Qualitative theory of differential equations

The (generally complex) quantity p plays a role analogous to the order parameter familiar from phase transitions. The fact that only one such parameter survives in the final equations illustrates the enormous reduction of degrees of freedom associated with the first bifurcation. Note also the similarity between equations (8) and (9) and the normal forms at which one arrives in the qualitative theory of differential equations in the vicinity of resonance points.3... 

Chapter 1 presents all the necessary information concerning linear algebra and the qualitative theory of differential equations in terms of which we construct and analyze kinetic models of heterogeneous catalytic reactions. 

Chapter 2 describes the evolution in fundamental concepts of chemical kinetics (in particular, that of heterogeneous catalysis) and the "prehis-tory of the problem, i.e. the period before the construction of the formal kinetics apparatus. Data are presented concerning the ideal adsorbed layer model and the Horiuti-Temkin theory of steady-state reactions. In what follows (Chapter 3), an apparatus for the modern formal kinetics is represented. This is based on the qualitative theory of differential equations, linear algebra and graphs theory. Closed and open systems are discussed separately (as a rule, only for isothermal cases). We will draw the reader s attention to the two results of considerable importance. 

In conclusion of our short excursion into the qualitative theory of differential equations, we shall discussed the often-used term "bifurcation . It is ascribed to the systems depending on some parameter and is applied to point to a fundamental reconstruction of phase portrait when a given parameter attains its critical value. The simplest examples of bifurcation are the appearance of a new singular point in the phase plane, its loss of stability, the appearance (birth) of a limit cycle, etc. Typical cases on the plane have been discussed in detail in refs. [11, 12, and 14]. For higher dimensions, no such studies have been carried out (and we doubt the possibility of this). 

To repeat the route of chemistry in the kinetic aspect , that was the formulation of the problem. To our mind, however, in the 1930s "the rational classification principle , whose appearance was predicted by Semenov, could not be realized. The possibility of solving this problem appeared only in recent times in terms of the concepts of the graph theory and the qualitative theory of differential equations. The analysis of the effect of the mechanism structure on the kinetic regularities of catalytic reactions is one of the connecting subjects in the present study. 

A.I. Vol pert, E.A. Gel man and A.N. Ivanova, Some Problems of the Qualitative Theory of Differential Equations on Graphs, Preprint, OIKhF AN SSSR, Chernogolovka, 1975 (in... 

V.V. Nemytskii and V.V. Stepanov, Qualitative Theory of Differential Equations, Gos. Izd. Teor. Tekh. Lit., Moscow, Leningrad, 1949 (in Russian) Princeton University Press, Princeton, 1960. 

Analysis of critical phenomena. Here, a working method will rapidly become the evolving qualitative theory of differential equations. It will be the basis for the elaboration of methods for the experimental search for critical phenomena. To carry out a detailed qualitative analysis of complex dynamics, special programs will be used. 

To interpret new experimental chemical kinetic data characterized by complex dynamic behaviour (hysteresis, self-oscillations) proved to be vitally important for the adoption of new general scientific ideas. The methods of the qualitative theory of differential equations and of graph theory permitted us to perform the analysis for the effect of mechanism structures on the kinetic peculiarities of catalytic reactions [6,10,11]. This tendency will be deepened. To our mind, fast progress is to be expected in studying distributed systems. Despite the complexity of the processes observed (wave and autowave), their interpretation is ensured by a new apparatus that is both effective and simple. 

QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS. V.V. Nemytskii and V.V. Stepanov. Classic graduate-level text by two prominent Soviet mathematicians covers classical differential equations as well as topological dynamics and erqodic theory. Bibliographies. 523pp. 5X x 8X. 65954-2 Pa. 10.95... 

Mathematical models of catalytic systems in the general form are rather sophisticated. Often, they consist of nonlinear systems of differential equations containing both conventional equations and equations with partial derivatives of parabolic, hyperbolic, and other forms. Efficient simulation is only possible if a well developed qualitative theory of differential equations (mainly, equations with partial derivatives) and high performance programs for computational experiments exist. 

Eventually, aU of them are based on the methods of general qualitative theory of differential equations developed by Poincare more than a century ago [47]. This theory was essentially developed by Andronov in 1930s [48] and, finally, after Hopf s theorem on bifurcation appeared in 1942 [49] it became a self-consistent branch of mathematics. This subject is currently known luider several names Poincare-Andronov s general theory of dynamic systems theory of non-linear systems theory of bifurcation in dynamic systems. Although the first notion is, in our opinion, the most exact one, we will use the term bifurcation theory , or BT, for the sake of brevity. 

