Note that the axes mean different things depending on which object you're referencing in the plot. If you're looking at the blue curve, the horizontal axis is x and the vertical axis is V. If you're looking at the red dot or orange curves, the horizontal axis is R and the vertical axis is H.
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CATASTROPHE EXAMPLE:
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Set (R,H) = (2,-2). Imagine that you initialize the system at the only fixed point, the global minimum of V(x) around x = -1.8. Now increase H while leaving R unchanged. If H is increased sufficiently slowly, the system would stay close to the fixed point and "track" it as the fixed point shifts to the right because the fixed point is stable. If you continue to increase H, a new stable fixed point appears around x = 1 once the (R,H) point passes through the lower bifurcation curve (roughly H = -1.09). This new fixed point is irrelevant to the system's behavior at the moment. The system was initialized at the x < 0 fixed point, and it follows that point as H increases because the fixed point is stable. If you increase H even more, you'll notice the value of the potential at the x < 0 fixed point increasing. The fixed point the system has been tracking appears to be becoming "less stable". Finally, if you increase H so much that the (R,H) point passes through the upper bifurcation curve (roughly H = 1.09), the x < 0 fixed point suddenly disappears completely. The system is no longer at a fixed point, so it immediately starts evolving toward the stable x > 0 fixed point. This is the catastrophe. A small change in a parameter (increasing H from 1.08 to 1.09) causes a large, sudden change in the state of the system. This is analogous to the act of continually stacking small weights on a glass table, where the total weight on the table is like H and the state of the table is like the state of the system. The table will be able to hold some number of weights without anything happening to it. Eventually though, one small weight will be added to the pile on the table and it will suddenly shatter.
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HYSTERESIS EXAMPLE:
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Continuing from the previous example, what happens if you now decrease H? Set (R,H) = (2,2) and assume the system has had time to settle close to the x > 0 fixed point after the catastrophe. Again, keep R fixed throughout the following process. As you decrease H below about 1.09, the x < 0 fixed point reappears. However, the system continues to track the x > 0 fixed point because it's stable and the system has settled near it. The system doesn't suddenly jump back to the x < 0 fixed point once it reappears. To recap: increasing H just above 1.09 caused the system to suddenly jump from x < 0 to x > 0, however subsequently decreasing H just below 1.09 didn't cause the system to jump back to x < 0. This lack of reversibility as a parameter is varied is called hysteresis. In order to "reset" the system back to x < 0, H would have to be substantially decreased to below about -1.09. This would cause the x > 0 fixed point to disappear, and the system would suddenly move back to the x < 0 fixed point.
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EXTRA STUFF:
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Play around with the parameter values and observe how changing the parameters changes the potential.
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Verify on your own that you can also get catastrophes and hysteresis if you hold H fixed and change R instead.
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Look at slides 21 and 22 in Lecture 4. Try to replicate the green paths by dragging the (R,H) point around the RH plane. Can you see the sudden changes in system state that are marked on the diagrams?
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