Constructive methods for investigating limit cycles of autonomous systems of second order (numerical-algebraic approach) (Q2839168)

The monograph deals with the problem of the investigation of limit cycles of smooth autonomous systems of ODEs. The main focus is on the case of planar systems, however, systems with cylindrical phase space are considered as well. Classical methods of the qualitative theory of ODEs and the theory of bifurcations are discussed, however, the emphasis is on the presentation of new constructive methods developed recently by A. A. Grin in collaboration with K. R. Schneider under scientific guidance of the renowned expert in the field of the theory of limit cycles of planar autonomous systems of ODEs, L. A. Cherkas. The methods allows one in some cases to find the precise number of limit cycles and their location as well as to determine the stability and multiplicity of limit cycles, which is crucial for understanding the phase portrait of planar systems of ODEs (the problem is directly related to the second part of Hilbert's 16th problem, which is the problem of determining the number of limit cycles and finding their localization in the case of polynomial systems). The new approach is presented systematically and illustrated by many examples, which confirm its efficiency.NEWLINENEWLINEThe book consists of six chapters. The first chapter lays the groundwork of the qualitative theory of ODEs and the theory of bifurcations. Main notions and facts are presented but most proofs are omitted. The 16th Hilbert problem and its simplified versions are discussed.NEWLINENEWLINEThe second chapter is devoted to the generalization of Dulac's criterion. A modified Dulac function, called the Dulac-Cherkas function, is introduced and algebraic aspects of its construction are discussed. A regular approach for the construction of such a function is presented (it is based on a reduction to a problem of linear programming).NEWLINENEWLINEIn the third chapter, the notion of the function of limit cycles (called also the Andronov-Hopf function) is introduced and methods of its construction are presented. Extrema of the function are studied making use of the Dulac and Poincaré functions. The possibility to use the Andronov-Hopf function to study limit cycles of families of autonomous systems depending on one parameter is discussed.NEWLINENEWLINEVarious methods to construct the Dulac-Cherkas function are presented in the fourth chapter. In particular, the spline approximation is used for constructing Dulac-Cherkas functions for both polynomial and non-polynomial systems. Methods to find such functions as a product of functions and using the critical points of some extrema are discussed as well.NEWLINENEWLINEIn the fifth chapter, a numerical method for constructing the curves of limit cycles of multiplicity two and three is described. It is shown how to use the Dulac and Poincaré functions to determine the multiplicity of a limit cycle.NEWLINENEWLINEFinally, in the six chapter, the developed theory is applied to study limit cycles of some families of autonomous polynomial systems in the plane and on the cylinder. In particular, the Kukles system, the Abel equation, the Liénard system with small parameters and systems admitting relaxation oscillations are studied.NEWLINENEWLINETo summarize, this interesting book presents the reader recent advances in the methods of precise estimation and localization of limit cycles of two-dimensional autonomous systems of ODEs. It is also suitable as a textbook for a graduate course in the theory of limit cycles of such systems.