The main topic of the dissertation is the stability of a crtical point at the origin in autonomous homogeneous quadratic systems of ordinary differential equations. The work is divided i nine ...chapters. In the first chapter the autonomous homogeneous quadratic systems of ordinary differential equations (quadratic systems in the sequel) are defined. Some relations between quadratic systems and commutative (in general monassociative) algebras, due toa german mathematician Markus [14], are described. In this chapter one can also find how to make an arbitrary polynomial sytem to be a autonomous homogeneous quadratic system. The procedures are called the autonomization, the homogenization and the quadratization, respectively. In the second chapter the stability in general systems in the sense of Lyapunov is considered. The only general theorem on stability, related with Markus theory which concerns the quadratic systems and the corresponding algebras, is resumed. The theorem states, that a quadratic system with a stable origin corresponds to an algebra, which contain no idempotents, but does contain some nilpotents of rank two. This theorem is frequently used as a necessary condition for stability of the origin in quadratic systems. In this chapter also some main theorems, concerning the linearization around the considered critical point of a nonlinear systems, are resumed. The basic information on perturbations of a Hamiltonian system is also included in this chapter. The reminder of the present dissertation is based on several original papers [18, 19, 20, 21, 22] and [16]. The linearization of a nonlinear system around some particular critical point helps ones to consider the stability in the hyperbolic critical point. It is well known, that the stability in this case depends on the spectrum of the corresponding Jacobian. In the third chapter two kinds of nonhyperbolic critical points are defined: the nonhyperbolic critical points of type 1 or of type 2; the corresponding Jacobian being nonzero or zero, respectively. Obviously, the origin in homogeneous quadratic systems always is a nonhyperbolic critical point of type 2. In the third chapter, which is totally based on original papers [16] and [18], the stability of nonhyperbolic critical points of type 1 in nonhomogeneous quadratic systems is considered. In the next chapters the stability in the nonhyperbolic critical points of type 2 is considered. In the fourth chapter we consider the case of dimension two. In the following chapters, however, we consider the case of R3 and of Rn for an arbitrary n. Irrespective of the dimension of the system, the origin can be stable only if the system contains a line of critical points. The stability analysis of all systems in R3 seems to be to complicated, thus we restrict our intention to a slightly simpler problem. We consider the stability of the origin in the systems which contain a plane of critical points. Thus, in the sixth chapter we classify the algebras corresponding to systems with a plane of critical points. The classification is based on the original paper [21]. In the seventh chapter a case-by-case stability analysis is performed. It is based on the original paper [22]. In the eighth chapter (see [20]) we seek for the algebraic properties which would characterize the stability of the origin in a quadratic system with a plane of critical point in R3. We are focussed on the spectra and the Jordan form of the matrix corresponding to the (left) multiplication by same particular nilpotents, which we call diagonal essential nilpotents. The second important algebraic property which characterizes the stability in these systems is related to the existence of a nontrivial idempotent in the complexification of the corresponding (real) algebra. The theorem on stability via algebraic properties can be proved for n=3. In the case of general n=3 the statement of the theorem can easily be amended, so in the ninth chapter we offer it as a conjecture. The axamples andtheorems which are supporting the validity of the conjecture are also taken from the the original paper [20]
Type of material - dissertation
; adult, serious
Publication and manufacture - Maribor : [M. Mencinger], 2003
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