It's possible that by slightly magnetizing asteroids we can
steer them out of earth's path by predicting interactions with
the magnetic fields out there.

If the orbital dynamics are chaotic, then arbitrarily small
magnetic
perturbations should suffice to save the earth. Problem is,
it may require arbitrarily large computers.

Are you sure? My understanding was that a chaotic
system's only unpredictable in principle if you try to
predict for all future time. Since states diverge
exponentially, if you're predicting a finite time into the
future, there is, for any desired prediction accuracy, a
finite accuracy (for measurement and computation) which
suffices to achieve it. Of course, that accuracy will, in
general, not be practically achievable (hence the crack
about arbitrarily large computers).

Also, there are control systems for some chaotic systems
that don't rely on actually solving the system.

The point is, if an arbitrarily small perturbation would suffice to save the earth, then an similarly small perturbation could make it hit the earth. Perturbations such as a knot in the solar wind, a solar flare, or a micrometeorite strike. These can't be calculated, no matter how much computing power you have.

Since magnets are bipolar, any turning motion, or other perturbations would only increase the chances of bringing the asteroid down towards one of the poles. Perhaps slightly preferable to an equatorial crash, but still pretty bad.

//Since a magnetic field can only impart a turning moment//

//Physics double-take moment//

A uniform magnetic field, that is, which it will be if we are considering the magnetic field of the earth out at the distance of a typical asteroid orbit.