Quantitative jump theorem

Author:
P. I. Kaleda

Translated by:
E. Khukhro

Original publication:
Trudy Moskovskogo Matematicheskogo Obshchestva, tom **72** (2011), vypusk 2.

Journal:
Trans. Moscow Math. Soc. **2011**, 171-191

MSC (2010):
Primary 34E15; Secondary 34C26, 34E05, 34E20, 37C10, 37G10

Published electronically:
January 12, 2012

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The jump theorem proved by Mishchenko and Pontryagin more than fifty years ago is one of the fundamental results in the theory of relaxation oscillations. Its statement is asymptotic in character. In this paper we present a quantitative analogue of it. This means the following. The jump theorem describes the map along trajectories (the Poincaré map) from a transversal `before the jump' to a transversal `after the jump'. This map is exponentially contracting, and its deviation from the jump point with respect to the slow coordinate is of order , where is the small parameter in the fast-slow system. These estimates are asymptotic. Normalizing the system by choosing the scale, we prove that for all no greater than , the Poincaré map is defined, its deviation lies in the interval , and the map itself is a contraction with a coefficient that does not exceed , where . The main tool used in the investigation is the method of blowup with different weights, in the form described by Krupa and Szmolyan.

**1.**V. I. Arnol′d, V. S. Afrajmovich, Yu. S. Il′yashenko, and L. P. Shil′nikov,*Bifurcation theory*, Current problems in mathematics. Fundamental directions, Vol. 5 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1986, pp. 5–218, i (Russian). MR**895653****2.**Shui-Nee Chow, Cheng Zhi Li, and Duo Wang,*Normal forms and bifurcation of planar vector fields*, Cambridge University Press, Cambridge, 1994. MR**1290117****3.**John Guckenheimer and Philip Holmes,*Nonlinear oscillations, dynamical systems, and bifurcations of vector fields*, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1990. Revised and corrected reprint of the 1983 original. MR**1139515****4.**E. F. Miščenko and L. S. Pontrjagin,*Derivation of certain asymptotic estimates for solutions of differential equations with a small parameter in the derivatives*, Izv. Akad. Nauk SSSR. Ser. Mat.**23**(1959), 643–660 (Russian). MR**0118916****5.**E. F. Mishchenko and N. Kh. Rozov,*Differential equations with small parameters and relaxation oscillations*, Mathematical Concepts and Methods in Science and Engineering, vol. 13, Plenum Press, New York, 1980. Translated from the Russian by F. M. C. Goodspeed. MR**750298****6.**Freddy Dumortier and Robert Roussarie,*Canard cycles and center manifolds*, Mem. Amer. Math. Soc.**121**(1996), no. 577, x+100. With an appendix by Cheng Zhi Li. MR**1327208**, 10.1090/memo/0577**7.**Neil Fenichel,*Geometric singular perturbation theory for ordinary differential equations*, J. Differential Equations**31**(1979), no. 1, 53–98. MR**524817**, 10.1016/0022-0396(79)90152-9**8.**M. Krupa and P. Szmolyan,*Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions*, SIAM J. Math. Anal.**33**(2001), no. 2, 286–314 (electronic). MR**1857972**, 10.1137/S0036141099360919

Retrieve articles in *Transactions of the Moscow Mathematical Society*
with MSC (2010):
34E15,
34C26,
34E05,
34E20,
37C10,
37G10

Retrieve articles in all journals with MSC (2010): 34E15, 34C26, 34E05, 34E20, 37C10, 37G10

Additional Information

**P. I. Kaleda**

Affiliation:
Research and Development Institute of Power Engineering, Moscow, Russia

Email:
pkaleda@yandex.ru

DOI:
https://doi.org/10.1090/S0077-1554-2012-00187-3

Keywords:
Relaxation oscillations,
fast-slow system,
jump point,
resolution of singularities,
normal form

Published electronically:
January 12, 2012

Additional Notes:
This research was supported by the Russian Foundation for Basic Research (grant no. 7-01-00017-a) and by RFBR/CNRS (grant no. 05-01-02801-CNRSa).

Article copyright:
© Copyright 2012
American Mathematical Society