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## Graphical Solutions: Phase Plane Analysis

Oftentimes we can gather a good deal of qualitative information about a solution to a differential equation without going through the trouble of finding an analytic or numerical solution. Instead, we can simply look for equilibrium points, or points where the derivative is zero, and determine whether the function moves toward or away from these points as time passes giving us the asymptotic behavior of the function without having to solve for it.

One Stable Pointno_title

Example 2.10.1.2 (One Stable Point)

Consider the differential equation: (2.10.1)

When , Thus, if , then for any value of , , since the derivative will always be zero. Thus is called an equilibrium since it will not change. However if , then the derivative will be negative, and thus, as grows larger and larger, will converge to . This is easily seen by simply plugging in different values for that are greater than . For example, if , then . If , then Likewise, if , than the derivative will be positive for all values of and will approach from below as goes to infinity. Since the line is approached from above when and below when , it is called a node or a stable state. This is because small perturbations to the system at this point will only lead back to it. That is, if the system is at and some outside force knocks it to or , it will asymptotically return to .

Figure 2.10.1 shows an actual plot of the phase lines, or slopes for various values of and . Due to the fact that there are no free instances of in Equation 2.10.1, the slopes are the same for each value of . In this illustration, it is easy to see the stable point and how the slope of any point above or below this line points toward it. Figure 2.10.2 demonstrates how that regardless of the initial condition, as gets larger, the solution will converge on the stable equilibrium. Figure 2.10.3 shows how a nullcline graph represents the same information.    1 Stable and 2 Unstableno_title

Example 2.10.1.4 (1 Stable and 2 Unstable)

Consider the cubic differential equation: (2.10.2)

where when , and . By plugging in different values for , we can determine the slope at different points. In this case, we end up with one stable point, and two unstable points, and . By unstable, we mean that if the system is at or , and it is perturbed slightly, it will not return to its original state. Instead, it will either move toward or . This is illustrated in Figure 2.10.4. Figure 2.10.5 shows the equivalent phase information contained in a plot of the nullcline. When the initial conditions are known, specific solutions can be plotted and this is shown in Figure 2.10.6.    2 Stable and 1 Unstableno_title

Example 2.10.1.6 (2 Stable and 1 Unstable)

Consider the cubic differential equation: (2.10.3)

where when , and . Again, by plugging in different values for , we can determine the slope at different points. In this case, we end up with two stable points and one unstable point. This is illustrated in Figure 2.10.7.      Next: Separation of Variables Up: Methods for Solving ODEs Previous: Methods for Solving ODEs   Index

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Frank Starmer 2004-05-19
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