Frequency Methods in Oscillation Theory

1. 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 Poincaré–Bendixson principle.- §3.9. A frequency realization of the generalized Poincaré-Bendixson principle.- §3.10. Frequency estimates of the period of a cycle.- 4. Yakubovich auto–oscillation.- §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.