Preface |
Preliminaries / 1: |
Introduction / 1.1: |
Nonlinear Continuous Systems / 1.2: |
General equations of nonlinear dynamics / 1.2.1: |
Perturbed motion equations / 1.2.2: |
Definitions of Stability / 1.3: |
Scalar, Vector and Matrix-Valued Liapunov Functions / 1.4: |
Auxiliary scalar functions / 1.4.1: |
Comparison functions / 1.4.2: |
Vector Liapunov functions / 1.4.3: |
Matrix-valued metafunction / 1.4.4: |
Comparison Principle / 1.5: |
Liapunov-Like Theorems / 1.6: |
Matrix-valued function and its properties / 1.6.1: |
A version of the original theorems of Liapunov / 1.6.2: |
Advantages of Cone-Valued Liapunov Functions / 1.7: |
Stability with respect to two measures / 1.7.1: |
Stability analysis of large scale systems / 1.7.2: |
Liapunov's Theorems for Large Scale Systems in General / 1.8: |
Why are matrix-valued Liapunov functions needed? / 1.8.1: |
Stability and instability of large scale systems / 1.8.2: |
Notes / 1.9: |
Qualitative Analysis of Continuous Systems / 2: |
Nonlinear Systems with Mixed Hierarchy of Subsystems / 2.1: |
Mixed hierarchical structures / 2.2.1: |
Hierarchical matrix function structure / 2.2.2: |
Structure of hierarchical matrix function derivative / 2.2.3: |
Stability and instability conditions / 2.2.4: |
Linear autonomous system / 2.2.5: |
Examples of third order systems / 2.2.6: |
Dynamics of the Systems with Regular Hierarchy Subsystems / 2.3: |
Ikeda-Siljak hierarchical decomposition / 2.3.1: |
Hierarchical Liapunov's matrix-valued functions / 2.3.2: |
Linear nonautonomous systems / 2.3.3: |
Stability Analysis of Large Scale Systems / 2.4: |
A class of large scale systems / 2.4.1: |
Construction of nondiagonal elements of matrix-valued function / 2.4.2: |
Test for stability analysis / 2.4.3: |
Linear large scale system / 2.4.4: |
Discussion and numerical example / 2.4.5: |
Overlapping Decomposition and Matrix-Valued Function Construction / 2.5: |
Dynamical system extension / 2.5.1: |
Liapunov matrix-valued function construction / 2.5.2: |
Test for stability of system (2.5.1) / 2.5.3: |
Numerical example / 2.5.4: |
Exponential Polystability Analysis of Separable Motions / 2.6: |
Statement of the Problem / 2.6.1: |
A method for the solution of the problem / 2.6.2: |
Autonomous system / 2.6.3: |
Polystability by the first order approximations / 2.6.4: |
Integral and Lipschitz Stability / 2.7: |
Definitions / 2.7.1: |
Sufficient conditions for integral and asymptotic integral stability / 2.7.2: |
Uniform Lipschitz stability / 2.7.3: |
Qualitative Analysis of Discrete-Time Systems / 2.8: |
Systems Described by Difference Equations / 3.1: |
Matrix-Valued Liapunov Functions Method / 3.3: |
Auxiliary results / 3.3.1: |
Comparison principle application / 3.3.2: |
General theorems on stability / 3.3.3: |
Large Scale System Decomposition / 3.4: |
Stability and Instability of Large Scale Systems / 3.5: |
Auxiliary estimates / 3.5.1: |
Autonomous Large Scale Systems / 3.5.2: |
Hierarchical Analysis of Stability / 3.7: |
Hierarchical decomposition and stability conditions / 3.7.1: |
Novel tests for connective stability / 3.7.2: |
Controlled Systems / 3.8: |
Nonlinear Dynamics of Impulsive Systems / 3.9: |
Large Scale Impulsive Systems in General / 4.1: |
Notations and definitions / 4.2.1: |
Sufficient stability conditions / 4.2.2: |
Instability conditions / 4.2.4: |
Hierarchical Impulsive Systems / 4.3: |
Analytical Construction of Liapunov Function / 4.4: |
Structure of hierarchical matrix-valued Liapunov function / 4.4.1: |
Structure of the total derivative of hierarchical matrix-valued function / 4.4.2: |
Uniqueness and Continuability of Solutions / 4.5: |
On Boundedness of the Solutions / 4.6: |
Novel Methodology for Stability / 4.7: |
Stability conditions / 4.7.1: |
Applications / 4.8: |
Estimations of Asymptotic Stability Domains in General / 5.1: |
A fundamental Zubov's result / 5.2.1: |
Some estimates for quadratic matrix-valued functions / 5.2.2: |
Algorithm of constructing a point network covering boundary of domain E / 5.2.3: |
Numerical realization and discussion of the algorithm / 5.2.4: |
Illustrative examples / 5.2.5: |
Construction of Estimate for the Domain E of Power System / 5.3: |
Oscillations and Stability of Some Mechanical Systems / 5.4: |
Three-mass systems / 5.4.1: |
Nonautonomous oscillator / 5.4.2: |
Absolute Stability of Discrete Systems / 5.5: |
References / 5.6: |
Subject Index |