Frequency Methods in Oscillation Theory

As mentioned in the Preface, the starting point for the development of frequency methods of investigating nonlinear oscillations is the qualitative theory of two-dimensional dynamical systems and such elements of absolute stability theory as the Popov method of a priori integral estimates and frequency theorems on solvability and properties of solutions of quadratic matrix inequalities.

The properties of stability and dichotomy of the nonlinear systems under consideration are of interest for us because they eliminate the existence of bounded solutions not tending to equilibrium in such systems, in particular, cycles.

As we have already noted in §1.1, by a multidimensional analogue of the van der Pol equation we mean a dynamical system with a minimal global attractor, containing a cycle and a unique Lyapunov unstable state of equilibrium. Various concrete three — dimensional systems possessing this property have been studied in the classical works of K.O.Friedrichs [123], L.L.Rauch [317], B.V.Shirokorad [334] and others.

The mathematical concepts of auto-oscillation and a self-oscillating system go back to the works of the A. A. Andronov school [11]. A. A. Andronov was the first to connect the property of nonlinear dynamical systems of generating undamped oscillations with the concept of a Poincaré cycle. A cycle, i.e. a closed phase trajectory stable in the large or in the small, is the mathematical image with which the concept of auto-oscillation is usually associated.

All frequency criteria for the existence of cycles of the second kind and circular motions of dynamical systems with cylindrical phase space having equilibria are based on various comparison systems.

The present chapter is devoted to the study of the existence of cycles of the first and second kind of the third-order equation and its various multidimensional analogue. Hera α and β are positive numbers, and φ(σ) is a Δ—periodic function.

All the results given in this chapter are in one way or another with a conjecture of M. A. Aizerman. Let us recall its essence. To begin with, we consider along with the linear system

The physical realizability of a cycle depends on its stability. Therefore the theory of local stability of closed trajectories [109, 259, 277, 301] operating in terms of multiplicators of equations in variations was formulated long ago. At the begining of the 60-s G.Borg, P.Hartman and C.Olech [73, 142, 143] suggested a new approach to the investigation of orbital stability which, firstly, is not restricted by closed trajectories and, secondly, is much more effective in performing computational procedures. The further development of this approach [66, 201, 205, 208, 209, 210, 212] led to the understanding that it has much in common with the proof of upper estimates of Hausdorff dimension of attractors [21–23, 112, 148–150, 152, 340, 352]. Therefore, it is by no means accidental that the application of these methods to the problem of stability in the large gives a remarkably similar results. Mutual penetration of the present directions helped to formulate a point of view on the Hausdorff measure of a compact set, represented along trajectories, as an analogue of the Lyapunov function. One of the main results obtained was the introduction of a Lyapunov function into estimates of the Hausdorff dimension of attractors [214, 215], which made it possible to generalize well-known theorems of Douady and Oesterlé [112] and Smith [340]. Application of the frequency theorem of Yakubovich and Kalman increases the effectiveness of estimates of dimension and stability. This chapter is devoted to a consistent presentation of these problems.