# Angular Values of Nonautonomous and Random Linear Dynamical Systems: Part I - Fundamentals

@article{Beyn2022AngularVO, title={Angular Values of Nonautonomous and Random Linear Dynamical Systems: Part I - Fundamentals}, author={Wolf-J{\"u}rgen Beyn and Gary Froyland and Thorsten H{\"u}ls}, journal={SIAM J. Appl. Dyn. Syst.}, year={2022}, volume={21}, pages={1245-1286} }

. 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…

## Figures and Tables from this paper

## One Citation

Angular values of linear dynamical systems and their connection to the dichotomy spectrum

- Mathematics
- 2020

This work continues the investigation of the previously defined angular values (cf. [6]) for linear nonautonomous dynamical systems in finite dimensions. The s-th angular value measures the maximal…

## References

SHOWING 1-10 OF 38 REFERENCES

Angular values of linear dynamical systems and their connection to the dichotomy spectrum

- Mathematics
- 2020

This work continues the investigation of the previously defined angular values (cf. [6]) for linear nonautonomous dynamical systems in finite dimensions. The s-th angular value measures the maximal…

Semi-uniform ergodic theorems and applications to forced systems

- Mathematics
- 2000

In nonlinear dynamics an important distinction exists between uniform bounds on growth rates, as in the definition of hyperbolic sets, and non-uniform bounds as in the theory of Liapunov exponents.…

Dichotomy spectra of triangular equations

- Mathematics
- 2015

Without question, the dichotomy spectrum is a central tool in the stability, qualitative and geometric theory of nonautonomous dynamical systems. In this context, when dealing with time-variant…

A multiplicative ergodic theorem for rotation numbers

- Mathematics
- 1989

Given a vector fieldX on a Riemannian manifoldM of dimension at least 2 whose flow leaves a probability measureμ invariant, the multiplicative ergodic theorem tells us thatμ-a.s. every tangent vector…

Exponential dichotomy and rotation number for linear Hamiltonian systems

- Mathematics
- 1994

Abstract Let x ′ = H ( t ) x be a non-autonomous Hamiltonian linear differential system. We show that if the rotation number α(λ) for the associated family of systems x ′ = ( H ( t ) + λ J γ( t )) x…

Lyapunov and Sacker–Sell Spectral Intervals

- Mathematics
- 2007

In this work, we show that for linear upper triangular systems of differential equations, we can use the diagonal entries to obtain the Sacker and Sell, or Exponential Dichotomy, and also –under some…

Two dimensional mapping with strange attractor

- Mathematics
- 1976

A system of three first-order differential equations, whose solutions tend toward a “strange attractor”, is investigated, and it is shown that the same properties can be observed in a simple mapping of the plane defined by:xi+1=yi+1−axi2,yi-1=bxi.

Matrix analysis

- MathematicsStatistical Inference for Engineers and Data Scientists
- 2018

This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications.

Classification of linear time-varying difference equations under kinematic similarity

- Mathematics
- 1996

This paper concerns the problem to classify linear time-varying finite dimensional systems of difference equations under kinematic similarity, i.e., under a uniformly bounded time-varying change of…