Nonlinear Iwasawa decomposition of control flows
Fritz Colonius Paulo Régis C. Ruffino
Let $\varphi(t,\cdot,u)$ be the flow of a control system on a Riemannian manifold $M$ of constant curvature. For a given initial orthonormal frame $k$ in the tangent space $T_{x_{0}}M$ for some $x_{0}\in M$, there exists a unique decomposition $\varphi_{t}=\Theta_{t}\circ\rho_{t}$ where $\Theta_{t}$ is a control flow in the group of isometries of $M$ and the remainder component $\rho_{t}$ fixes $x_{0}$ with derivative $D\rho_{t}(k)=k\cdot s_{t}$ where $s_{t}$ are upper triangular matrices. Moreover, if $M$ is flat, an affine component can be extracted from the remainder.
keywords: Control flows isometries group of affine transformations non-linear Iwasawa decomposition.
Hartman-Grobman theorems along hyperbolic stationary trajectories
Edson A. Coayla-Teran Salah-Eldin A. Mohammed Paulo Régis C. Ruffino
We extend the Hartman-Grobman theorems for discrete random dynamical systems (RDS), proved in [7], in two directions: for continuous RDS and for hyperbolic stationary trajectories. In this last case there exists a conjugacy between travelling neighborhoods of trajectories and neighborhoods of the origin in the corresponding tangent bundle. We present applications to deterministic dynamical systems.
keywords: hyperbolic stationary trajectories. Random dynamical systems Hartman-Grobman theorems

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