Appendix 243
If 0 < K < +1, then Z
Z
t
t
+ <
1
and sign (Z t+1 ) = sign (Z t ). The system tends
to approach the equilibrium point (stability) with exponential convergence
(Figure A.5).
If K > +1, then Z
Z
t
t
+ >
1
and sign (Z t+1 ) = sign (Z t ). The system tends to
depart from the equilibrium point (instability) with exponential deviation
(Figure A.6).
Those conclusions result from the study of only one variable, but they are
tions. We should point out the value of a linear approach for small perturbations, given that according to mathematical analysis it is often acceptable
as a first approach when we deal with continuous differentiable functions.
A system’s response to an external perturbation can play a significant role
in the regulation and control of its behaviour. The case of negative feedback
is of special interest for cybernetics; in this case, the external perturbation
0
x
y
Figure A.4 Converging oscillations in the phase space. (From Hadjibiros, K. (2007).
Ecology. Ecosystems and Environmental Protection, 3rd edition. Symmetria,
Athens (in Greek). With permission.)
0
t
x
Figure A.5 Exponential convergence. (From Hadjibiros, K. (2007). Ecology. Ecosystems
and Environmental Protection, 3rd edition. Symmetria, Athens (in Greek).
With permission.)