Environmental Engineering Reference
In-Depth Information
Now we have the main conceptual ideas at hand to deal with more complex
cases. A strategic approach to represent more complex situations can move in two
directions - to increase the number of variables, and to replace constants by
particular algebraic expressions involving the variables; i.e. to introduce nonlinea-
rities. While the Lotka-Volterra equations can be solved analytically, equations
with higher nonlinearities usually require numeric approximation (simulation).
6.5 The Few Basic Types of Long-Term Behaviour
in Deterministic Dynamical Systems
The determinism implied in differential equations always results in the same
trajectory when identical functions and identical initial conditions are used. From
this general condition we can derive relevant restrictions concerning the types of
dynamics that can occur in these systems. The restrictions apply to systems which
do not receive external inputs and are completely described by the equations. Such
a model is called an
autonomous system
. The knowledge about these restrictions
can be used to determine the minimum dimensions for certain types of dynamic
behaviour to occur. In the following we introduce the most important types of
dynamics in differential equations. If the description of a system's behaviour
requires the use of external forcing functions, it is obvious that the system equations
themselves capture only a part of what determines the dynamics of the considered
variables. This usually invokes consideration of whether a more complete descrip-
tion could be achieved in further studies.
6.5.1 Dynamic Properties in One-Dimensional Systems
One-dimensional systems are the most restricted concerning the potential dyna-
mics. The only possibilities are to (a) approach zero, (b) approach infinity or (c)
approach a steady state (steady state equilibrium).
There can be more than one equilibrium point in a particular equation if non-
linearities are involved. If there is more than one equilibrium point, the initial
conditions are crucial which of the alternative equilibria is approximated (see
below: domain of attraction).
Collapse or Explosion
A simplistic way to describe collapsing or exploding dynamics is by exponential
increase or decline as the only possible alternatives of the dynamics. An example
was presented in (6.8) with
N
initial
>
0 and
c
>
0 for explosion,
c
<
0 for collapse,
and marginal stability for
c
0. This type of behaviour can also occur in systems
with a higher number of variables.
ΒΌ
