Equilibria and the Phase Line

Solving a differential equation can be done in three major ways: analytical, qualitative, and numerical. We have seen some examples of differential equations solved through analytical techniques (for example: linear, separable, and Bernoulli equations). Euler's Method (though very primitive) illustrates the use of numerical techniques in solving differential equations. For the qualitative approach, we have defined the slope field of a differential equation and showed how this can be helpful when other techniques fail.

When the differential equation is autonomous, more can be said about the solutions using qualitative techniques.

Consider the autonomous differential equation

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where f(y) is a function. Clearly we recognize a separable equation. So we know how to solve it via the analytical technique. First we look for the constant solutions as the roots of the equation

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Then to find the nonconstant solutions we separate the variable

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and integrate

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Note that the integration may be hard if not impossible to carry out. So it may not be possible to use the analytical technique.

In this case, one may try numerical techniques. Unfortunately, numerical techniques fail if f(y) contains a parameter and we need to make conclusions relative to this parameter. For example, recall the logistic equation with harvesting (with a constant rate H)

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where y(t) is the population of the given species at time t. In order to find the optimal rate H (which makes hunters happy as well as ecologists worried about preserving the species), numerical techniques may not be the best tool to use. Here we will see how new ideas based on a graphical approach will help us to say something about the solutions. Be aware that these ideas are only valid for autonomous equations.

Equilibria of Autonomous Equations

Consider the autonomous equation

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The equilibria or constant solutions of this differential equation are the roots of the equation

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Using the existence and uniqueness theorem, these constant solutions will cut the entire plane (where the solutions live) into independent regions. This means, if an initial condition belongs to one of the regions, then the solution satisfying the initial condition will stay in that region all the time.

Example. Consider the logistic equation

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Its equilibria are y=0 and y=1. Consider the solution y(t) to the IVP

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Since 0 <y(0) < 1, then we must have

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In the next picture, we draw few solutions associated to initial conditions in the different regions (0 < y < 1, y < 0, and 1 < y)

Note that in this example, any solution y(t) is increasing if 0 < y(0) < 1 is satisfied. So one may wonder whether a similar conclusion is true in general. Consider again the autonomous equation

displaymath138

Assume that tex2html_wrap_inline190 and tex2html_wrap_inline192 ( tex2html_wrap_inline194 ) are equilibria (that is tex2html_wrap_inline196 and tex2html_wrap_inline198 ). Assume moreover that f(y) has a constant sign on the interval tex2html_wrap_inline202 . Assume that f(y) is positive and consider the solution to the IVP

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where tex2html_wrap_inline208 . We know that we will have tex2html_wrap_inline210 for every t. Since

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we must have tex2html_wrap_inline216 for all t. This clearly implies that y(t) is always increasing. So the sign of f(y) helps us determine the behavior of the solutions (whether they are increasing or decreasing).

Example. Consider the autonomous equation

displaymath138

where the graph of f(y) is given below

From the graph, we can determine the equilibria. They are y=0, y=1, and y=2. Using the discussion above, we obtain the following information:

tex2html_wrap_inline234
if y(t) is a solution satisfying y(0) < 0, then y(t) is always increasing;
tex2html_wrap_inline234
if y(t) is a solution satisfying 0<y(0) < 1, then y(t) is always decreasing;
tex2html_wrap_inline234
if y(t) is a solution satisfying 1 <y(0) < 2, then y(t) is always increasing;
tex2html_wrap_inline234
if y(t) is a solution satisfying 2 <y(0), then y(t) is always decreasing.

In the picture below, we draw some solutions

Remark. Recall that if the function h(t) is increasing or decreasing, then the following hold :

tex2html_wrap_inline234
the limit of h(t) when tex2html_wrap_inline272 exists (or is a number) if and only if the function is bounded above; otherwise we have

displaymath274

tex2html_wrap_inline234
the limit of h(t) when tex2html_wrap_inline280 exists (or is a number) if and only if the function is bounded below; otherwise we have

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Note that in this case, if h(t) has a limit (as a number) when tex2html_wrap_inline286 , and h(t) is a "nice function", then we must have

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The next image illustrates this conclusion better than words can do.

Example. Consider the solution to the initial value problem

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where the graph of f(y) is given below

Discuss the limit of y(t) when tex2html_wrap_inline286 .

Answer. Using the above results, we know that for the given initial condition, y(t) is decreasing and for every t we have

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The above remark implies that tex2html_wrap_inline306 exist and we have

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Since y(t) is not a constant solution (because y(0)=0.5), then we have

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On the other hand, since

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and y is solution to the autonomous equation y'=f(y), then the only possibilities for both limits are the equilibria 0,1, and 2. Clearly if we put everything together, we conclude that

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Summary. Let us write down some general steps to follow when dealing with the autonomous equation

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Step 1. Find the constant solutions or equilibria through the algebraic equation

displaymath142

Draw the constant solutions. Note that here we are drawing y versus t.

