Normal hyperbolicity

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Author: Dr. Neil Fenichel, Microstar Laboratories, Inc., Bellevue, WA

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Inline Formulas

dynamical system on $M_1$

diffeomorphism $F$ of a manifold to itself or the flow $F^t$ defined by a vector field

point $m^* = F^T(m)$ that also is in $M$ . The derivative $DF^T(m)$

Dynamical system is $C^r, r \geq 1$ , and

where $\pi$ is the projection onto $N$ . Let $|| \ ||$ be a metric on tangent vectors

Then the system of differential equations can be transformed to:

$x' = ax$

$y' = by$

where $a > 0$ and $b < 0$ . I

The unstable manifold of $P$ is defined as the set of points $(x, y)$ near $P$ such that $F^T(x, y, \varepsilon) \rightarrow P$ as $t \rightarrow -\infty$ .

the contraction factor of $G^1$ is approximately $e^{(b-a)T}$ , and

The estimate of $|\delta y|/|\delta x|$ above

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Formulas for Exponential Rates

$v_{-t} = DF^{-t}(m) \cdot v_0$ and $w_{-t} = \pi D F^{-t}(m) \cdot w_0$

$\nu(m) = \inf \{a:(||w_0||/||w_{-t}||)/a^t \rightarrow 0$ as $t \rightarrow \infty$ for all $w_0 \in N_m \}$

$\sigma(m) = \inf \{ s: (||w_0||^s/||v_0||)/(||w_{-t}||^s/||v_{-t}||) \rightarrow 0$ as $t \rightarrow \infty$ for all $v_0 \in T_m M$ and $w_0 \in N_m \}$ .