Singular perturbations. Introduction to system order reduction methods with applications. With a foreword by Robert O'Malley
The review consist of the abstracts to each chapter. Chapter 1: Introduction -- Chapter 1 provides an easy introduction to perturbation methods. It begins with an algebraic equation and proceeds to a second-order ODE. The concept of an initial or boundary layer is introduced. This motivates the method of multiple scales. The idea of slow surfaces and slow integral manifolds is introduced and illustrative examples are given. Then a statement of Tikhonov's theorem is given which answers the question about the permissibility of the application of a ``degenerate'' system \((\varepsilon=0)\) as a zero-approximation to the full system. Chapter 2: Slow Integral Manifolds -- In the present chapter we use a method for the qualitative asymptotic analysis of singularly perturbed differential equations by reducing the order of the differential system under consideration. The method relies on the theory of integral manifolds. It essentially replaces the original system by another system on an integral manifold with a lower dimension that is equal to that of the slow subsystem. The emphasis in this chapter is on the study of autonomous systems. Chapter 3: The Book of Numbers -- In this chapter the first number in the title of a section denotes the dimension of the slow variable, the second one denotes the dimension of the fast variable. A series of examples, of increasing complexity, are given to illustrate the theoretical concepts. The main examples come from applications in enzyme kinetics. These examples illustrate the effectiveness of the order reduction method. Chapter 4: Representations of Slow Integral Manifolds -- In constructing the asymptotic expansions of slow integral manifolds it is assumed that the degenerate equation \((\varepsilon=0)\) allows one to find the slow surface explicitly. In many problems this is not possible due to the fact that the degenerate equation is either a high degree polynomial or transcendental. In this situation many authors suggest the use of numerical methods. However, in many problems the slow surface can be described in parametric form, and then the slow integral manifold can be found in parametric form as asymptotic expansions. If this is not possible, it is necessary to use an implicit slow surface and obtain asymptotic representations for the slow integral manifold in an implicit form. Model examples, as well as examples borrowed from combustion theory, are treated. Chapter 5: Singular Singularly Perturbed Systems -- In this chapter we consider singularly perturbed differential systems whose degenerate equations have an isolated but not simple solution. In that case, the standard theory to establish a slow integral manifold near this solution does not work. Applying scaling transformations and using the technique of gauge functions we reduce the original singularly perturbed problem to a regularized one such that the existence of slow integral manifolds can be established by means of the standard theory. We illustrate the method by several examples from control theory and chemical kinetics. Chapter 6: Reduction Methods for Chemical Systems -- Many systems studied in chemical kinetics can be posed as a high order nonlinear differential system with slow and fast variables. This has given an impetus to the development of methods that reduce the order of the differential systems but retain a desired degree of accuracy. This research has led to a rapidly expanding volume of papers devoted to reduction methods. All these methods are connected with the integral manifold method in one way or another. These connections were clearly given by \textit{H. G. Kaper} and \textit{T. J. Kaper} [Physica D 165, No. 1--2, 66--93 (2002; Zbl 1036.80007)], which also gives a good overview of reduction methods in chemical kinetics. In this chapter we will use results given previously in parallel with our interpretation of the connection hetween the two most often used reduction methods and demonstrate that both slow integral manifolds which were described in Chapter 5. Chapter 7: Specific Cases -- 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. Chapter 8: Canards and Black Swans -- The chapter is devoted to the investigation of the relationship between slow integral manifolds of singularly perturbed differential equations and critical phenomena in chemical kinetics. We consider different problems e.g., laser models, classical combustion models and gas combustion in a dust-laden medium models, 3-D autocatalator model, using the techniques of canards and black swans. The existence of canard cascades is stated for the van der Pol model and models of the Lotka-Volterra type. The language of singular perturbations seems to apply to all critical phenomena even in the most disparate chemical systems. Chapter 9: Appendix: Proofs -- In this chapter we give the proof of a number of assertions from previous chapters. The level of mathematical sophistication required of the reader is higher than earlier chapters. For this reason, and if the readers' primary interest is in the application of the techniques, this chapter may be skipped.
scientific article; zbMATH DE number 6340730