Static Stability
Let's say you push the nose down 15 degrees from horizon and let go of the yoke. What happens next or initally, will determine whether the plane is longitudinally stable.
Positive stability: The nose reverses (or comes up) and decreases in degrees.
Negative stability: The nose continues past 15 degrees.
Neutral stability: The nose remains at 15 degrees.
Dynamic Stability
Under the same conditions above, pushing the nose down 15 degrees.What happens over time determines the dynamic stabilty.
Positive dynamic stability: The oscillations begin to decrease. The nose moves up to say 10 degrees above the horizon. Then moves down 5 degrees below horizon and moves back up to 5 degrees above horzion. Eventually, the oscillations grow smaller and return to the orignal equilibrium.
Negative dynamic stabilty: The oscillations begin to increase. The nose continues past 15 degrees and goes to say, 18 degrees. It then goes up to 18 degees above the horizon and then back down below the horizon 20 degrees and so on.
Neutral dynamic stability: The oscillation remains where it is. After reaching 15 degrees below horizon, it goes back up to 15 degress above the horizon and then back down to 15 degress below and continues this cycle wihtout any change.
Not much to add, except that normally, you cannot have a dynamically stable aircraft without static stability. The exception is if you have a statically unstable aircraft with fly-by-wire (F-16, X-29, some other modern fighter aircraft). In these cases, the aircraft is statically unstable to make it more maneuverable, and the fly-by-wire computers can command the control surfaces quickly enough to keep the aircraft pointed in the correct direction.
Interestingly, the Wright Brothers were fans of neutrally statically stable aircraft (or even statically unstable aircraft), feeling that they gave more maneuverability. Of course, those aircraft flew slowly enough that the pilot could compensate.
EDIT - The The lack of stability of the Wrights' aircraft may be attributed more to lack of understanding of stability issues. Here's a paper I found, but haven't had a chance to read yet, except for a very cursory skim.
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