Section 5.2: Qualitative Analysis
Example 3: Consider the first order system
1.
Graph the nullclines for this system.
2.
Insert direction field "arrows" at the nullclines
3.
Determine the general direction of the direction field in each of the regions cut out by
the nullclines.
4.
Sketch possible phase portraits for the two initial conditions indicated by "dots" for as
long as the phase portraits stay within the graph's frame.
5.
Find all the equilibrium points of the system.
Answer:
1.
The x-nullclines can be obtained by setting x'=0, i.e.
The y-nullclines can be obtained by setting y'=0, i.e.
For 2, 3, 4, use the following graph:
5.
The equilibrium points of the system are the intersection points of the x-nullclines and ynullclines. Therefore, we get
,
which yields x=2 and
. Hence, we have two equilibrium points
.
Example 4: Consider the system
.
Draw the nullclines and find all the equilibrium points. Use this information to determine
the fate of the solutions corresponding to the following initial conditions:
Answer: Note that this system is non linear. The x-nullclines are given by
We recognize the two lines x=0 (y-axis) and the line -x-y+1=0.
The y-nullclines are given by
We recognize the line y=0 (x-axis) and the circle
with radius 2). In the graph below we give the nullclines:
(centered at (0,0)
The equilibrium points may be obtained as the intersection between the x-nullclines and
y-nullclines. There are 6 points (see the above graph). The solutions corresponding to
the given points are drawn in white in the graph below:
Example 5: Consider the system
Find the equilibrium points and the nullclines. Draw the vector field. Sketch some
solutions and specially the solutions around the equilibrium points.
Answer. The equilibrium points are given by the algebraic system
It is easy to see that we must have x=0 (from the second equation) and therefore y=0.
Hence the system has one equilibrium point (0,0). The x-nullcline are given by the
equation
which is the graph of the function
equation
. The y-nullcline are given by the
which reduces to the line x=0 (the y-axis). In the picture below, we draw the vector field
as well as the nullclines.
Clearly, the solutions spiral around the equilibrium point (see the picture below)
Notice that the solutions spiral and get closer to a cycle. We can see this better by
looking at the graphs of x versus t as well as the graphs of y versus t.
and
Example 6: Consider the system
Find the equilibrium points and the nullclines. Draw the vector field. Sketch some
solutions and specially the solutions around the equilibrium points.
Answer. The equilibrium points are given by the algebraic system
From the first equation, we get y=0. The second equation gives x=0 or x=1. Hence the
equilibrium points are (0,0) and (1,0). The x-nullcline are given by the equation y = 0
which is the x-axis and the y-nullcline are given by the equation
, which
reduces to the two vertical lines x=0 (the y-axis) and x=1. In the picture below, we draw
the vector field as well as the nullclines.
Clearly, the solutions spiral around the equilibrium point (1,0) and get away from the
other equilibrium point (0,0)(see the picture below)
A closer look at the solutions around the equilibrium point (1,0) gives
Clearly the solutions are close to cycles. The graphs of x versus t as well as the graphs
of y versus t illustrate better this remark:
and
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