The system x_ = y, y_ = y x( x 0.15)( x 2) results from an approximation to the

Chapter 9, Problem 10

(choose chapter or problem)

The system x_ = y, y_ = y x( x 0.15)( x 2) results from an approximation to the Hodgkin--Huxley6 equations, which model the transmission of neural impulses along an axon. a. Find the critical points, and classify them by investigating the approximate linear system near each one. G b. Draw phase portraits for = 0.8 and for = 1.5. G c. Consider the trajectory that leaves the critical point (2, 0). Find the value of for which this trajectory ultimately approaches the origin as t . Draw a phase portrait for this value of . Bifurcation Points. Consider the system x_ = F( x, y, ), y_ = G( x, y, ), (46) where is a parameter. The equations F( x, y, ) = 0, G( x, y, ) = 0 (47) determine the x- and y-nullclines, respectively; any point where an x-nullcline and a y-nullcline intersect is a critical point. As varies and the configuration of the nullclines changes, it may well happen that, at a certain value of , two critical points coalesce into one. For further variation in , the critical point may once again separate into two critical points, or it may disappear altogether. Or the process may occur in reverse: For a certain value of , two formerly nonintersecting nullclines may come together, creating a critical point, which, for further changes in , may split into two. A value of at which such phenomena occur is a bifurcation point. It is also common for a critical point to experience a change in its type and stability properties at a bifurcation point. Thus both the number and the kind of critical points may change abruptly as passes through a bifurcation point. Since a phase portrait of a system is very dependent on the location and nature of the critical points, an understanding of bifurcations is essential to an understanding of the global behavior of the systems solutions. 1