It may happen that it is hard to get an exact or approximate solution either by analytical or by numerical methods. In these cases one still wants to obtain at least qualitative information about the nature of the solutions. To obtain qualitative information about the solutions without solving the equation is the aim of a branch of the theory of differential equations of the qualitative theory of differential equations. 

A more natural phenomenon seems to be the oligo-oscillation, or the overshoot-undershoot phenomenon. These expressions denote the case when there is only a finite number of local extrema on the concentration versus time functions. Natural as it is, it has rarely been studied in a well-controlled experiment (see, however, Rabai et a/., 1979). It has also rarely been studied from the theoretical point of view. This situation can be explained by the fact that the qualitative theory of differential equations usually makes statements on long-range behaviour and much less on transient behaviour. The only exception seems to be that Pota (1981) has given a complete proof of the statement called Jost s theorem which says in a closed reversible compart-mental system of M components none of the concentrations can have more than M - 2 strict extrema. The methods used by Pota makes it possible to extend this result (see Problem 6 below). Another result of this type, relating nonlinear kinetic differential equations, can also be found among the Problems. 

This question has a certain mathematical interest, as gradient systems are relatively easy to deal with from the point of view of exotic phenomena (or, to put it in a more mathematical way, from the point of view of the qualitative theory of differential equations) in general (see, for example, Hirsch Smale, 1974) and from the point of view of catastrophe theory in particular (Thom, 1975, p. 55). 

Farkas, M. (ed.) (1981). Qualitative theory of differential equations. Coll. Math. Soc. Janos Bolyai 30, Vols. I, II. North-Holland, Amsterdam. 

In the qualitative theory of differential equations, especially in the so called "catastrophe theory", it has turned out that differential equations of the gradient type are relatively easy to deal with /Thom, 1975/- Eurthermore - and this may prove more relevant - it has been proposed sometimes that gradient systems are only worth studying in thermodynamics. /Gyarmati, 1961a, b Edelen, 1973/. Easy treatment and physical relevance has also come up in connection with equations with other kinds of symmetries too in connection with Hamiltonian systems and systems having similar but different specialities. 

Finally, we remark that the reader may find deeper insights to the above issues in the book Dynamical Systems by Birkhoff [31] and in the book Qualitative Theory of Differential Equations by Nemytskii and Stepanov [98]. 

Bykov, V. V. [1978] On the structure of a neighborhood of a separatrix contour with a saddle-focus, in Methods of Qualitative Theory of Differential Equation (Gorky Gorky State University), 3-32 [1980] On bifurcations of dynamical systems with a separatrix contour containing a saddle-focus, ibid. 44-72 [1988] On the birth of a non-trivial hyperbolic set from a contour formed by separatrices of a saddle, ibid. 22-32. 

Gavrilov, N. K. and Roshchin, N. V. [1983] On the stability of an equilibrium with one zero and a pair of pure imaginary eigenvalues, in Methods of Qualitative Theory of Differential Equations Leontovich-Andronova, ed., (Gorky State University Gorky), 41-49. 

Gavrilov, N. K. and Shilnikov, A. L. [1996] On a blue sky catastrophe model, Proc. Int. Conf. Comtemp. Problems of Dynamical Systems Theory, ed. Lerman, L. (Nizhny Novgorod State University Nizhny Novgorod). [1999] An Example of blue sky catastrophe, in Ams Transl Series II. Methods of qualitative theory of differential equations and related topics. (AMS, Providence, Rhode Island). 

Nemytskii, V. V. and Stepanov, V. V. [1960] Qualitative Theory of Differential Equations (Princeton, N.J., Princeton University Press. Series title Princeton Mathematical Series 22). 

Turaev, D. V. [1984] On a case of bifurcations of a contour composed by two homoclinic curves of a saddle, in Methods of Qualitative Theory of Differential Equations (Gorky State University), 162-175. 

Byragov, V. S. [1987] Bifurcations in a two-parameter family of conservative mappings that are close to the Henon mapping , in Methods of the Qualitative Theory of Differential Equations (Gorky Gorky State Univ. Press), 10-24. 

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