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Attractor Dimension Estimates for Dynamical Systems: Theory and ComputationSingular Values, Exterior Calculus and Logarithmic Norms

Attractor Dimension Estimates for Dynamical Systems: Theory and Computation: Singular Values,... [Global stability and dimension properties of nonlinear differential equations essentially depend on the contraction properties of k-parallelopipeds or k-ellipsoids under the flow of the associated variational equations. The goal of this second chapter is to develop some elements of multilinear algebra for the investigation of linear differential equations. This includes the discussion of singular value inequalities for linear operators in finite-dimensional spaces, the Fischer-Courant theorem as an extremal principle for eigenvalues of Hermitian matrices, exterior powers of operators and spaces, the logarithmic norm calculation and the use of the Kalman-Yakubovich frequency theorem for the effective estimation of time-dependent singular values of the solution operator to linear differential equations. The Kalman-Yakubovich frequency theorem is also used to get sufficient conditions for convergence in dynamical systems.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Attractor Dimension Estimates for Dynamical Systems: Theory and ComputationSingular Values, Exterior Calculus and Logarithmic Norms

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Publisher
Springer International Publishing
Copyright
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
ISBN
978-3-030-50986-6
Pages
41 –93
DOI
10.1007/978-3-030-50987-3_2
Publisher site
See Chapter on Publisher Site

Abstract

[Global stability and dimension properties of nonlinear differential equations essentially depend on the contraction properties of k-parallelopipeds or k-ellipsoids under the flow of the associated variational equations. The goal of this second chapter is to develop some elements of multilinear algebra for the investigation of linear differential equations. This includes the discussion of singular value inequalities for linear operators in finite-dimensional spaces, the Fischer-Courant theorem as an extremal principle for eigenvalues of Hermitian matrices, exterior powers of operators and spaces, the logarithmic norm calculation and the use of the Kalman-Yakubovich frequency theorem for the effective estimation of time-dependent singular values of the solution operator to linear differential equations. The Kalman-Yakubovich frequency theorem is also used to get sufficient conditions for convergence in dynamical systems.]

Published: Jul 2, 2020

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