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Solutions of pure-time- delay dynamical systems
Solutions of pure-time-
delay dynamical systems
Duncan K. Foley
In the Marxian circuit of capital analysis the following equation system arises:
(1)
P
@tD=
-¶
t
C
@tD a@t-tD t
S
@tD=H1+qL
-¶
t
P
@tD b@t-tD t
C
@tD=
-¶
t
H1+pqL
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
1 + q
S
@tD c@t-tD t+B@tD
Here C is the flow of capital outlays into the production process, P is the flow of value of finished com-
modities at cost emerging from the production process, a is the transfer function representing the produc-
tion process, S is the flow of sales valued at selling prices including the mark-up on cost q = e k , where e
is the ratio of surplus value (profits) to wages, and k is the ratio of wages to total capital outlays (costs), b
is the transfer function representing the realization lag, p is the proportion of surplus value recommitted to
capitalist production (assuming that all recovery of costs is recommitted), c is the transfer function repre-
senting the finance lag, and B is the flow of new borrowing to finance capital outlays.
Transforming equations (1) by Laplace transforms, where C
*
@sD is the transform of C@tD, and similarly
yields:
(2)
P
*
@sD=a
*
@sD C
*
@sD
S
*
@sD=H1+qL b
*
@sD P
*
@sD=H1+qL a
*
@sD b
*
@sD C
*
@sD
C
*
@sD=H1+pqL
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
1 + q
c
*
@sD S
*
@sD=H1+pqL a
*
@sD b
*
@sD c
*
@sD C
*
@sD+B
*
@sD
Solving for C
*
@sD, we get:
(3)
C
*
@sD=
B
*
@sD
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅ
1 -
H1+pqL a
*
@sD b
*
@sD c
*
@sD
The pure time-delay transfer functions have the form a
@tD=d@t-T
P
D where d@.D is the Dirac delta func-
tion, and T
P
is the production lag, and similarly for the realization and finance lags, T
R
and T
F
. The
Laplace transforms of these functions are a
*
@sD=
-s T
P
and similarly for the realization and finance lags.
In this case equation (3) takes the form:
Solutions of Pure-Time-Delay.nb
1 The pure time-delay transfer functions have the form a
@tD=d@t-T
P
D where d@.D is the Dirac delta func-
tion, and T
P
is the production lag, and similarly for the realization and finance lags, T
R
and T
F
. The
Laplace transforms of these functions are a
*
@sD=
-s T
P
and similarly for the realization and finance lags.
In this case equation (3) takes the form:
(4)
C
*
@sD= B
*
@sD
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
1 -
H1+pqL
-s T
P

-s T
R

-s T
F
=
B
*
@sD
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅ
1 -
H1+pqL
-s T
where T = T
P
+ T
R
+ T
F
is the total circuit of capital lag.
Using the method of residues, given an exogenous path of borrowing, the solutions for C
@tD are: (5)
C
@tD=
n=-¶
¶
B
*
@s
n
D
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
Q '
@s
n
D
s
n
t
where Q
@sD=1-
H1+pqL
-s T
is the denominator of equation (4), and , s
-n
, , s
0
, , s
n
, are the
roots of Q
@sD. These roots are:
(6)
s
n
= Log
@1+pqD+2pnÂ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
T
Simplify
@E^HLog@1+pqD+2PinIL,nIntegersD
1 + pq
At a root, Q '
@s
n
D=s
n
H1+pqL
-s
n
T
= s
n
, since
H1+pqL
-s
n
T
= 1.
s
0
=
Log
@1+pqD
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
T
= gis the only real root, which corresponds to paths of expanded reproduction. The other
roots
come
in
conjugate
pairs,
since
s
n
=
Log
@1+pqD+2pnÂ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
T
= g +
2 p n Â
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
T
and
s
-n
=
Log
@1+pqD-2pnÂ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
T
= g -
2 p n Â
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
T
. If, for example, we take B
@t¥0D=1,B@t<0D=0, then B
*
@sD=1,
and the coefficient of
s
n
t
in the solution (5) will be
1
ÅÅÅÅÅÅ
s
n
=
s
-n
ÅÅÅÅÅÅÅÅÅ
»s
n
»
. We can thus combine the terms correspond-
ing to s
n
and s
-n
to eliminate the imaginary components. In this case the solution is:
(7)
C
@tD=
g t

i
k
j
j
j
j
j
1
ÅÅÅÅ
g + 2 n=1
¶

1
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
»s
n
» JgCosA2pntÅÅÅÅTE-2pnÅÅÅÅÅÅÅÅÅÅÅÅTSinA2pntÅÅÅÅTEN
This example illustrates two points. First, systems of integral equations can have an infinite number of
roots, each of which will contribute to the solution path. Second, in the case of the circuit of capital,
starting the circuit from nothing with a sudden increase in capital outlays financed by borrowing will
excite all of the latent cyclical paths of the system. What these extra terms do is to suppress the growth
path for on the interval
@0,TD before the circuit can get started.
As an example, take a discrete-time system, in which there is a three-period total lag in production,
realization and finance.
CapOutlay
@p_,q_,B_D@capoutlays_D:=
8capoutlays,H1+pqL capoutlays@@-3DD+B<êêFlatten
Solutions of Pure-Time-Delay.nb
2 CapOutlay
@.5,1,1D@80,0,0<D
80,0,0,1<
CapPath
@CapOutlayF_,initcapoutlays_,n_D:=
NestList
@CapOutlayF,initcapoutlays,nD
CapPath
@CapOutlay@.5,1.,1D,80,0,0<,10D
880,0,0<,80,0,0,1<,80,0,0,1,1<,80,0,0,1,1,1<,
80,0,0,1,1,1,2.5<,80,0,0,1,1,1,2.5,2.5<,
80,0,0,1,1,1,2.5,2.5,2.5<,80,0,0,1,1,1,2.5,2.5,2.5,4.75<,
80,0,0,1,1,1,2.5,2.5,2.5,4.75,4.75<,
80,0,0,1,1,1,2.5,2.5,2.5,4.75,4.75,4.75<,
80,0,0,1,1,1,2.5,2.5,2.5,4.75,4.75,4.75,8.125<<
ListPlot
@CapPath@CapOutlay@.5,1,.1D,80,0,0<,100DêêLast,
PlotJoined Ø True, PlotRange Ø All
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Solutions of Pure-Time-Delay.nb
3