Averaging in Stability Theory

Inhaltsverzeichnis

• 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.