Angular values of nonautonomous and random linear dynamical systems. I: Fundamentals. (English) Zbl 1498.37040
Summary: We introduce the notion of angular values for deterministic linear difference equations and random linear cocycles. We measure the principal angles between subspaces of fixed dimension as they evolve under nonautonomous or random linear dynamics. The focus is on long-term averages of these principal angles, which we call angular values: we demonstrate relationships between different types of angular values and prove their existence for random dynamical systems. For one-dimensional subspaces in two-dimensional systems our angular values agree with the classical theory of rotation numbers for orientation-preserving circle homeomorphisms if the matrix has positive determinant and does not rotate vectors by more than \(\frac{\pi}{2} \). Because our notion of angular values ignores orientation by looking at subspaces rather than vectors, our results apply to dynamical systems of any dimension and to subspaces of arbitrary dimension. The second part of the paper delves deeper into the theory of the autonomous case. We explore the relation to (generalized) eigenspaces, provide some explicit formulas for angular values, and set up a general numerical algorithm for computing angular values via Schur decompositions.
MSC:
| 37C60 | Nonautonomous smooth dynamical systems |
| 37H10 | Generation, random and stochastic difference and differential equations |
| 37A05 | Dynamical aspects of measure-preserving transformations |
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
| 39A06 | Linear difference equations |
| 65Q10 | Numerical methods for difference equations |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
Keywords:
nonautonomous dynamical systems; random dynamical systems; angular value; ergodic average; principal angles of subspaces; numerical algorithmReferences:
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