On stability loss delay for dynamical bifurcations

Discrete and Continuous Dynamical Systems - Series S

Volumn 2, Issue 4, 2009, Pages 897-909

  Neishtadt, Anatoly ^a,b  

^a LOUGHBOROUGH UNIVERSITY   (United Kingdom)

^b Space Research Institute ^*  (Russian Federation)

Author keywords

Dynamical bifurcations; Slow fast systems

Indexed keywords

EID: 77953515266     PISSN: 19371632     EISSN: 19371179     Source Type: Journal    
DOI: 10.3934/dcdss.2009.2.897     Document Type: Article

References (24)

1. V. I. Arnold, "Geometrical Methods in the Theory of Ordinary Differential Equations," Springer-Verlag, New York, 1988.
2. V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Russian Math. Surveys, 18 (1963), 85-191.
3. C. Baesens, Slow sweep through a period-doubling cascade: Delayed bifurcations and renor-malisation, Physica D, 53 (1991), 319-375.
4. S. M. Baer, T. Erneux and J. Rinzel, The slow passage through a Hopf bifurcation: delay, memory effects and resonance, SIAM J. Appl. Math., 49 (1989), 1-17.
5. E. Benoit (ed.), "Dynamic Bifurcations," Lecture Notes in Mathematics, Vol. 1493, Springer-Verlag, Berlin, 1991.
6. J. L. Callot, Champs lents-rapides complexes à une dimension lente, Annales scientifiques de l'Ecole Normale Supérieure, 4 (1993), 149-173.
7. F. Diener and M. Diener, Maximal delay, in "Dynamic Bifurcations" (ed. E. Benoit), Lecture Notes in Mathematics, Vol. 1493. Springer-Verlag, Berlin, (1991), 71-86.
8. T. Erneux and P. Mandel, Stationary, harmonic and pulsed operations of an optically bistable laser with saturable absorber. II, Phys. Rev. A, 18 (1984), 1902-1909.
9. A. Fruchard, Canards et r̂ateaux, Annales de L'Institut Fourier, 42 (1992), 825-855.
10. A. V.Gaponov-Grekhov and V. I. Nekorkin (eds.), "Nonlinear Waves-2006," IPF, N.Novgorod, 2007 (in Russian).
11. S. Karimov, Asymptotics of solutions of some classes of differential equations with small parameter multiplying derivatives, when there is change of stability of a stationary point in the "fast motion" plane, Differ. Equations, 21 (1985), 1136-1139.
12. E. F. Mishchenko and N. Kh. Rozov, "Differential Equations With Small Parameters and Relaxation Oscillations," New York-London: Plenum Press, 1980.
13. A. I. Neishtadt, Asymptotic study of stability loss of equilibrium under slow transition of two eigenvalues through imaginary axis, Usp. Mat. Nauk, 40 (1985), 300-301 (Russian).
14. A. I. Neishtadt, Prolongation of the loss of stability in the case of dynamic bifurcations, I, II, Differ. Equations, 23 (1987), 1385-1390; 24 (1988), 171-176.
15. A. I. Neishtadt, On calculation of stability loss delay time for dynamical bifurcations, in "XI International Congress on Mathematical Physics" (Paris, 1994) (ed. D. Yagolnitzer), International Press, Cambridge, MA (1995), 280-287.
16. A. I Neishtadt and V. V. Sidorenko, Delay of loss of stability in Ziegler's system, J. Appl. Math. Mech., 61 (1997), 15-25. (Pubitemid 127163958)
17. A. I. Neishtadt, C. Simó and D. V. Treschev, On stability loss delay for a periodic trajectory, Progress in Nonlinear Differential Equations and Their Applications, 19 (1996), 253-278. (Pubitemid 126010640)
18. L. S. Pontryagin and L. V. Rodygin, Approximate solution of a system of ordinary differential equations involving a small parameter in the derivatives, Sov. Math. Dokl., 1 (1960), 237-240.
19. M. A. Shishkova, Examination of a system of differential equations with a small parameter in the highest derivatives, Sov. Math. Dokl., 14 (1973), 483-487.
20. Y. Sibuya, Sur réduction analytique d' un système d' equations différentiales ordinares linéaires contentant un papamètre, J. Fac. Sci. Univ. Tokio. Sect. 1, 10 (1958), 527-540.
21. J. Su, Effects of periodic forcing on delayed bifurcations, J. Dynam. Differential Equations, 9 (1997), 561-625.
22. T. T. Tsotis R. C. Sane and T. H. Lindstrom, The bifurcation behaviour of catalytic reaction system due to slowly varying control parameter, AIChE, 34 (1987), 383-388. (Pubitemid 18577738)
23. H. Ziegler, Die Stabilitätskriterien der Elastomechanik, Ing. Archiv, 20 (1952), 49-56.
24. A. K. Zvonkin and M. A. Shubin, Nonstandard analysis and singular perturbations of ordinary differential equations, Russian Math. Surveys, 39 (1984), 69-131.