User:Neil Fenichel/Proposed/Normal hyperbolicity
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
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 \( w_0 \in N_m \} \ .\)
\[\nu(m) < 1\] and \( \sigma(m) < 1/r \) for all \( m \in M \ .\)
\[\lambda^+(m) = \lim_{t \rightarrow \infty} ||\pi^+ DF^{-t}(m) | N^+ ||^{1/t}\ ,\] \[\nu^-(m) = \overline{\lim_{t \rightarrow \infty}} ||\pi^- DF^t(F^{-t}(m)) | N^- ||^{1/t}\ ,\] \[\sigma^-(m) = \overline{\lim_{t\rightarrow \infty}} \frac{\log || D(F^{-t} | M)(m)||}{-\log ||\pi^- DF^t(F^{-t}(m)) | N^- ||}\ .\]
\[\lambda^+(m) < 1 \] and \( \nu^-(m) < 1 \) for all \( m \in M \)
\[\sigma^-(m) < 1/r \] for all \( m \in M\ .\)
