Slow passage through a saddle-center bifurcation

Journal of Nonlinear Science

Volumn 10, Issue 2, 2000, Pages 197-221

  Diminnie, D C ^a   Haberman, R ^a  

^a SOUTHERN METHODIST UNIVERSITY   (United States)

Author keywords

Averaging; Change in action; First Painlev transcendent; Generalized Riemann zeta function; Homoclinic orbits; Nonautonomous saddle approach; Nonlinear saddle point; Passage through a separatrix; Saddle center bifurcation

Indexed keywords

EID: 0001460135     PISSN: 09388974     EISSN: None     Source Type: Journal    
DOI: 10.1007/s003329910009     Document Type: Article

References (32)

1. Arnold, V. I., Kozlov, V. V., and Neishtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, Dynamical Systems III, Springer-Verlag, New York, 1988.
2. Bourland, F. J., and Haberman, R., Separatrix crossing: Time-invariant potentials with dissipation, SIAM J. Appl. Math., 50 (1990), pp. 1716-1744.
3. Bourland, F. J., Haberman, R., and Kath, W. L., Averaging methods for the phase shift of arbitrarily perturbed strongly nonlinear oscillators with an application to capture, SIAM J. Appl. Math., 51 (1991), pp. 1150-1167.
4. Bourland, F. J., and Haberman, R., Connection across a separatrix with dissipation, Stud. Appl. Math., 91 (1994), pp. 95-124.
5. Carrier, G. F., Krook, M., and Pearson, C. E., Functions of a Complex Variable, McGraw-Hill, New York, 1966, pp. 192-194.
6. Cary, J. R., Escande, D. F., and Tennyson, J. L., Adiabatic-invariant change due to separatrix crossing, Phys. Rev. A, 34 (1986), pp. 4256-4275.
7. Cary, J. R., and Skodje, R. T., Phase change between separatrix crossings, Physica D, 36 (1989), pp. 287-316.
8. Edwards, H. M., Riemann's Zeta Function, Academic Press, New York, 1974, pp. 9-11.
9. Fokas, A. S., Mugan, U., and Zhou, X., On the solvability of Painlevé I, III and V, Inverse Problems, 8 (1992), pp. 757-785.
10. Golubitsky, M., and Schaeffer, D. G., Singularities and Groups in Bifurcation Theory, Vol. 1, Springer-Verlag, New York, 1985.
11. Golubitsky, M., Stewart, I., and Schaeffer, D. G., Singularities and Groups in Bifurcation Theory, Vol. 2, Springer-Verlag, New York, 1988.
12. Guckenheimer, J., and Holmes, P. J., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
13. Haberman, R., Slowly varying jump and transition phenomena associated with algebraic bifurcation problems, SIAM J. Appl. Math., 37 (1979), pp. 69-109.
14. Henrard, J. The adiabatic invariant in classical mechanics, in Dynamics Reported Expositions in Dynamical Systems New Series: Volume 2, eds. C. K. R. T. Jones, U. Kirchgraber, H. O. Walther, Springer-Verlag, New York, 1993, pp. 117-235.
15. Ince, E. L., Ordinary Differential Equations, Dover, New York, 1956.
16. Its, A. R., and Novokshenov, V. Y., The Isomonodromic Deformation Method in the Theory of Painlevé Equations, in Lecture Notes in Mathematics Vol. 1191, Springer-Verlag, New York, 1986.
17. Kapaev, A. A., Asymptotics of solutions of the Painlevé equation of the first kind, Diff. Eq. 24 (1988), pp. 1107-1115 [Diff. Urav., 24 (1988), pp. 1684-1695].
18. Kapaev, A. A., Asymptotics of solutions of the Painlevé equation of the first kind, Diff. Eq. 24 (1988), pp. 1107-1115 [Diff. Urav., 24 (1988), pp. 1684-1695].
19. Lebovitz, N. R., and Schaar, R. J., Exchange of stabilities in autonomous systems, Stud. Appl. Math., 54 (1975), pp. 229-260.
20. Lebovitz, N. R., and Schaar, R. J., Exchange of stabilities in autonomous systems - II. Vertical bifurcation, Stud. Appl. Math., 56 (1977), pp. 1-50.
21. Mareé, G. J. M., Slow passage through a pitchfork bifurcation, SIAM J. Appl. Math., 56 (1996), pp. 889-918.
22. Neishtadt, A. I., Passage through a separatrix in a resonance problem with a slowly varying parameter, Prikl. Mat. Mekh., 39 (1975), pp. 621-622 [J. Appl. Math. Mech., 39 (1975), pp. 594-605].
23. Neishtadt, A. I., Passage through a separatrix in a resonance problem with a slowly varying parameter, Prikl. Mat. Mekh., 39 (1975), pp. 621-622 [J. Appl. Math. Mech., 39 (1975), pp. 594-605].
24. Neishtadt, A. I., Change of an adiabatic invariant at a separatrix, Fiz. Plazmy, 12 (1986), pp. 992-1001 [J. Plasma Phys., 12 (1986), pp. 568-573].
25. Neishtadt, A. I., Change of an adiabatic invariant at a separatrix, Fiz. Plazmy, 12 (1986), pp. 992-1001 [J. Plasma Phys., 12 (1986), pp. 568-573].
26. Neishtadt, A. I., Persistence of stability loss for dynamical bifurcations. I, Diff. Urav., 23 (1987), pp. 2060-2067 [Diff. Eq., 23 (1987), pp. 1385-1391].
27. Neishtadt, A. I., Persistence of stability loss for dynamical bifurcations. I, Diff. Urav., 23 (1987), pp. 2060-2067 [Diff. Eq., 23 (1987), pp. 1385-1391].
28. Neishtadt, A. I., Persistence of stability loss for dynamical bifurcations. II, Diff. Urav., 24 (1988), pp. 226-233 [Diff. Eq., 24 (1988), pp. 171-176].
29. Neishtadt, A. I., Persistence of stability loss for dynamical bifurcations. II, Diff. Urav., 24 (1988), pp. 226-233 [Diff. Eq., 24 (1988), pp. 171-176].
30. Neishtadt, A. I., Vainshtein, D. L., and Vasiliev, A. A., Chaotic advection in a cubic Stokes flow, Physica D, 111 (1998), pp. 227-242.
31. Sanders, J. A., and Verhulst, F., Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, New York, 1985.
32. Vainshtein, D. L., Vasiliev, A. A., and Neishtadt, A. I., Changes in the adiabatic invariant and streamline chaos in confined incompressible Stokes flow, Chaos, 6 (1996), pp. 67-77.