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I. The concept of stability and systems with constant coefficients.- § 1. Some remarks on the concept of stability.- 1.1. Existence, uniqueness, continuity.- 1.2. Stability in the sense of Lyapunov.- 1.3. Examples.- 1.4. Boundedness.- 1.5. Other types of requirements and comments.- 1.6. Stability of equilibrium.- 1.7. Variational systems.- 1.8. Orbital stability.- 1.9. Stability and change of coordinates.- 1.10. Stability of the m-th order in the sense of G. D. Birkhoff.- 1.11. A general remark and bibliographical notes.- § 2. Linear systems with constant coefficients.- 2.1. Matrix notations.- 2.2. First applications to differential systems.- 2.3. Systems with constant coefficients.- 2.4. The Routh-Hurwitz and other criteria.- 2.5. Systems of order 2.- 2.6. Nonhomogeneous systems.- 2.7. Linear resonance.- 2.8. Servomechanisms.- 2.9. Bibliographical notes.- II. General linear systems.- § 3. Linear systems with variable coefficients.- 3.1. A theorem of Lyapunov.- 3.2. A proof of (3.1.i).- 3.3. Boundedness of the solutions.- 3.4. Further conditions for boundedness.- 3.5. The reduction to L-diagonal form and an outline of the proofs of (3.4. iii) and (3.4. iv).- 3.6. Other conditions.- 3.7. Asymptotic behavior.- 3.8. Linear asymptotic equilibrium.- 3.9. Systems with variable coefficients.- 3.10. Matrix conditions.- 3.11. Nonhomogeneous systems.- 3.12. Lyapunov’s type numbers.- 3.13. First application of type numbers to differential equations.- 3.14. Normal systems of solutions.- 3.15. Regular differential systems.- 3.16. A relation between type numbers and generalized characteristic roots.- 3.17. Bibliographical notes.- § 4. Linear systems with periodic coefficients.- 4.1. Floquet theory.- 4.2. Some important applications.- 4.3. Further results concerning equation (4.2.1) and extensions.- 4.4. Mathieu equation.- 4.5. Small periodic perturbations.- 4.6. Bibliographical notes.- § 5. The second order linear differential equation and generalizations.- 5.1. Oscillatory and non-oscillatory solutions.- 5.2. Fubini’s theorems.- 5.3. Some transformations.- 5.4. Bellman’s and Prodi’s theorems.- 5.5. The case f(t) ? + ?.- 5.6. Solutions of class L2.- 5.7. Parseval relation for functions of class L2.- 5.8. Some properties of the spectrum S.- 5.9. Bibliographical notes.- III. Nonlinear systems.- § 6. Some basic theorems on nonlinear systems and the first method of Lyapunov.- 6.1. General considerations.- 6.2. A theorem of existence and uniqueness.- 6.3. Periodic solutions of periodic systems.- 6.4. Periodic solutions of autonomous systems.- 6.5. A method of successive approximations and the first method of Lyapunov.- 6.6. Some results of Bylov and Vinograd.- 6.7. The theorems of Bellman.- 6.8. Invariant measure.- 6.9. Differential equations on a torus.- 6.10. Bibliographical notes.- § 7. The second method of Lyapunov.- 7.1. The function V of Lyapunov.- 7.2. The theorems of Lyapunov.- 7.3. More recent results.- 7.4. A particular partial differential equation.- 7.5. Autonomous systems.- 7.6. Bibliographical notes.- § 8. Analytical methods.- 8.1. Introductory considerations.- 8.2. Method of Lindstedt.- 8.3. Method of Poincaré.- 8.4. Method of Krylov and Bogolyubov, and of van der Pol.- 8.5. A convergent method for periodic solutions and existence theorems.- 8.6. The perturbation method.- 8.7. The Liénard equation and its periodic solutions.- 8.8. An oscillation theorem for the equation (8.7.1).- 8.9. Existence of a periodic solution of equation (8.7.1).- 8.10. Nonlinear free oscillations.- 8.11. Invariant surfaces.- 8.12. Bibliographical notes.- 8.13. Nonlinear resonance.- 8.14. Prime movers.- 8.15. Relaxation oscillation.- §9. Analytical-topological methods.- 9.1. Poincaré theory of the critical points.- 9.2. Poincaré-Bendixson theory.- 9.3. Indices.- 9.4. A configuration concerning Liénard’s equation.- 9.5. Another existence theorem for the Liénard equation.- 9.6. The method of the fixed point.- 9.7. The method of M. L. Cartwright.- 9.8. The method of T. Wazewski.- IV. Asymptotic developments.- § 10. Asymtotic developments in general.- 10.1. Poincaré’s concept of asymptotic development.- 10.2. Ordinary, regular and irregular singular points.- 10.3. Asymptotic expansions for an irregular singular point of finite type.- 10.4. Asymptotic developments deduced from Taylor expansions.- 10.5. Equations containing a large parameter.- 10.6. Turning points and the theory of R. E. Langer.- 10.7. Singular perturbation.
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