A4 Phase Plane Analysis

Obtaining a "solution" to first-order ODEs means that you have expressed all of the dependent variables as functions of time. In the case of the 2 X 2 linear equations in Section A.3, this means that we have the time series for x, and x2. A great deal can be learned about these solutions by plotting the dependent variables as a function of time as done in Figure A.1. However, there are other ways of plotting solutions that give additional insight. For example, one can plot x, versus time, or some function of x, and x2 versus time. Perhaps the most useful plot is a phase plane plot, in which x2 is plotted versus x, with time serving only as a parameter, as shown in Figure A.2 for the numerical solutions shown in Figure A.1. This type of plot represents the trajectory of the solution, just as the arc of a baseball thrown in the air is a trajectory in three dimensional space.

Technically, the phase plane (or phase space for more than two variables) is a Cartesian plane with coordinates (x,, x2). Since the initial condition for the ODEs is arbitrary, any one of these points could be the initial point of a trajectory like those in Figure A.2. Continuing the analogy of phase space trajectories to the trajectory of a baseball, it makes sense to associate a velocity with the trajectory that goes through a point in phase space. This can be done directly using the differential equations, since

Figure A.2 The three solutions in Figure A.1(A)-(C) represented here in phase plane plots in corresponding (A)-(C). The arrow represents the direction of the initial point on the trajectory, which is given by the full line. The dashed line is the x2 nullcline, the broken dashed line is the xi nullcline, and their intersection is the steady state, which is unstable in (A), marginally stable in (B), and stable in (C).

Figure A.2 The three solutions in Figure A.1(A)-(C) represented here in phase plane plots in corresponding (A)-(C). The arrow represents the direction of the initial point on the trajectory, which is given by the full line. The dashed line is the x2 nullcline, the broken dashed line is the xi nullcline, and their intersection is the steady state, which is unstable in (A), marginally stable in (B), and stable in (C).

the right—hand sides of the equations are explicit expressions for a 1 and a2 as functions

of and ^2. Thus for the matrix A — I 4 1 1 that gives rise to the trajectory in Figure A.2C, the a1 component of the velocity at the point (a1 ,a2) is —2a1 — a2, whereas the a2 component of the velocity is 4a1 +x2. For the initial point (0.5, 0.5) of the trajectory in Figure A.2C the velocity vector at that point has components (—1.5, 2.5). In the figure, the head of the arrow on the velocity vector indicates its direction, and the length is proportional to its magnitude. Just as the velocity of a baseball is parallel to its trajectory, so is the velocity vector in phase space parallel to its trajectory.

There are a number of important curves and points in the phase plane that are defined by the differential equations. Isoclines are lines in the phase plane where the time rate of change of a variable is constant. For example, for the matrix in the previous paragraph, the isoclines for x1 are defined by c = —2x1 — x2, i.e., x2 = —2x1 + c, and the isoclines for x1 are given by x2 = —4x1 + c, where c is a constant. A particularly useful isocline is the nullcline, for which the time rate of change is zero, i.e., c = 0. So for this special case the nullclines are given by the straight lines through the origin, x2 = —2x1 and x2 = —4x1, shown in Figure A.2C. It is straightforward to show that the nullclines for the general 2 X 2 linear equations (A.17)—(A.18) are also straight lines. Since x 1 = 0 on the x1 nullcline, x1 cannot decrease if the trajectory crosses the nullcline from the right and cannot increase if the trajectory crosses it from the left. This means, as can be verified by looking at Figure A.2, that the trajectory must cross the x1 nullcline perpendicular to the x1 axis. Similary, the trajectory crosses the x2 nullcline perpendicular to the x2 axis.

Steady states are defined as points in the phase space at which both x 1 =0 and x2 = 0. These points, which are also known as singular points, equilibrium points, or stationary points, have the property that neither variable changes as a function of time. They are determined graphically by the intersection of the nullclines. However, just because the variables do not change in time at a steady state does not mean that trajectories starting from nearby points will end up at the steady state. Three different situations are illustrated in Figure A.2. In panel A the steady state is at the origin, (0, 0). However, the trajectory starting at (0.5, 0.5) grows without bound. In panels B and C the steady states are also at the origin, but the trajectory in B circles the origin periodically, whereas in C it spirals into the steady state.

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