Mathematical definition

Consider a differential equation
\begin{displaymath}
\frac{du}{dt} = f(u,\alpha), \quad u\in \ensuremath{\mathbf...
...\ensuremath{\mathbf{R}}^{n+1}\to \ensuremath{\mathbf{R}}^n\ .
\end{displaymath} (33)

We are interested in an equilibrium curve, i.e. $f(u,\alpha)=0$. The defining function is therefore:
\begin{displaymath}
F(x) = f(u,\alpha) = 0
\end{displaymath} (34)

with $x=(u,\alpha)\in \ensuremath{\mathbf{R}}^{n+1}$. Denote by $v\in \ensuremath{\mathbf{R}}^{n+1}$ the tangent vector to the equilibrium curve at $x$.