## B22 Nullclines and Fixed Points

As discussed in earlier chapters, a powerful technique for the analysis of planar differential equations and related to the direction fields is the use of nullclines. Nullclines are curves in the plane along which the rate of change of one or the other variable is zero. The x-nullcline is the curve where dx/dt = 0, that is, f (x, y) = 0. Similarly, the w w

Figure B.9 Direction fields and some trajectories for the FitzHugh-Nagumo equations.

y-nullcline is the curve where g(x, y) = 0. The usefulness of these curves is that they break the plane up into regions along which the derivatives of each variable have a constant sign. Thus, the general direction of the flow is easy to determine. Furthermore, any point where they intersect represents a fixed point of the differential equation. XPPAUT can compute the nullclines for planar systems. To do this, just click on

You should see two curves appear: a red one representing

Nullcline

the V-nullcline and a green one representing the W-nullcline. The green one is a straight line, and the red is a cubic. They intersect just once: There is a single fixed point. Move the mouse into the phase plane area and hold it down as you move it. At the bottom of the Main Window you will see the x and y coordinates of the mouse. The intersection of the nullclines appears to be at (0, 0).

The stability of fixed points is determined by linearizing the system of equations about them and then finding the eigenvalues of the resulting linear matrix. XPPAUT will do this for you quite easily. XPPAUT uses Newton's method to find the fixed points and then numerically linearizes the system about them to determine stability. To use Newton's method, a decent guess needs to be provided. For planar systems this is easy to do; it is just the intersection of the nullclines. In XPPAUT fixed points and their stability are found using the Sing pts command, since "singular points"

is a term sometimes used for fixed points or equilibrium points. Click on Sing pts v

Mouse

) and move the mouse to near the intersction of the nullclines. Click the button, and a message box will appear on the screen. Click on No, since we do not need the eigenvalues. A new window will appear that contains information about the fixed points. The stability is shown at the top of the window.

The nature of the eigenvalues follows: c+ denotes the number of complex eigenvalues with positive real part; c- is the number of complex eigenvalues with negative real part; im is the number of purely imaginary eigenvalues; r+ is the number of positive real eigenvalues; and r- is the number of negative real eigenvalues. Recall that a fixed point is linearly stable if all of the eigenvalues have negative real parts. Finally, the value of the fixed points is shown under the line. As can be seen from this example, there are two complex eigenvalues with negative real parts: the fixed point is (0, 0). ( XPPAUT reports a very small nonzero fixed point due to numerical error.) Integrate the system using the mouse, starting with initial conditions near the fixed point. (In the Main Window, tap H [T].) Note how solutions spiral into the origin, as is expected when there are complex eigenvalues with negative real parts.

For nonplanar systems of differential equations you must provide a direct guess.

_ in the Initial Data click on

Type your guess into the Initial Data Window and click on Window. Then from the Main Window

Sing Pts

Change the parameter I from 0 to 0.4 in the Parameter Window and click on in the Parameter Window. In the Main Window erase the screen and redraw

(HE [N IN). The fixed point has moved The fixed point should the nullclines:

Erase

Nullclines

New up. Check its stability using the mouse ( Sing pts

Mouse be (0.1, 0.4). Use the mouse to choose a bunch of initial conditions in the plane. All solutions go to a nice limit cycle. That is, they converge to a closed curve in the plane representing a stable periodic solution.

We can make a nice picture that has the nullclines, the direction fields, and a few representative trajectories. Since XPPAUT keeps only the last trajectory computed, we will "freeze" the solutions we compute. We can freeze trajectories automatically or one at a time, and we will do the former. Click on

Graphic stuff

(F)reeze

(O)n freeze ( G 11 F 11 O |) to permanently save computed curves. Up to 26 can be saved in any window. Frist we use the mouse to compute a bunch of trajectories. Draw the direction fields by clicking Dir.field/flow (D)irect Field | ( D 11 D |).

Viewaxes

We can label the axes as follows: Click on view dialog will come up. Change nothing but the labels (the last two entries), and put V as the Xlabel and w as the Ylabel. Click on Ok to close the dialog. Finally, since the

Graphic stuff aXes opts

Click on axes are confusing in the already busy picture, click on

(| G 11 X |) and in the dialog box change the 1's in the entries X-org(1=on) and Y-org(1=on) to 0's to turn off the plotting of the X and Y axes. Click are done.

Ok when you w w

Figure B.10 Nullclines,direction fields, trajectories for I = 0.4 in the Fitzhugh-Nagumo equations.

To create a PostScript file, follow Graphic stuff (P)ostscript (| G 11 P ]) and accept all the defaults. Name the file whatever you want and click on | Ok | in the file selection box. Figure B.10 shows the version that we made. Yours will be slightly different. If you want to play around some more, turn off the automatic freeze option,

G F O ), and delete all the frozen curves,

Graphic |
stuff |
Freeze |
Off freeze| | |

Graphic |
stuff |
Freeze |

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