Math 1A: introduction to functions and calculus Oliver Knill, 2011Lecture 17: CatastrophesIn this lecture, we once more cover extrema problems. We are interested how extrema change whena parameter changes. Nature, economies, processes favor extrema. Extrema change smoothly withparameters. How come tha t the outcome is often not smooth? What is the reason that politicalchange can go so fast once a tipping point is reached? One can explain this with mathemati-cal models. We look at a simple example, which explains it. In reality, the situation is morecomplicated. In the ”New York Times” of February 24, 2011, Jennifer E. Sims, the directorof intelligence studies at Georgetown University’s School of Foreign Service and senior fellow atthe Chicago Council on Global Affairs asked: Why, with the U.S. spending 80 billion dollars onintelligence, were we apparently surprised by recent regime changes in the Middle East? Why didchange happen at all? These are complex questions. Obviously, some tipping point has beenreached and the smallest event like the confiscation of a fruit stand in Tunesia or increasing foodprizes in Egypt has produced change. In these complex examples, it will never be possible to un-derstand everything. Lets look here at a simple mathematical model which illustrates the generalprinciple that:If a local minimum seizes to become a local minimum, a new stable position isfavored. This can be far away from the original situation.To get started, lets look at a n extremization problem1Find all the extrema of the function f(x) = x4−x2. So-lution: f′(x) = 4x3−2x is zero for x = 0, 1/√2, −1/√2.The second derivative is 12x2−2. It is negative for x = 0and positive at the other two points. We have two lo calminima and one local maximum.x4- x2x2Now find all the extrema of the function f(x) = x4−x2− 2x. There is only one critical point. It is x = 1.x4- x2- 2 xxSomething has happened from the first example to the second example. The local minimum tothe left has disappeared. Assume the function f measures the prosperity of some kind and c is12a parameter. We look at the position of the first equilibrium point of the function. Catastrophtheorists usually assume the so called Delay assumption.A stable equilibrium is here used as an other name for a local minimum. A systemstate remains in a stable equilibrium until it disappears. If that happens, the systemsettles in a neighboring stable equilibrium.A pa r ameter value for which a stable minimum disappears is called a catastrophe.Here is the position of the equilibrium point plotted in dependence of c.3cfA parameter value for which a local minimum disappears is called a catastrophe.cBifurcation diagram: The picture shows the equilibrium points as they change in dependenceof the parameter c. The vertical axes is the parameter c, the horizontal axes is x. At the bottomfor c = 0, we have three equilibrium points, two local minima and one local maximum. At thetop for c = 1 we have only one local minimum.Catastrophes always go for the worse in the sense that the value decreases. It is notpossible to reverse the process and have a catastrophe, where the minimum jumpsup.Look again at the above ”movie” of graphs. But run it backwards and use the same principle.We do not end up at t he position we started with. The new equilibrium stays the equilibrium.Decreasing the food prizes again did not reverse the process of change in Egypt for example.Catastrophes are in general irreversible.We see that in real life: It is easy to screw up a relationship, get sick, have a ligament torn orlose trust. Building up a relationship, getting healthy or gaining trust o n the other hand happens4slowly. Ruining a country or a company or losing a good reputation of a brand is very easy. Ittakes a long time to regain it.Local minima can change discontinuously, when a parameter is changed. This canhapp en with perfectly smooth functions and smooth parameter changes.3 Lets look a t f(x) = x4+ cx2, where −1 ≤ c ≤ 1. We will look at that in class.cHomeworkIn this homework, we study a catastrophe for the functionf(x) = x6− x4+ cx2,where c is a parameter between 0 and 1.1 a) Find all the critical points in the case c = 0 and analyze their stability. b) Find all thecritical points in the case c = 1 and analyze their stability.2 Plot the graph of f for at least 10 values of c between 0 and 1. You can of course usesoftware, a graphing calculator or Wolfram alpha. Mathematica code is below.3 If you change from c = −0.3 to 0.6 pinpoint the value for the catastrophe and show a roughplot of c → f(xc), the value at the first local minimum xcin dependence o f c. The textabove provides this graph for an other function. It is the graph with a discontinuity.4 If you change back from c = 0.6 to 0.3 pinpoint the value for the catastrophe (it will bedifferent from the one in the previous question).5 Sketch the bifurcation diagram. That is, if xk(c) is the k’th equilibrium point, then drawthe union of all graphs of xk(c) as a function of c (the c- axes pointing upwards). As in thetwo example provided, draw the local maximum with dotted lines.✞Manipulate [ Plot [ xˆ6 − xˆ4 + c x ˆ 2 , {x , −1, 1 }] , {c , 0 , 1 }]✝