I don't believe there is a mathematical reason, especially if there is latitude in reverse-engineering the field theory or stat mech system to evince such a behavior. Indeed, if Lorentz-nonivariant systems are examined, things like limit cycles , e.g. this one are not hard to concoct. As for physical reasons, they might well be easy to bypass/moot if one argued for them. I don't know of any systems, however, with this property, which might not say much.

As a mathematical wisecrack, I could manufacture a simple toy system with two couplings, *x* and *y* and logarithmic scale variable *t* :
$$
\dot{x}=-x + ay, \qquad \dot{y}= -y ,
$$
with evident solutions stable around the fixed point (0,0),
$$
y= e^{-t}, \qquad x= (c +at) e^{-t} .
$$
The stability matrix of the ODE system is
$$
\left( \begin{array}{cc}
-1 & a \\
0 & -1 \\
\end{array} \right)
$$
which is not diagonalizable, with only one eigenvector,
$$
\left( \begin{array}{c}
1 \\
0 \\
\end{array} \right)
$$
of eigenvalue -1. This is not to say the system is not stable, however, if one could solve the ODE, somehow, as here.

This post imported from StackExchange Physics at 2017-10-16 12:31 (UTC), posted by SE-user Cosmas Zachos