Environmental Engineering Reference
In-Depth Information
dN
1
dt
¼
C
11
N
1
K
N
1
ð
C
2
N
1
N
2
Þ
C
1
þ
N
1
(6.18)
Cs
þ
N
1
K
ð
AþN
1
Þ
N
1
dN
2
dt
ð
b
C
2
N
1
N
2
Þ
¼
C
3
N
2
ð
A
þ
N
1
Þ
N
1
For the following parameter values (
6.18
) exhibits a globally stable limit cycle:
C
1
¼
0.1,
C
11
¼
0.2,
Cs
¼
1,000,
K
¼
10,000,
C
2
¼
0.001,
A
¼
50,
b
¼
0.1,
C
3
¼
0.1.
6.5.3 Dynamic Properties in Three-Dimensional or Higher
Dimensional Systems
There are dynamic phenomena that cannot be observed in one-dimensional or two-
dimensional systems. In systems with three or more variables additional phenom-
ena can occur. These are chaos and quasi-periodic oscillations. The latter type of
behaviour can occur, if two or more oscillators are overlaid or intertwined. Trajec-
tories of initially close starting points remain close during the subsequent dynamics;
however, in a certain domain of the state space, every point will eventually be
approached as time proceeds. More material on quasi-periodic oscillators can be
found at
http://www.mced-ecology.org
.
Deterministic Chaos
Unlike in quasi-periodic oscillations, in systems exhibiting deterministic chaos we
do not find single distinct oscillation frequencies. Instead, a continuous spectrum of
frequencies occurs over time. This implies that originally closely neighbouring
starting points will gradually lose coherence. A criterion for chaos is that in any
domain of the state space there are trajectories that approximate the other areas in
the overall domain of attraction and lose correlation with each other. The correla-
tion between trajectories that are initially close to each other decays over the long
term. Periodic trajectories also exist. The resulting dynamics of chaotic systems are
complex. In three or higher dimensions, such a deterministic, non-periodic flow is
possible, even though the trajectories do not cross. Otherwise, this phenomenon
would not be possible in deterministic systems. An example of a chaotic system
inherent in the equations of a predator-prey system was discovered by Gilpin 1979.
It was published some time after Lorenz (1963), who had discovered chaotic
behaviour in dynamic systems for the first time when modelling turbulent atmo-
spheric processes.
