Recall that only the solutions of linear systems may be found
explicitly. The problem is that in general real life problems may
only be modeled by nonlinear systems. In this case, we only know how
to describe the solutions globally (via nullclines). What happens
around an equilibrium point remains a mystery so far. Here we propose
the to discuss this problem. The main idea is to approximate a
nonlinear system by a linear one (around the equilibrium point). Of
course, we do hope that the behavior of the solutions of the linear
system will be the same as the nonlinear one. This is the case most
of the time (not all the time!).

**Example.** Consider the Van der Pol equation

This is a nonlinear equation. Let us translate this equation into a system. Set . Then we have

The equilibrium points reduce to the only point (0,0). Let us find the nullclines and the direction of the velocity vectors along them.

- The x-nullcline is given by
Hence the x-nullcline is the x-axis.

- The y-nullcline is given by
Hence the y-nullcline is the curve .

Note that the arrangement of these curves tell us that the solutions
``circles'' around the origin. But it is not clear whether the
solutions circle and dye at the origin, circle away from the origin,
or keep on circling periodically. A very rough approach to this
problem suggests that if we rewrite the term as , then when (*x*,*y*) is close to (0,0), the term is
very small compared to -*x*+*y*. Hence a close system to the original
nonlinear system is

which happens to be a linear system. The eigenvalues of this system are . Hence the solutions of the linear system spiral away from the origin (since the real part is positive). So we suggest that the solutions of nonlinear system spiral away from the origin (look at the picture below)

The solution started close to the equilibrium point, then it moved away. Notice that in this case, the trajectory is getting close to what looks like a cycle. To better see this, let us consider the graphs of the function *x(t)* and *y(t)*:

and

So what if we want to
generalize this to different systems. Is there a technique that mimic
what we did? The answer is yes. It is called **linearization**.

**Linearization Technique.**

Consider the autonomous system

And assume that is an equilibrium point. So we would like
to find the closest linear system when (*x*,*y*) is close to
. In order to do that we need to approximate the functions
*f*(*x*,*y*) and *g*(*x*,*y*) when (*x*,*y*) is close to . This is
a similar problem to approximating a real valued function by its
tangent (around a point of course). From multivariable calculus, we
get

and

when (*x*,*y*) is close to . Then the nonlinear system may
be approximated by the system

But since is an equilibrium point, then we have . Hence we have

This is a linear system. Its coefficient matrix is

This matrix is called the **Jacobian matrix** of the system at the
point .

**Summary of the linearization technique.**

Consider the autonomous system

and an equilibrium point.

- Find the partial derivatives
Write down the Jacobian matrix

- Find the eigenvalues of the Jacobian matrix.
- Deduce the fate of the solutions around the
equilibrium point from the eigenvalues. For example,
- if the eigenvalues are negative or complex with negative real part, then the equilibrium point is a sink (that is all the solutions will dye at the equilibrium point). Note that if the eigenvalues are complex, then the solutions will spiral around the equilibrium point.
- If the eigenvalues are positive or complex with positive real part, then the equilibrium point is a source (that is all the solutions will move away from the equilibrium point). Note that if the eigenvalues are complex, then the solutions will spiral away from the equilibrium point.
- If the eigenvalues are real number with different
sign (one positive and one negative), then the the equilibrium point
is a saddle. In fact, there will be two solutions which approach the
equilibrium point as , and two more solutions
which approach the equilibrium point as . For
the linear system theses solutions are lines, but for the nonlinear
system they are not in general. These four solutions are called
**separatrix**.

**Remark.** When dealing with an autonomous system without prior
knowledge of the equilibrium point, then we advice to first find the
Jacobian matrix and plug the values for every equilibrium point. This
way you don't repeat the calculations over and over again.

**Example.** Consider the equation of the pendulum

where is the damping coefficient. See the picture below.

The equivalent system is

The equilibrium points are , where . The angles , for , correspond to the pendulum at its lowest position, while , for , correspond to the pendulum at its highest position. The Jacobian matrix of the system

Let us concentrate on the equilibrium positions (0,0) and .

- For (0,0), the Jacobian matrix is
For the sake of illustration let us fix the parameters. For example,

- if we take (undamped pendulum), then the eigenvalues are which implies that the mass will oscillate around the lowest position in a periodic fashion.
- If (dumped pendulum),
*m*=1, and*l*=1. Then the eigenvalues areSince the real part is negative, the solutions will sink (dye) while oscillating around the equilibrium point. Here we have the same behavior for the linear and nonlinear system.

- For , the Jacobian matrix is
The eigenvalues are

Clearly we have two real eigenvalues with one positive and one negative. So the solutions will always get away from the equilibrium position except along one curve (the separatrix).

For more examples, click on Example.

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