Slow passage through a pitchfork bifurcation

SIAM Journal on Applied Mathematics

Volumn 56, Issue 3, 1996, Pages 889-918

  Maree G J M ^a  

^a WAGENINGEN UNIVERSITY   (Netherlands)

Author keywords

Averaging; Matched asymptotic expansions; Nonlinear oscillator; Pitchfork bifurcation; Second Painlev transcendent; Slowly varying parameter

Indexed keywords

COMPRESSIVE STRENGTH; DAMPING; DEGREES OF FREEDOM (MECHANICS); DIFFERENTIAL EQUATIONS; ELASTICITY; OSCILLATORS (MECHANICAL);

AVERAGING; MATCHED ASYMPTOTIC EXPANSIONS; NONLINEAR OSCILLATOR; PITCHFORK BIFURCATION; SECOND PAINLEVE TRANSCENDENT; SLOWLY VARYING PARAMETER;

BIFURCATION (MATHEMATICS);

EID: 0030167803     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139993257399     Document Type: Article

References (28)

1. S. M. BAER, T. ERNEUX, AND J. RINZEL (1989), The slow passage through a Hopf bifurcation: Delay, memory effects and resonance, SIAM J. Appl. Math., 49, pp. 55-71.
2. D. L. BOSLEY AND J. KEVORKIAN, Adiabatic invariance and transient resonance in very slowly varying oscillatory Hamiltonian systems, SIAM J. Appl. Math., 52, pp. 494-527.
3. F. J. BOURLAND AND R. HABERMAN (1990), Separatrix crossing: Time-invariant potentials with dissipation, SIAM J. Appl. Math., 50, pp. 1716-1744.
4. _ (1994), Connection across a separatrix with dissipation, Stud. Appl. Math., 91, pp. 95-124.
5. F. J. BOURLAND, R. HABERMAN, AND W. L. KATH (1991), Averaging methods for the phase shift of arbitrarily perturbed strongly nonlinear oscillators with an application to capture, SIAM J. Appl. Math., 51, pp. 1150-1167.
6. D. R. J. CHILLINGWORTH (1976), Differentiable Topology with a View to Applications, Pitman, London.
7. P. A. DEIFT AND X. ZHOU (1993), Asymptotics for the Painlevé II Equation: Announcement of results, in Spectral and Scattering Theory and Its Applications, Adv. Stud. Pure Math., Kinokiniya-Shoten, Tokyo, American Mathematical Society.
8. J. P. DENIER AND R. GRIMSHAW (1990), Slowly-varying bifurcation theory in dissipative systems, J. Austral. Math. Soc. Ser. B, 3, pp. 301-318.
9. W. ECKHAUS (1979), Asymptotic Analysis of Singular Perburbations, North-Holland, Amsterdam, New York, Oxford.
10. H. FLASCHKA AND A. C. NEWELL (1980), Monodromy- and spectrum-preserving deformations I, Comm. Math. Phys., 76, pp. 65-116.
11. R. HABERMAN (1979), Jump and transition phenomena, SIAM J. Appl. Math., 37, pp. 69-106.
12. J. HENRARD (1993), The adiabatic invariant in classical mechanics, in Dynamics Reported, Vol. 2, New Series, Springer-Verlag.
13. L. HOLDEN AND T. ERNEUX (1993), Slow passage through a Hopf bifurcation: From oscillatory to steady state solutions, SIAM J. Appl. Math., 53, pp. 1045-1058.
14. A. R. ITS, A. S. FOKAS, AND A. A. KAPAEV (1994), On the asymptotic analysis of the Painlevé equations via the isomonodromy method, Nonlinearity, 7, pp. 1291-1325.
15. A. R. ITS AND V. YU. NOVOKSHENOV, The Isomonodromic Deformations Method in the Theory of Painlevé Equations, Lecture Notes in Math. 1191, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1986.
16. A. V. KITAEV (1989), The justification of the asymptotic formulae obtained by the isomonodromic deformation method, LOMI 179, Mathematical Questions of the Wave Propagation Theory 19, Zap. Nauch. Semin. (In Russian.)
17. D. LEVI AND P. WINTERNITZ (1992), Painlevé transcendents: Their asymptotics and physical applications, in Proceedings of a NATO Advanced Research Workshop, September 3-7, 1990, Sainte Adele, Quebec, Canada.
18. G. J. M. MARÉE (1993), Sudden change in a second-order non-linear system with a slowly varying parameter, Internat. J. Non-Linear Mech., 28, pp. 409-426.
19. A. L. NEISHTADT (1986), Change in adiabatic invariant at a separatrix, Sov. J. Plasma Phys., 12, pp. 568-573.
20. _ (1987), Persistence of stability loss for dynamical bifurcations I, Differential Equations, 23, pp. 1385-1391.
21. _ (1988), Persistence of stability loss for dynamical bifurcations II, Differential Equations, 24, pp. 171-176.
22. _ (1993), On probabilistic phenomena in perturbed systems, Selecta Mechanica, 12, pp. 195-201.
23. 2 + x, Bull. Soc. Math. France, 28, pp. 227-252.
24. J. SANDERS AND F. VERHULST (1985), Averaging Methods in Nonlinear Dynamical Systems, Appl. Math. Sci. 59, Springer-Verlag, New York.
25. J. J. STOKER (1950), Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience, New York.
26. F. VERHULST (1990a), Basic perturbation theory and asymptotic methods, Lecture notes, State University Utrecht.
27. _ (1990), Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, Berlin.
28. V. E. ZAKHAROV (1991), What Is Integrability?, Springer Ser. Nonlinear Dynamics, Springer-Verlag, New York.