This book is devoted to the influence of parametric perturbations and uncertainties on the Lyapunov exponents of discrete linear systems with timevarying coefficients. Different types of perturbations are considered, limited in terms of certain norms, tending to zero with a specified rate. For each of them, we describe their impact on the values of Lyapunov exponents. Results of such considerations, as presented in this book, while being widely known for stationary systems, are new and original when referred to the systems with variable coefficients. The book is addressed primarily to persons interested in the theory of Lyapunov exponents, which well characterize the behavior of the trajectory of time-varying systems. It is also important that the results of this work can be used by graduate students and researchers in new areas of research in dynamics of discrete systems. Analysis of discrete-time systems is also important from cognitive point of view and well established part of control theory and applied computer science.
- Contents
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1 Definitions of characteristic exponents and basic properties 1
2 Bounded perturbations 7
2.1 Stability of Lyapunov exponents 7
2.2 Two-dimensional system 11
2.3 Generalized spectral radius and subradius 18
2.4 Time-invariant system 20
3 Arbitrary small perturbations 23
3.1 Central exponents of families of sequences 24
3.2 Central exponents of linear system. Definitions and basic properties 29
3.3 Central exponents of triangular system 32
3.4 Relations between central and Lyapunov exponents 34
4 Regular systems and regularity coefficients 41
4.1 Coefficients of regularity and basic properties of regular systems 41
4.2 Grobman perturbation 44
4.3 Grobman perturbation for diagonal system 45
4.4 Characterization of regularity in terms of nonhomogeneous system 53
5 Perturbations exponentially tending to zero 61
5.1 Definitions and basic properties of exponential exponent 62
5.2 Formula for the Exponential Exponents 66
5.3 Basic properties of sigma exponents 70
5.4 Relations between sigma exponents and Lyapunov exponents 73
Contents
6 Stability by the Linear Approximation 77
6.1 Preliminaries 77
6.2 Perron examples 78
6.3 Conditions for exponential stability 80
7 Perron exponents 85
7.1 Numbers of Perron exponents 86
7.2 Formula for Perron exponent 89
7.3 Bounds for Lyapunov and Perron Exponents 93
8 Conclusion 99
References 103