Step 2. Find the sign of the function f(y) through its graph (versus y).
Step 3. In the region between any two equilibria, the solutions will be all increasing (if f(y) is positive) or decreasing (if f(y) is negative).
Step 4. If a solution is increasing and bounded above by a constant solution y = L, then the limit of the solution when tex2html_wrap_inline272 is the number L. Otherwise the limit is tex2html_wrap_inline346 .
If this solution is bounded below by a constant solution y = l, then the limit of the solution when tex2html_wrap_inline280 is the number l. Otherwise the limit is tex2html_wrap_inline354 .
Step 5. If a solution is decreasing and bounded below by a constant solution y = L, then the limit of the solution when tex2html_wrap_inline272 is the number L. Otherwise the limit is tex2html_wrap_inline354 .
If this solution is bounded above by a constant solution y = l, then the limit of the solution when tex2html_wrap_inline280 is the number l. Otherwise the limit is tex2html_wrap_inline346 .
Step 6. Draw some solutions.

Remark. You must be very careful here, since two graphs will be drawn: one for the function f(y) (versus y) and another one for the solutions, where this time y is on the vertical axis while t is on the horizontal axis. Keep in mind that the second graph is the most important one, since it deals with what we are looking for: the solutions of the differential equation.

Example. Draw some solutions for the equation

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Answer. Note that this equation models the logistic growth with threshold. The equilibria or constant solutions are given by

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or y=0, y=2, and y=5. The graph of the function f(y) = .2y(5-y)(y-2) is given in the next picture

So we have:

tex2html_wrap_inline234
the solution which satisfies the initial condition tex2html_wrap_inline394 , with tex2html_wrap_inline396 , is increasing. Moreover we have

displaymath398

tex2html_wrap_inline234
the solution which satisfies the initial condition tex2html_wrap_inline394 , with tex2html_wrap_inline404 , is decreasing. Moreover we have

displaymath406

tex2html_wrap_inline234
the solution which satisfies the initial condition tex2html_wrap_inline394 , with tex2html_wrap_inline412 , is increasing. Moreover we have

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tex2html_wrap_inline234
the solution which satisfies the initial condition tex2html_wrap_inline394 , with tex2html_wrap_inline420 , is decreasing. Moreover we have

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In the picture below we draw some solutions

Phase Line

Let us reconsider the above example. Let us focus our attention on the solutions which satisfy the initial condition tex2html_wrap_inline394 and tex2html_wrap_inline412 . We already know that

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It is somehow amazing that this conclusion is valid regardless whether tex2html_wrap_inline190 is close to 5 or close to 2. In this case we say that the equilibrium point 5 is attractive from below. In fact, in this example, this equilibrium point is attractive from below and above. On the other hand, the solutions get away from the equilibrium point 2 from below and above. We will say that the equilibrium point 2 is repelling. Note that there is no general agreement on the words used to describe these phenomena.

Definition. Classification of Equilibrium Points.
Consider the autonomous equation

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Assume that tex2html_wrap_inline190 is an equilibrium point (that is tex2html_wrap_inline196 ).

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The equilibrium point tex2html_wrap_inline190 is called a sink if and only if any solution y(t) to the autonomous equation such that y(0) is close to tex2html_wrap_inline190 , then we have

displaymath448

tex2html_wrap_inline234
The equilibrium point tex2html_wrap_inline190 is called a source if and only if any solution y(t) to the autonomous equation such that y(0) is (or not) close to tex2html_wrap_inline190 , then y(t) will move away from tex2html_wrap_inline190 when tex2html_wrap_inline272 .
tex2html_wrap_inline234
The equilibrium point tex2html_wrap_inline190 is called a node if and only if it is neither a source nor a sink.

Example. Classify the equilibrium points of the equation

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as source, sink, or node.
Answer. The equilibrium points are 0, 2, and 5. Using the above results, we see that 0 and 5 are sinks while 2 is a source.

Example. Find and classify the equilibrium points of the equation

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as source, sink, or node.
Answer. It is easy to see that the only equilibrium point is 0. Since tex2html_wrap_inline474 is always positive, then the solutions will always be increasing. Using previous results, we conclude:

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if y(t) is a solution satisfying y(0) < 0, then we have

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tex2html_wrap_inline234
if y(t) is a solution satisfying y(0) > 0, then we have

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Hence, the equilibrium point 0 is a node. In the picture below we draw some solutions.

There is a very nice way to put all this information together. Indeed, draw a vertical line (where the variable describing it is y) and start by marking the equilibrium points of the equation

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on the y-axis.

Then using the sign of f(y), we draw arrows pointing upward in a region where f(y) is positive, and downward in a region where f(y) is negative. This vertical line is called the phase line of the equation.

An equilibrium point is a sink, if the arrows on both sides point towards the equilibrium point, and it is a source, if both arrows point away from it.

Example. Draw the phase line of the equations

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and

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Answer. We will use our previous knowledge to get the two phase lines

and

You can readily see, that an equilibrium point is a sink, if the arrows on both sides point towards the equilibrium point, and that it is a source, if both arrows point away from it.

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Author: Mohamed Amine Khamsi
Last Update 6-22-98

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