Frequency Methods in Oscillation Theory - A. I. Shepeljavyi

. .

Résumé :
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 Poincar?-Bendixson principle.- ?3.9. A frequency realization of the generalized Poincar?-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 Poincar? stability of cycles of the second kind.- ?8.6. Frequency estimates for the Hausdorff dimension and conditions for global asymptotic stability....

Sommaire:
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 Poincar?-Bendixson principle.- ?3.9. A frequency realization of the generalized Poincar?-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 Poincar? stability of cycles of the second kind.- ?8.6. Frequency estimates for the Hausdorff dimension and conditions for global asymptotic stability.