Construction of minimal cocycles arising from specific differential equations (original) (raw)

Construction of ergodic cocycles that are fundamental solutions to linear systems of a special form

Journal of Modern Dynamics, 2007

If T = {T t } t∈R is an aperiodic measure-preserving jointly continuous flow on a compact metric space Ω endowed with a Borel probability measure m, and G is a compact Lie group with Lie algebra L, then to each continuous map A : Ω → L we can associate the fundamental matrix solution Ω × R (ω, t) → X A (ω, t) ∈ G of the family of time-dependent ordinary differential equations X (t) = A(T t ω)(X(t)) , x ∈ G , ω ∈ Ω .

On the solution of linear differential equations in Lie groups

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1999

On the solution of linear differential equations in Lie groups Email alerting service here corner of the article or click Receive free email alerts when new articles cite this article-sign up in the box at the top right-hand http://rsta.royalsocietypublishing.org/subscriptions

Motion on lie groups and its applications in control theory

Reports on Mathematical Physics, 2003

The usefulness in control theory of the geometric theory of motion on Lie groups and homogeneous spaces will be shown. We quickly review some recent results concerning two methods to deal with these systems, namely, a generalization of the method proposed by Wei and Norman for linear systems, and a reduction procedure. This last method allows us to reduce the equation on a Lie group G to that on a subgroup H, provided a particular solution of an associated problem in G/H is known. These methods are shown to be very appropriate to deal with control systems on Lie groups and homogeneous spaces, through the specific examples of the planar rigid body with two oscillators and the front-wheel driven kinematic car.

Solution Curve for Linear Control Systems on Lie Groups

Cornell University - arXiv, 2019

The purpose of this paper is to describe explicitly the solution for linear control systems on Lie groups. In case of linear control systems with inner derivations, the solution is given basically by the product of the exponential of the associated invariant system and the exponential of the associated invariant drift field. We present the solutions in low dimensional cases and apply the results to obtain some controllability results.

Iterated Commutators, Lie’s Reduction Method and Ordinary Differential Equations on Matrix Lie Groups

Foundations of Computational Mathematics, 1997

In the context of devising geometrical integrators that retain qualitative features of the underlying solution, we present a family of numerical methods the method of iterated c ommutators, 5, 13 to integrate ordinary di erential equations that evolve on matrix Lie groups. The schemes apply to the problem of nding a numerical approximation to the solution of Y 0 = At;Y Y; Y0 = Y0; whereby the exact solution Y evolves in a matrix Lie group G and A is a matrix function on the associated Lie algebra g. W e show that the method of iterated commutators, in a linear setting, is intrinsically related to Lie's reduction method for nding the fundamental solution of the Lie-group equation Y 0 = AtY .

Almost reduction and perturbation of matrix cocycles

Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2013

In this note, we show that if all Lyapunov exponents of a matrix cocycle vanish, then it can be perturbed to become cohomologous to a cocycle taking values in the orthogonal group. This extends a result of Avila, Bochi and Damanik to general base dynamics and arbitrary dimension. We actually prove a fibered version of this result, and apply it to study the existence of dominated splittings into conformal subbundles for general matrix cocycles.

Remarks concerning the Lyapunov exponents of linear cocycles

We impose a condition of pointwise convergence on the Lyapunov exponents of a d-dimensional cocycle over a compact metric minimal flow. This condition turns out to have significant consequences for the dynamics of the cocycle. We make use of such classical ODE techniques as the Lyapunov-Perron triangularization method, and the ergodic-theoretical techniques of Krylov and Bogoliubov.

Iterated Commutators, Lie's Reduction Method and Ordinary Differential Equations on Matrix Lie Groups

1997

Construction of minimal cocycles arising from specific differential equations (original) (raw)

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