*x*and

*y*for dependent variables. Of course, the above system can be written in compact vector form

*t*by dot: \( \dot{\bf x} = {\text d}{\bf x} / {\text d} t. \)

If an initial position of the vector \( {\bf x} (t) \) is known, we get an initial value problem:

*t*in our case). It is of the form

*t*and \( {\bf f} ({\bf x}) \) takes values in

*n*-dimensional Euclidean space and

*t*is usually time.

# 2.4.1. Planar Case

**Example**.

Consider the following non-linear system of ordinary differential equations

*x = 0*. Then the first equation becomes \( 2y \left( 2y-4 \right) =0 , \) so

*y = 0*or

*y = 2*. More solutions can be found by choosing

*y = -1*in the second equation. Then the first equation becomes \( \left( x+2 \right) \left( x-6 \right) =0 , \) so

*x = -2*or

*x = 6*. Thus, we have obtained the four critical points: (0,0), (0,2), (-2,-1), and (6,-1).

Figure shows a direction field ■

For a two-dimensional autonomous system with at least one asymptotically stable critical point, it is often of interest to determine where in the phase plane the trajectories lie that ultimately approach a given critical point. Let *P* be a point in the *xy*-plane with the property that a trajectory passing through *P* ultimately approaches the critical point as \( t \mapsto \infty . \) Then this trajectory is said to be attracted by the critical point. Further, the set of all such points is called the **basin of attraction** or the **region of asymptotic stability** of the critical point. A trajectory that bounds a basin of attraction is called a **separatrix** because it separates trajectories that approach a particular critical point from other trajectories that do not do so. Determination of basin of attraction is important in understanding the large-scale behavior of the solutions of an autonomus system.

**Example**.
Consider the system

The trajectoreis of a two-dimensional autonomous system

*x*and

*y*. Observe that such a reduction is not usually possible for nonautonomous system when

*f*and

*g*depend also on time varibale

*t*. If the above equation \( {\text d}y / {\text d}x = g(x,y) / f(x,y) \) can be solved by any of the methods of Part I, and if we write solutions (implicitly) in the form

*c*, then it defines trajectories of the given autonomous system. In other words, the trajectories lie on the level curves of

*H(x,y)*. Keep in mind that there is no general way of solving first order equation \( y' = g(x,y)/f(x,y) \) to obtain the function

*H(x,y)*, so this approach is applicable only in special cases.

**Example**. Find trajectories of the system

*c*is an arbitrary constant. A computer plotting routine is helpful in displaying the level curves of

*H(x,y)*. The point (-3,2) is center anad (3,2) is a saddle point.

```
Complete
```

Since Matlab has no friendly subroutine to plot direction fileds for ODEs, we present several codes that allow to plot these fields directly. Many others can found on the Internet.

```
end
axis([tval(1)-dt/2,tval(end)+dt/2,yval(1)-dy/2,yval(end)+dy/2])
```