Abstract
The next two chapters consist of a contribution to advancing the geometrical approach to the investigation of singularly perturbed systems in cases when the main hypothesis is violated, i.e., when the real parts of some or all of the eigenvalues of the matrix of the linearized fast subsystem are no longer strictly negative. This means that the hypotheses of the Tikhonov’s theorem are violated. This chapter is organized as follows. The first section is concerned with weakly attractive slow integral manifolds. The examples are borrowed from the theory of gyroscopic systems and flexible-joints manipulators. The next section is devoted to the application of repulsive slow invariant manifolds to thermal explosion problems. In the last section, the case when the slow integral manifold is conditionally stable is discussed and an optimal control problem is given as an application.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Babushok, V.I., Goldshtein, V.M., Sobolev, V.A.: Critical condition for the thermal explosion with reactant consumption. Combust. Sci. Tech. 70, 81–89 (1990)
Bogatyrev, S.V., Sobolev, V.A.: Separation of rapid and slow motions in problems of the dynamics of systems of rigid bodies and gyroscopes. J. Appl. Math. Mech. 52(1), 34–41 (1988)
Ghorbel, F., Spong, M.W.: Integral manifolds of singularly perturbed systems with application to rigid-link flexible-joint multibody systems. Int. J. Non Linear Mech. 35, 133–155 (2000)
Gray, B.F.: Critical behaviour in chemical reacting systems: 2. An exactly soluble model. Combust. Flame 21, 317–325 (1973)
Kobrin, A.I., Martynenko, Yu.G.: Application of the theory of singularly perturbed equations to the study of gyroscopic systems (in Russian). Sov. Phys. Dokl. 21(7–12), 498–500 (1976)
Kokotović, P.V., Khalil, K.H., O’Reilly, J.: Singular Perturbation Methods in Control: Analysis and Design. SIAM, Philadelphia (1986)
Koshlyakov, V.N., Sobolev, V.A.: Permissibility of use of precessional equations of gyroscopic compasses. Mech. Solids 4, 17–22 (1998)
Magnus, K.: The Gyroscope. Göttingen, Phywe (1983)
McIntosh, A.C.: Semenov approach to the modelling of thermal runawway of damp combustible material. IMA J. Appl. Math. 51, 217–237 (1993)
Merkin, D.R.: Introduction to the Theory of Stability. Texts in Applied Mathematics, vol. 24. Springer, New-York (1996)
Mishchenko, E.F., Rozov, N.Kh.: Differential Equations with Small Parameters and Relaxation Oscillations. Plenum Press, New York (1980)
Mitropol’skii, Yu.: The method of integral manifolds in the theory of nonlinear oscillations. In: International Symposium on Nonlinear Differential Equations and Non-Linear Mechanics, pp. 1–15. Academic, New York/London (1963)
Mortell, M.P., O’Malley, R.E., Pokrovskii, A., Sobolev, V.A. (eds.): Singular Perturbation and Hysteresis. SIAM, Philadelphia (2005)
Osintsev, M.S., Sobolev, V.A.: Dimensionality reduction in optimal control and estimation problems for systems of solid bodies with low dissipation. Autom. Remote Control 74(8), 1334–1347 (2013)
Osintsev, M.S., Sobolev, V.A.: Reduction of dimension of optimal estimation problems for dynamical systems with singular perturbations. Comput. Math. Math. Phys. 54(1), 45–58 (2014)
Pendyukhova, N.V., Sobolev, V.A., Strygin, V.V.: Motion of rigid body with gyroscope and moving mass points. Mech. Solids 3, 12–18 (1986)
Semenov, N.N.: Zur theorie des verbrennungsprozesses (in German). Z. Phys. Chem. 48, 571–581 (1928)
Sobolev, V.A.: Integral manifolds and decomposition of singularly perturbed systems. Syst. Control Lett. 5, 169–179 (1984)
Sobolev, V.: Black swans and canards in laser and combustion models. In: Mortell, M., O’Malley, R., Pokrovskii, A., Sobolev, V. (eds.) Singular Perturbations and Hysteresis, pp. 153–206. SIAM, Philadelphia (2005)
Spong, M.W.: Modeling and control of elastic joint robots. J. Dyn. Syst. Meas. Control 109(4), 310–319 (1987)
Strygin, V.V., Sobolev, V.A.: Separation of Motions by the Integral Manifolds Method (in Russian, MR 89k:34071). Nauka, Moscow (1988)
Voropaeva, N.V., Sobolev, V.A.: Geometrical Decomposition of Singularly Perturbed Systems (in Russian). Moscow, Fizmatlit (2009)
Zeldovich, Ya.B., Barenblatt, G.I., Librovich, V.B., Makhviladze, G.M.: The Mathematical Theory of Combustion and Explosions. Consultants Bureau, New York (1985)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Shchepakina, E., Sobolev, V., Mortell, M.P. (2014). Specific Cases. In: Singular Perturbations. Lecture Notes in Mathematics, vol 2114. Springer, Cham. https://doi.org/10.1007/978-3-319-09570-7_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-09570-7_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-09569-1
Online ISBN: 978-3-319-09570-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)