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Singular Values, Exterior Calculus and Logarithmic Norms

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Attractor Dimension Estimates for Dynamical Systems: Theory and Computation

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 38))

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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.

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Notes

  1. 1.

    Suppose \(({\mathbb {E}}, (\cdot ,\cdot )_{\mathbb {E}})\) is an m-dimensional Euclidean space and \(\{u_i\}^m_{i=1}\) is an orthonormal basis of \({\mathbb {E}}\). If \(a_1 \ge a_2\ge \cdots a_m > 0\) are arbitrary positive numbers the set \(\mathcal{E}:= \{ \sum ^m_{i=1} \xi _i u_i \,|\,\sum ^m_{i=1} ({\xi _i \over a_i})^2 \le 1\}\) is called (non-degenerated) ellipsoid in \({\mathbb {E}}\) with semi-axes \(u_1, \ldots , u_m\) and length of semi-axes \(a_1, \ldots , a_m\).

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  1. Department of Applied Cybernetics, Faculty of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia

    Nikolay Kuznetsov & Volker Reitmann

  2. Faculty of Information Technology, University of Jyväskylä, Jyväskylä, Finland

    Nikolay Kuznetsov

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  1. Nikolay Kuznetsov
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Kuznetsov, N., Reitmann, V. (2021). Singular Values, Exterior Calculus and Logarithmic Norms. In: Attractor Dimension Estimates for Dynamical Systems: Theory and Computation. Emergence, Complexity and Computation, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-030-50987-3_2

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  • DOI: https://doi.org/10.1007/978-3-030-50987-3_2

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