This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up**This preview shows
pages
1–3. Sign up
to
view the full content.*

682
CHAPTER
7
Nonlinear Systems of
Differential Equations
7.1
Nonlinear Systems
±
Review of Classifications
1.
2s
i
n
xx
t
y
yx
y
t
γ
′
=+
′
+
Dependent variables:
,
x y
Parameter:
Nonautonomous linear system
Nonhomogeneous
(s
i
n)
t
2.
34
i
n
uu
ut
υ
′
′=−
+
Dependent variables: ,
u
Parameters: none
Nonautonomous linear system
Nonhomogeneous (sin )
t
3.
12
21
=
=—sin
x
x
κ
′
′
Dependent variables:
,
x
x
Parameter:
Autonomous nonlinear
1
(sin
)
x
system
4.
sin
pq
qp
q
t
′ =
′
=−
Dependent variables:
p
,
q
Parameters: none
Nonautonomous nonlinear (
)
pq
system
5.
Sr
S
I
Ir
S
II
RI
′ = −
−
′ =
Dependent variables:
R
,
S
,
I
Parameters: ,
r
Autonomous nonlinear (
)
SI
system
±
Verification Review
6.
=
x
x
yy
′
′ =
Substituting
t
x
ye
=
=
into the two
differential equations, we get
=
=
tt
ee
7.
x
y
′ =
′ = −
Substituting
sin
x
t
=
and
sin
yt
=
into the
two differential equations, we get
cos
cos
sin
sin
=
−=
−
8, 9.
Use direct substitution, as in Problems 6 & 7.

This
** preview**
has intentionally

SECTION 7.1
Nonlinear Systems
683
±
A Habit to Acquire
For the phase portraits of Problems 10-13 we focus on the
slope information
contained in the DEs, using
the following general principles:
•
Setting
0
x
′ =
gives the v-nullcline of vertical slopes
•
Setting
0
y
′ =
gives the v-nullcline of vertical slopes
•
The equilibria are located where an h-nullcline intersects a v-nullcline, i.e., where
0
x
′ =
and
0
y
′ =
simultaneously
•
In the regions between nullclines the DEs tell whether trajectories move left or right (sign of
x
′
), up
or down (sign of
y
′
)
•
The direction picture that results shows the stability of the equilibria
Note: if the trajectories circle around an equilibrium, further argument is necessary to distinguish
between a center and a spiral (which could be either stable or unstable).
Note: For computer-drawn trajectories, the Runge-Kutta method will be far more accurate than Euler's
method at answering these questions.
Note: Recall from Section 6.5 that an equilibrium with trajectories that head toward it in one direction
and others that head away in
another
direction is a
saddle
. Unique trajectories (
separatrices
) head to or
from a saddle and
separate
the behaviors.
10.
(1
)
xy
yx
x
′ =
=−
Equilibria: (0,0), (1,0)
v-nullcline:
0
y
=
h-nullclines:
0
x
=
and
1
x
=
Because some direction arrows point away from (1,0) the second equilibrium is unstable; in fact because
other direction arrows point toward it, this equilibrium is more precisely a saddle.
Because direction arrows circle around (0,0) the first equilibrium could be either a center or a spiral. We
argue that the symmetry of the direction field for positive and negative
y
implies trajectories must circle
rather than spiral, so the equilibrium is a center point.
The trajectories and vector field confirm all of the above information; see figures. Most trajectories come
from the lower right, bend around the equilibria, and leave at the upper right, except those that circle
through points between (0,0) and (1,0), the equilibria, the separatrices of the saddle.

This is the end of the preview. Sign up
to
access the rest of the document.

Ask a homework question
- tutors are online