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

Chapter 9, Problem 13

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The system x = y, y = y x(x 0.15)(x 2) results from an approximation to the HodgkinHuxley5 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. (b) Draw phase portraits for = 0.8 and for = 1.5. (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, ), (i) where is a parameter. The equations F(x, y, ) = 0, G(x, y, ) = 0 (ii) 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 kind of critical points may change abruptly as passes through a bifurcation point. Since a phase portrait of asystem is very dependent on the location and nature of the critical points, an understanding ofbifurcations is essential to an understanding of the global behavior of the systems solutions.In each of 13 through 16:(a) Sketch the nullclines and describe how the critical points move as increases.(b) Find the critical points.(c) Let = 2. Classify each critical point by investigating the corresponding approximatelinear system. Draw a phase portrait in a rectangle containing the critical points.(d) Find the bifurcation point 0 at which the critical points coincide. Locate this critical pointand find the eigenvalues of the approximate linear system. Draw a phase portrait.(e) For >0 there are no critical points. Choose such a value of and draw a phase portrait.