Environmental Engineering Reference
In-Depth Information
Fig. 19.7 The Lorenz
attractor, produced using
MATLAB
®
Attractor of Lorenz system
the listing. Then the trajectory in the phase space, which is spanned by the first two
variables, is plotted (computation and output are not strictly separated in this M-
file). The advantage of the chosen procedure is that the storage space for the
solutions remains small. If the
span
is small, only a limited number of storage is
required: the old solution values are overwritten after the corresponding part of the
trajectory has been plotted.
The example shows that even simple nonlinear systems may exhibit chaotic
behavior. Also systems of chemical or biological species may lead to observations
similar to the Lorenz system (see for example: Fussmann and Heber
2002
).
If a model system is chaotic, it can not be used for predictions. In fact, chaos in
the mentioned sense can also be characterized by the property that tiny deviations in
an initial state lead to significant deviations at later times (Devaney
1987
). The
modeler, working with dynamical systems, should be aware that models may show
such a 'strange' behavior.
References
Devaney RL (1987) An introduction to chaotic dynamical systems. Addison-Wesley, Redwood
City, p 320
Fussmann G, Heber (2002) Food web complexity and chaotic population dynamics. Ecol Lett
5:394-401
Goldschmidt T (1998) Darwin's Dreampond. MIT Press, Boston, p 274
Hale J, Ko¸ak H (1991) Dynamics and bifurcations. Springer, New York, p 568
Jordan DW, Smith P (1977) Nonlinear ordinary differential equations. Clarendon, Oxford, p 360
Lorenz EN (1963) Deterministic nonperiodic flow. J Atmos Sci 20:130-141
Lotka AJ (1956) Elements of mathematical biology. Dover Publ, New York, p 465
Murray JD (2002) Mathematical biology I: introduction. Springer, New York, p 551
Polking J (2004) Ordinary differential equations using MATLAB. Prentice Hall, Upper Saddle
River, p 264
Richter O (1985) Simulation des Verhaltens okologischer Systeme, VCH Verlagsgesellschaft,
Weinheim, 219 p (in German)
