3 ms·

Seems a bit of a leap to tag that with catastrophe theory? Surely that's just hysteresis, and that's found everywhere.

ColinWright 13y ago
Yes, I need to expand that bit. The individual craft simply displays hysteresis, but the point here is that hysteresis is what you get if your system is a slice through a fold (or other) catastrophe. By looking at the parametrized systems you get the fold emerging, even though each one individually is "just" a system with (or without) hysteresis. Understanding hysteresis as a part of a larger topological structure is a rather deep insight, and one that I've struggled to convey completely. So thanks for your comment, it's helped me think again about how this should be expanded and enhanced to make that point clearer.
- tlarkworthy 13y ago
  Hmmm. Is there really a fold though? If you added velocity and position above the water into the mix then the state transition is unique and smooth. The fold appears because important quantities are not plotted and resolved elsewhere. The only real discontinuity is the when the hull touches the water. Maybe I don't really know the definition of a catastrophe. To me a PID controllor exhibiting ringing would look "folded" in position space (smooth in position/velocity space).
  - ColinWright 13y ago
    If you plot "Steady State Driving Force" against "Steady Speed" then for any given craft there is a curve. This curve the changes from simple quadratic(ish) through to the version discussed for the hydrofoil. We can think of this as adding every more efficient and effective foils to a more-or-less standard craft. Now in the space "Foil Size" x "Driving Force" we plot "Resultant Speed" and we get a classic folded manifold, including the singularity. So yes, there really is a fold. Physics tells us that if you include lots and lots of things in your phase space then all the transitions are "smooth." Even here, when the craft touches the water and decelerates dramatically, almost "catastrophically", the transition (when considered in sufficient detail) is smooth. We can plot keel height above water, and as that gets to -1 mm then the deceleration is increasing smoothly, etc. However, dealing with this on a macro level, it's more useful to consider the dynamics as being a fold catastrophe.
- tlarkworthy 13y ago
  Oh I think I understand better now. Something flips and moves the attractor for the system. Like an elastic band snapping, but with boats. Thanks for your reply

Yes, I need to expand that bit. The individual craft simply displays hysteresis, but the point here is that hysteresis is what you get if your system is a slice through a fold (or other) catastrophe. By looking at the parametrized systems you get the fold emerging, even though each one individually is "just" a system with (or without) hysteresis. Understanding hysteresis as a part of a larger topological structure is a rather deep insight, and one that I've struggled to convey completely. So thanks for your comment, it's helped me think again about how this should be expanded and enhanced to make that point clearer.

tlarkworthy 13y ago
Hmmm. Is there really a fold though? If you added velocity and position above the water into the mix then the state transition is unique and smooth. The fold appears because important quantities are not plotted and resolved elsewhere. The only real discontinuity is the when the hull touches the water. Maybe I don't really know the definition of a catastrophe. To me a PID controllor exhibiting ringing would look "folded" in position space (smooth in position/velocity space).
- ColinWright 13y ago
  If you plot "Steady State Driving Force" against "Steady Speed" then for any given craft there is a curve. This curve the changes from simple quadratic(ish) through to the version discussed for the hydrofoil. We can think of this as adding every more efficient and effective foils to a more-or-less standard craft. Now in the space "Foil Size" x "Driving Force" we plot "Resultant Speed" and we get a classic folded manifold, including the singularity. So yes, there really is a fold. Physics tells us that if you include lots and lots of things in your phase space then all the transitions are "smooth." Even here, when the craft touches the water and decelerates dramatically, almost "catastrophically", the transition (when considered in sufficient detail) is smooth. We can plot keel height above water, and as that gets to -1 mm then the deceleration is increasing smoothly, etc. However, dealing with this on a macro level, it's more useful to consider the dynamics as being a fold catastrophe.
tlarkworthy 13y ago
Oh I think I understand better now. Something flips and moves the attractor for the system. Like an elastic band snapping, but with boats. Thanks for your reply