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