zbMATH — the first resource for mathematics

Geometry in the neighborhood of invariant manifolds of maps and flows in linearization. (English) Zbl 0746.58008
Pitman Research Notes in Mathematics. 233. Harlow, New York: Longman Scientific & Technical, John Wiley & Sons, Inc. 89 p. (1990).
The book consists of two parts of approximately equal lengths each written by one of the authors. In the first part homomorphisms $$P: \mathbb{R}^ r\times \mathbb{R}^ t\to\mathbb{R}^ r\times\mathbb{R}^ t$$ given by $$P(z,y)=(f(z,y),L(y)+Y(z,y))$$ are considered where $$L: \mathbb{R}^ t\to\mathbb{R}^ t$$ is linear and expanding, and $$P$$ is (by Lipschitz constraints) sufficiently close to a mapping $$P_ z\times L$$ with $$P_ z: \mathbb{R}^ r\to\mathbb{R}^ r$$ less expanding than $$L$$. Under these assumptions a horizontal and a vertical foliation $$H$$, $$V$$, respectively of $$\mathbb{R}^ r\times\mathbb{R}^ t$$ are constructed such that $$P$$ maps leaves to leaves. There is an invariant leaf of $$H$$ which is the graph of a function $$G: \mathbb{R}^ r\to \mathbb{R}^ t$$. Using these foliations new coordinates $$\bar z,\;\bar y$$ can be introduced. (The leaves correspond to the manifolds given by $$\bar y=\text{const.}$$ or $$\bar z=\text{const.}$$, respectively.) With respect to the new coordinates $$P$$ splits as $P(\bar z,\bar y)=(f(\bar z,G(\bar z)),L(\bar y))$ and is therefore linearized with respect to $$y$$. This is applied to get a corresponding linearization for flows given by autonomous differential equations on $$\mathbb{R}^ r\times\mathbb{R}^ t$$. In the second part this linearization for the solution of differential equations is constructed directly (also using a horizontal and a vertical foliation).
Reviewer: H.G.Bothe (Berlin)

MSC:
 58-02 Research exposition (monographs, survey articles) pertaining to global analysis 37D99 Dynamical systems with hyperbolic behavior 37C10 Dynamics induced by flows and semiflows 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$