0, dynamic instability is
tantamount to a chaotic system. Deterministic chaos is a property of some
nonlinear systems whereby even simple deterministic systems can produce
complex, pseudorandom patterns, independently of stochastic forcing or
environmental heterogeneity. In chaotic systems complexity and unpredictability
are inherent in system dynamics. Such systems are strongly sensitive to initial
conditions, in that initially similar states diverge exponentially, on average,
and become increasingly different over time. Chaotic systems are also sensitive
to perturbations of all magnitudes.Â The Kolmogorov (K-)
entropy of a nonlinear system measures its 'chaoticity', because K-entropy is
equal to the sum of the positive Lyapunov exponents. In real landscapes,
measured entropy can be due to deterministic complexity, or to 'colored noise',
the combination of randomness and deterministic order. Culling (1988b) was
apparently the first to suggest exploiting the relationship between K-entropy
(estimated using standard statistical or information theoretic entropy
measures) and chaos in geomorphic systems.Â There are three forms of entropy referred to in geomorphology.
Thermodynamic entropy is a measure of the amount of thermal energy unavailable
to do work, or the disorder in a closed system. Statistical (information
theoretic) entropy measures the loss of information in a transmission, or the
degree of disorder in a statistical distribution.Â Kolmogorov (K-) entropy
measures the expansion of a system's phase space (the n-dimensional space
defining all possible system states or combinations of values of the n
components).Â Â Â ConclusionÂ Many geomorphic
systems show clear evidence of chaotic dynamics and deterministic complexity.
These phenomena cause many nonlinear dynamical systems to behave unpredictably
(at certain scales), to exhibit extraordinary sensitivity to initial
conditions, and to show complicated, pseudorandom patterns even in the absence
of environmental heterogeneity and stochastic forcing. The positive Î» reflect
the K-entropy or 'chaoticity', and the rate of disorganization the negative Î»
give the rate of organization. If an open, dissipative geomorphic system is to
organize itself, there must be at least one positive Lyapunov exponent, but the
sum of Î» must be negative. The sum of the diagonal elements of the system
interaction matrix is equal to the sum of real parts of the complex
eigenvalues, and to the Lyapunov exponents.]]>