The linear theory of oscillations traditionally operates with frequency representa tions based on the concepts of a transfer function and a frequency response. The universality of the critria of Nyquist and Mikhailov and the simplicity and obvi ousness of the application of frequency and amplitude - frequency characteristics in analysing forced linear oscillations greatly encouraged the development of practi cally important nonlinear theories based on various forms of the harmonic balance hypothesis [303]. Therefore mathematically rigorous frequency methods of investi gating nonlinear systems, which appeared in the 60s, also began to influence many areas of nonlinear theory of oscillations. First in this sphere of influence was a wide range of problems connected with multidimensional analogues of the famous van der Pol equation describing auto oscillations of generators of various radiotechnical devices. Such analogues have as a rule a unique unstable stationary point in the phase space and are Levinson dis sipative. One of the pioneering works in this field, which started the investigation of a three-dimensional analogue of the van der Pol equation, was K. O. Friedrichs's paper [123]. The author suggested a scheme for constructing a positively invariant set homeomorphic to a torus, by means of which the existence of non-trivial periodic solutions was established. That scheme was then developed and improved for dif ferent classes of multidimensional dynamical systems [131, 132, 297, 317, 334, 357, 358]. The method of Poincare mapping [12, 13, 17] in piecewise linear systems was another intensively developed direction.

Inhalt

Classical two-dimensional oscillating systems and their multidimensional analogues.- §1.1. The van der Pol equation.- §1.2. The equation of oscillations of a pendulum.- §1.3. Oscillations in two-dimensional systems with hysteresis.- §1.4. Lower estimates of the number of cycles of a two-dimensional system.- 2. Frequency criteria for stability and properties of solutions of special matrix inequalities.- §2.1. Frequency criteria for stability and dichotomy.- §2.2. Theorems on solvability and properties of special matrix inequalities.- 3. Multidimensional analogues of the van der Pol equation.- §3.1. Dissipative systems. Frequency criteria for dissipativity.- §3.2. Second-order systems. Frequency realization of the annulus principle.- §3.3. Third-order systems. The torus principle.- §3.4. The main ideas of applying frequency methods for multidimensional systems.- §3.5. The criterion for the existence of a periodic solution in a system with tachometric feedback.- §3.6. The method of transition into the "space of derivatives".- §3.7. A positively invariant torus and the function "quadratic form plus integral of nonlinearity".- §3.8. The generalized PoincaréBendixson principle.- §3.9. A frequency realization of the generalized Poincaré-Bendixson principle.- §3.10. Frequency estimates of the period of a cycle.- 4. Yakubovich autooscillation.- §4.1. Frequency criteria for oscillation of systems with one differentiable nonlinearity.- §4.2. Examples of oscillatory systems.- 5. Cycles in systems with cylindrical phase space.- §5.1. The simplest case of application of the nonlocal reduction method for the equation of a synchronous machine.- §5.2. Circular motions and cycles of the second kind in systems with one nonlinearity.- §5.3. The method ofsystems of comparison.- §5.4. Examples.- §5.5. Frequency criteria for the existence of cycles of the second kind in systems with several nonlinearities.- §5.6. Estimation of the period of cycles of the second kind.- 6. The Barbashin-Ezeilo problem.- §6.1. The existence of cycles of the second kind.- §6.2. Bakaev stability. The method of invariant conical grids.- §6.3. The existence of cycles of the first kind in phase systems.- §6.4. A criterion for the existence of nontrivial periodic solutions of a third-order nonlinear system.- 7. Oscillations in systems satisfying generalized Routh-Hurwitz conditions. Aizerman conjecture.- §7.1. The existence of periodic solutions of systems with nonlinearity from a Hurwitzian sector.- §7.2. Necessary conditions for global stability in the critical case of two zero roots.- §7.3. Lemmas on estimates of solutions in the critical case of one zero root.- §7.4. Necessary conditions for absolute stability of nonautonomous systems.- §7.5. The existence of oscillatory and periodic solutions of systems with hysteretic nonlinearities.- 8. Frequency estimates of the Hausdorff dimension of attractors and orbital stability of cycles.- §8.1. Upper estimates of the Hausdorff measure of compact sets under differentiable mappings.- §8.2. Estimate of the Hausdorff dimension of attractors of systems of differential equations.- §8.3. Global asymptotic stability of autonomous systems.- §8.4. Zhukovsky stability of trajectories.- §8.5. A frequency criterion for Poincaré stability of cycles of the second kind.- §8.6. Frequency estimates for the Hausdorff dimension and conditions for global asymptotic stability.