from you are slowly changing of the diagram, at the stable equilibrium at the bottom until you reach a critical threshold at r as 0.4. As soon as you cross this threshold, the system's state suddenly jumps up to another stable eglin Jum point at the top of the diagram. Such a sudden jump in the system's state is often alled a catastrophe. You get upset, and try to bring the system's state back to where "was, by reducing r. However, counter to your expectation, the system's state remains high even after you reduce r below 0.4. This is hysteresis; the system's asymptotic wats depends not just on r, but also on where its state was in the immediate past. In dine words, the system's state works as a memory of its history. In order to bring the system state back down to the original value, you have to spend extra effort to reduce . all the way below another critical threshold , r ~ -0.4 . Such hysteresis could be useful; every bit (binary digit) of computer memory has this kind of bifurcation dynamics, which is why we can store information in it. But in other con- texts, hysteresis could be devastating-if an ecosystem's state has this property (mary studies indicate it does), it takes a huge amount of effort and resources to revert a de serted ecosystem back to a habitat with vegetation, for example. Exercise 8.1 Conduct a bifurcation analysis of the following dynamical system with parameter r: dx =rx(x+1) -x (8.17) dt Find the critical threshold of r at which a bifurcation occurs. Draw a bifurcation diagram and determine what kind of bifurcation it is. Exercise 8.2 Assume that two companies, A and B, are competing against each other for the market share in a local region. Let x and y be the market share of A and B, respectively. Assuming that there are no other third-party competitors, xty = 1 (100%), and therefore this system can be understood as a one-variable system. The growth/decay of A's market share can thus be modeled as (8.18) dx at = ax(1 -x) (x -y), where x is the current market share of A, 1 - a is the size of the available potential customer base, and x - y is the relative competitive edge of A, which can b140 CHAPTER 8. BIFURCATIONS rewritten as x - (1 - x) = 2x -1. Obtain equilibrium points of this system and their stabilities. Then make an additional assumption that this regional market is connected to and influenced by a much larger global market, where company A's market share IS somehow kept at p (whose change is very slow so we can consider it constant): = ar(1 - x)(x-y) +r (p -x) (8.19) Here r is the strength of influence from the global to the local market. Determine a critical condition regarding r and p at which a bifurcation occurs in this system. Draw its bifurcation diagram over varying r with a = 1 and p = 0.5, and determine what kind of bifurcation it is. Finally, using the results of the bifurcation analysis, discuss what kind of market- ing strategy you would take if you were a director of a marketing department of a company that is currently overwhelmed by its competitor in the local market. How can you "flip" the market? 8.3 Hopf Bifurcations in 2-D Continuous-Time Models For dynamical systems with two or more variables, the dominant eigenvalues of the Ja cobian matrix at an equilibrium point could be complex conjugates. If such an equilibrium point, showing an oscillatory behavior around it, switches its stability, the resulting bitur cation is called a Hopf bifurcation. A Hopf bifurcation typically causes the appearance (or disappearance) of a limit cycle around the equilibrium point. A limit cycle is a cycle closed trajectory in the phase space that is defined as an asymptotic limit of other oscila tory trajectories nearby. You can check whether the bifurcation is Hopf or not by looking the imaginary components of the dominant eigenvalues whose real parts are at a critical value (zero); if there are non-zero imaginary components, it must be a Hopf bifurcation oscillator: Here is an example, a dynamical model of a nonlinear oscillator, called the van der