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Note: 1 lecture, §6.1–§6.2 in [EP], §9.2–§9.3 in [BD]

Except for a few brief detours in chapter 1, we considered mostly linear equations. Linear equations suﬃce in many applications, but in reality most phenomena require nonlinear equations. Nonlinear equations, however, are notoriously more diﬃcult to understand than linear ones, and many strange new phenomena appear when we allow our equations to be nonlinear.

Not to worry, we did not waste all this time studying linear equations. Nonlinear equations can often be approximated by linear ones if we only need a solution “locally,” for example, only for a short period of time, or only for certain parameters. Understanding linear equations can also give us qualitative understanding about a more general nonlinear problem. The idea is similar to what you did in calculus in trying to approximate a function by a line with the right slope.

In § 2.4 we looked at the pendulum of length . The goal was to solve for the angle as a function of the time . The equation for the setup is the nonlinear equation

Instead of solving this equation, we solved the rather easier linear equation

While the solution to the linear equation is not exactly what we were looking for, it is rather close to the original, as long as the angle is small and the time period involved is short.

You might ask: Why don’t we just solve the nonlinear problem? Well, it might be very diﬃcult, impractical, or impossible to solve analytically, depending on the equation in question. We may not even be interested in the actual solution, we might only be interested in some qualitative idea of what the solution is doing. For example, what happens as time goes to inﬁnity?

We restrict our attention to a two dimensional autonomous system

where and are functions of two variables, and the derivatives are taken with respect to time . Solutions are functions and such that

The way we will analyze the system is very similar to § 1.6, where we studied a single autonomous equation. The ideas in two dimensions are the same, but the behavior can be far more complicated.

It may be best to think of the system of equations as the single vector equation

(8.1) |

As in § 3.1 we draw the phase portrait (or phase diagram), where each point corresponds to a speciﬁc state of the system. We draw the vector ﬁeld given at each point by the vector . And as before if we ﬁnd solutions, we draw the trajectories by plotting all points for a certain range of .

Example 8.1.1: Consider the second order equation . Write this equation as a ﬁrst order nonlinear system

The phase portrait with some trajectories is drawn in Figure 8.1.

From the phase portrait it should be clear that even this simple system has fairly complicated behavior. Some trajectories keep oscillating around the origin, and some go oﬀ towards inﬁnity. We will return to this example often, and analyze it completely in this (and the next) section.

Let us concentrate on those points in the phase diagram above where the trajectories seem to start, end, or go around. We see two such points: and . The trajectories seem to go around the point , and they seem to either go in or out of the point . These points are precisely those points where the derivatives of both and are zero. Let us deﬁne the critical points as the points such that

In other words, the points where both and .

The critical points are where the behavior of the system is in some sense the most complicated. If is zero, then nearby, the vector can point in any direction whatsoever. Also, the trajectories are either going towards, away from, or around these points, so if we are looking for long term behavior of the system, we should look at what happens there.

Critical points are also sometimes called equilibria, since we have so-called equilibrium solutions at critical points. If is a critical point, then we have the solutions

In Example 8.1.1, there are two equilibrium solutions:

Compare this discussion on equilibria to the discussion in § 1.6. The underlying concept is exactly the same.

In § 3.5 we studied the behavior of a homogeneous linear system of two equations near a critical point. For a linear system of two variables the only critical point is generally the origin . Let us put the understanding we gained in that section to good use understanding what happens near critical points of nonlinear systems.

In calculus we learned to estimate a function by taking its derivative and linearizing. We work similarly with nonlinear systems of ODE. Suppose is a critical point. First change variables to , so that corresponds to . That is,

Next we need to ﬁnd the derivative. In multivariable calculus you may have seen that the several variables version of the derivative is the Jacobian matrix^{1}. The Jacobian matrix of the vector-valued function at is

This matrix gives the best linear approximation as and (and therefore and ) vary. We deﬁne the linearization of the equation (8.1) as the linear system

Example 8.1.2: Let us keep with the same equations as Example 8.1.1: , . There are two critical points, and . The Jacobian matrix at any point is

Therefore at the linearization is

where and .

At the point , we have and , and the linearization is

The phase diagrams of the two linearizations at the point and are given in Figure 8.2. Note that the variables are now and . Compare Figure 8.2 with Figure 8.1, and look especially at the behavior near the critical points.

Exercise 8.1.3: Find the critical points and linearizations of the following systems.

a) , ,

b) , ,

c) , .

Exercise 8.1.4: For the following systems, verify they have critical point at , and ﬁnd the linearization at .

a) ,

b) ,

c) , , where , , and all ﬁrst partial derivatives of and are also zero at , that is, .

Exercise 8.1.5: Take , .

a) Find the set of critical points.

b) Sketch a phase diagram and describe the behavior near the critical point(s).

c) Find the linearization. Is it helpful in understanding the system?

Exercise 8.1.6: Take , .

a) Find the set of critical points.

b) Sketch a phase diagram and describe the behavior near the critical point(s).

c) Find the linearization. Is it helpful in understanding the system?

Exercise 8.1.101: Find the critical points and linearizations of the following systems.

a) , ,

b) , ,

c) , .

Exercise 8.1.103: The idea of critical points and linearization works in higher dimensions as well. You simply make the Jacobian matrix bigger by adding more functions and more variables. For the following system of 3 equations ﬁnd the critical points and their linearizations:

,

,

.

Exercise 8.1.104: Any two-dimensional non-autonomous system , can be written as a three-dimensional autonomous system (three equations). Write down this autonomous system using the variables , , .

^{1}Named for the German mathematician Carl Gustav Jacob Jacobi (1804–1851).