Averaging in Stability Theory, 1992 A Study of Resonance Multi-Frequency Systems Mathematics and its Applications Series, Vol. 79

This volume presents a generalization of the second Lyapunov method involving its combination with the asymptotic averaging method. This generalized method can be applied to multifrequency systems having resonance harmonics. A new method is also described for estimating small denominators in multifrequency systems which makes use of the nonlinear properties of the system frequencies. The methods derived can also be extended to integro-differential equations, delay differential equations, and to partial differential equations having small nonlinear terms. One of the various applications relating to multifrequency and resonance problems which are discussed in this book is the stability of the classical three-body problem. For specialists in stability theory, nonlinear oscillation theory and asymptotic methods in mathematics and celestial mechanics.

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.