested to us by K. Meyer [see K. R. Meyer and D. S. Schmidt, Funkcialaj Ekvacioj 20, 171 (1977), Eq. (3.16)]; cf. G. E. O. Giacaglia, Perturbation Methods in Nonlinear Systems, Applied Mathematics Sciences Vol. 8 (Springer, New York, 1972), Chap. 11.8.
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