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AuthorLeonov, G. A. author
TitleFrequency Methods in Oscillation Theory [electronic resource] / by G. A. Leonov, I. M. Burkin, A. I. Shepeljavyi
ImprintDordrecht : Springer Netherlands, 1996
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Descript XII, 404 p. online resource


1. Classical two-dimensional oscillating systems and their multidimensional analogues -- ยง1.1. The van der Pol equation -- ยง1.2. The equation of oscillations of a pendulum -- ยง1.3. Oscillations in two-dimensional systems with hysteresis -- ยง1.4. 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 -- ยง2.1. Frequency criteria for stability and dichotomy -- ยง2.2. Theorems on solvability and properties of special matrix inequalities -- 3. Multidimensional analogues of the van der Pol equation -- ยง3.1. Dissipative systems. Frequency criteria for dissipativity -- ยง3.2. Second-order systems. Frequency realization of the annulus principle -- ยง3.3. Third-order systems. The torus principle -- ยง3.4. The main ideas of applying frequency methods for multidimensional systems -- ยง3.5. The criterion for the existence of a periodic solution in a system with tachometric feedback -- ยง3.6. The method of transition into the "space of derivatives" -- ยง3.7. A positively invariant torus and the function "quadratic form plus integral of nonlinearity" -- ยง3.8. The generalized Poincarรฉโ{128}{147}Bendixson principle -- ยง3.9. A frequency realization of the generalized Poincarรฉ-Bendixson principle -- ยง3.10. Frequency estimates of the period of a cycle -- 4. Yakubovich autoโ{128}{147}oscillation -- ยง4.1. Frequency criteria for oscillation of systems with one differentiable nonlinearity -- ยง4.2. Examples of oscillatory systems -- 5. Cycles in systems with cylindrical phase space -- ยง5.1. The simplest case of application of the nonlocal reduction method for the equation of a synchronous machine -- ยง5.2. Circular motions and cycles of the second kind in systems with one nonlinearity -- ยง5.3. The method of systems of comparison -- ยง5.4. Examples -- ยง5.5. Frequency criteria for the existence of cycles of the second kind in systems with several nonlinearities -- ยง5.6. Estimation of the period of cycles of the second kind -- 6. The Barbashin-Ezeilo problem -- ยง6.1. The existence of cycles of the second kind -- ยง6.2. Bakaev stability. The method of invariant conical grids -- ยง6.3. The existence of cycles of the first kind in phase systems -- ยง6.4. 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 -- ยง7.1. The existence of periodic solutions of systems with nonlinearity from a Hurwitzian sector -- ยง7.2. Necessary conditions for global stability in the critical case of two zero roots -- ยง7.3. Lemmas on estimates of solutions in the critical case of one zero root -- ยง7.4. Necessary conditions for absolute stability of nonautonomous systems -- ยง7.5. 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 -- ยง8.1. Upper estimates of the Hausdorff measure of compact sets under differentiable mappings -- ยง8.2. Estimate of the Hausdorff dimension of attractors of systems of differential equations -- ยง8.3. Global asymptotic stability of autonomous systems -- ยง8.4. Zhukovsky stability of trajectories -- ยง8.5. A frequency criterion for Poincarรฉ stability of cycles of the second kind -- ยง8.6. Frequency estimates for the Hausdorff dimension and conditions for global asymptotic stability

Mathematics Fourier analysis Global analysis (Mathematics) Manifolds (Mathematics) Differential equations Applied mathematics Engineering mathematics Mathematics Ordinary Differential Equations Fourier Analysis Global Analysis and Analysis on Manifolds Applications of Mathematics


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