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.
Hence the x-nullcline is the x-axis.
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):
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.
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
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
Summary of the linearization technique.
Consider the autonomous system
and an equilibrium point.
Write down the Jacobian matrix
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 the sake of illustration let us fix the parameters. For example,
Since 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.
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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Author: Mohamed Amine Khamsi