Averaging in Stability Theory

1 Averaging of Ordinary Differential Equations in One — and Multi — Frequency Systems.- 1.1 Introduction.- 1.2 Averaging in Standard — Form Systems.- 1.3 Averaging in Systems with Slow and Fast Variables.- 1.4 Averaging in Multi — Frequency Systems.- 2 Generalization of Lyapunov Second Method and Averaging in Stability Theory.- 2.1 Introduction.- 2.2 Lyapunov Functions Positive — Definite in Subset of Variables.- 2.3 Equipotential Surfaces of Perturbed Lyapunov Function. Proximity of Solutions of Complete and Unperturbed Systems.- 2.4 Perturbed Lyapunov Function in Annular Region. Theorem on Stability.- 2.5 Theorem on Attraction of Solutions to Equilibrium Point.- 2.6 Investigation of Stability on Finite Interval.- 2.7 Investigation of Stability in Higher Approximations.- 2.8 Theorem on Asymptotic Stability of Perturbed Nonlinear Systems in Neutral Case.- 2.9 Theorems on Asymptotic Stability of Standard — Form Systems and Systems with Small Perturbations.- 2.10 Theorem on Stability of Systems Splitting without Perturbations.- 2.11 Stability of Systems with Additional Correlations between Properties of Mean and Derivative of Lyapunov Function.- 2.12 Investigation of Stability by Averaging over Explicit Time Dependence.- 2.13 Investigation of Stability by Averaging along Solutions of Linear System.- 2.14 Investigation of Stability of Integro — Differential Systems.- 2.15 On Numerical Realization of Theorems of Generalized Lyapunov Second Method.- 2.16 Theorems on Instability.- 2.17 Study of Stability of Perturbed Systems Using Positive — Definite Function which is not Lyapunov Function.- 3 Stability of Systems of Ordinary Differential Equations with Quasi — Periodic Coefficients.- 3.1 Investigation of Stability by Means of Lyapunov Function of LinearSystem.- 3.2 Construction of Perturbed Lyapunov Function for Higher Order Resonances.- 4 Stability of Multi — Frequency Systems 114.- 4.1 Statement of the Problem.- 4.2 Stability of Single — Frequency Systems of Equations with Asymptotically Stable Averaged System.- 4.3 Stability of Multi — Frequency Systems of Equations with Asymptotically Stable Averaged System.- 4.4 Stability of Multi — Frequency Systems on Finite Time Interval.- 4.5 Stability of Multi — Frequency Problems of Nonlinear Mechanics.- 5 Stability of Orbits in Three — Body Problem.- 5.1 Orbit Stability in Three — Body Problem and Description of the Models.- 5.2 Canonical Variable, Equations and Integrals of Motion in the Point-like Three — Body Problem.- 5.3 Resonance Curves and Choice of New Variables.- 5.4 Construction of Perturbed Lyapunov Function and Stability of Point — Like Model of Three — Body Problem.- 5.5 Corrections to Force Function in Hydrodynamic Model of Planets.- 5.6 Theorem on Stability of Planetary Systems.- 5.7 Evolution of Planetary Orbits.- 6 Stability of Systems with Admissible Region of Motion. Stability of Gyroscope with No — Contact Suspension.- 6.1 Estimation of Region of Motion of the System.- 6.2 Stability of Systems with Known Region of Motions.- 6.3 Stability of Multi — Frequency Systems with Known Region of Motions.- 6.4 Stability of Gyroscope with No — Contact Suspension.- 7 Averaging and Stability in Systems of Equations with Delay.- 7.1 Averaging in Systems with Delay.- 7.2 Stability of Systems with Deviating Argument.- 7.3 Stability in Multi — Frequency Systems with Delay.- 7.4 Effect of Variable Tide Delay on the Evolution of Orbital Elements of a Tide - Forming Body.- 8 Stability of Partial Differential Equations.- 8.1 Statement of theProblem.- 8.2 Theorem on Stability.- 8.3 Theorem on Stability over Finite Interval.- 8.4 Theorem on Instability.- 8.5 Stability of Some Hyperbolic Systems.- 8.6 Stability of Nonlinear Evolutionary Differential Equation with Perturbation.- 9 Stability of Stable System Influenced by Small Random Perturbations.- 9.1 Construction of Perturbations of Lyapunov Functions under Small Random Perturbations.- 9.2 Averaging in Some Stochastic Systems.