Chemistry Reference
In-Depth Information
(4.5) is defined only for positive values of the dynamic variables x
1
and x
2
. Conse-
quently, the first question that arises is, under which conditions does the sequences
of efforts remain positive? Given initial efforts x
1
.0/;x
2
.0/, we will say that a tra-
jectory is
feasible
if .x
1
.t/;x
2
.t//
D
T
t
.x
1
.0/;x
2
.0//; t
D
0;1;2;::: is entirely
contained in the positive orthant
R
2
C
D
f
.x
1
;x
2
/
j
x
1
>0and x
2
>0
g
.The
feasible
set
is the subset of
whose points generate feasible trajectories.
For our dynamical system it is obvious that under the conditions
v
i
c
i
<1for
i
D
1;2, it follows that if the efforts in period t are positive, then the efforts in
the subsequent period are positive as well. That is, if .x
1
.t/;x
2
.t//
2 R
R
2
C
2
C
then
2
C
.x
1
.t
C
1/;x
2
.t
C
1//
2 R
: Furthermore, it is easy to check that any feasible tra-
jectory of our dynamical system is bounded (Bischi and Kopel (2003
b
)). Hence, the
conditions
v
1
c
1
<1and
v
2
c
2
<1are sufficient for the feasibility and boundedness
of all points in R
2
C
. It turns out that these conditions are also necessary for the feasi-
bility of the whole region
2
C
R
. If at least one of these two inequalities does not hold,
2
C
then points of
exist that generate infeasible trajectories. In order to see this,
note first that the coordinate axes are invariant: x
i
.t/
D
0 implies x
i
.t
C
1/
D
0.
The dynamics along the invariant x
i
-axis is governed by the one-dimensional linear
map
R
x
i
.t
C
1/
D
.1
v
i
c
i
/x
i
.t/:
(4.6)
For example, if
v
1
c
1
<1, then given a point .x
1
;0/, with x
1
>0, the map (4.6)
generates a sequence of points on the x
1
-axis with x
1
>0. By continuity, the same
holds for points .x
1
;x
2
/ with arbitrarily small x
2
. Hence, in this case the feasible
region includes the x
1
-axis. Instead, if
v
1
c
1
>1, then a point .x
1
;0/, with x
1
>0,
generates a negative point after the first iteration of (4.6). In this case the whole x
1
-
axis must belong to the set of infeasible points. Clearly, the same reasoning applies
to the x
2
-axis.
In order to obtain an exact delineation of the boundary of the feasible region, we
consider the invariant coordinate axes and their preimages. The map T is a nonin-
vertible map. If we consider a generic point
0;x
0
2
, x
0
2
>0,onthex
2
-axis, then its
preimages are the positive solutions of the system
.1
v
1
c
1
/x
1
x
ˇ
1
1
C
kx
ˇ
2
2
C
v
1
ˇ
1
Akx
ˇ
1
x
ˇ
2
D
0;
2
..1
v
2
c
2
/x
2
x
0
2
/
x
ˇ
1
C
kx
ˇ
2
2
C
v
2
ˇ
2
Akx
ˇ
1
x
ˇ
2
D
0;
1
2
obtained from (4.5) with x
i
.t/
D
x
i
as unknowns and x
1
.t
C
1/
D
0, x
2
.t
C
1/
D
x
0
2
taken as parameters. If
v
2
c
2
<1;then one solution always exists on the x
2
-axis. It
is given by x
1
1
1
v
2
c
2
x
0
2
. Solutions with x
1
>0cannot exist if
v
1
c
1
<1,
because in this case the first equation can never be satisfied. On the other hand,
if
v
1
c
1
>1, two preimages with x
1
>0exist. They are located on the curves with
equation
D
0, x
2
D
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