**Authors**

**Abstract**

Â Extended

AbstractÂ 1-

Introduction[1]Â In geomorphology literature, equilibrium refers to researches of Davis and

Gilbert who presented steady-state and dynamics equilibrium. To analyze chaotic

geosystems and pattern formation in non-equlibrium conditions, the nonlinear

thermodynamics concepts is needed. In this article, have been analyzed

geosystems responses in non-equilibrium states by Lyapunov theory.

Self-organization of dissipative systems leads to form regular pattern in

non-equilibrium which can be a sign to forecast geosystem evolutions.Â Â Â Non-Equilibrium Geosystems: Â The second law of thermodynamics asserts that if a spontaneous reaction

occurs, the reaction moves towards an irreversible state of equilibrium and in

the process, becomes increasingly random or disordered. It is this increasing

disorder or entropy of a system that forces a spontaneous reaction to persist

but, once a system attains maximum entropy or equilibrium, the spontaneous

reaction ceases to continue.Â Non-equilibrium systems

are maintained in a state away from thermodynamic equilibrium by the steady

injection and transport of energy. Most interesting to us is systems displaying

regular or nearly regular spatial structures same ripple marks and river

pattern in geosystems.Â A particular

nonequilibrium system can be thought of as occupying a point in a

three-dimensional parameter space with axes labeled by three dimensionless

parameters R, Î, and N.Â The parameter R is some

dimensionless parameter like the Rayleigh number that measures the strength of

driving compared to dissipation. For many systems, driving a system further

from equilibrium by increasing R to larger values leads to chaos and then to

ever-more complicated spatiotemporal states for which there is ever finer

spatial structure and ever faster temporal dynamics.Â Â Â Lyapunov theory and regular patternÂ An n-dimensional (where the number of dimensions equals the number of

components) system has n Lyapunov exponents, which determine the rate of

convergence or divergence of initially similar system states in the system

phase space, and thus the sensitivity to perturbations or to variations in

initial conditions.Â The system is not, and

cannot be, chaotic unless there is at least one positive Lyapunov exponent.

Because an unstable system has at least one Î» > 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.

AbstractÂ 1-

Introduction[1]Â In geomorphology literature, equilibrium refers to researches of Davis and

Gilbert who presented steady-state and dynamics equilibrium. To analyze chaotic

geosystems and pattern formation in non-equlibrium conditions, the nonlinear

thermodynamics concepts is needed. In this article, have been analyzed

geosystems responses in non-equilibrium states by Lyapunov theory.

Self-organization of dissipative systems leads to form regular pattern in

non-equilibrium which can be a sign to forecast geosystem evolutions.Â Â Â Non-Equilibrium Geosystems: Â The second law of thermodynamics asserts that if a spontaneous reaction

occurs, the reaction moves towards an irreversible state of equilibrium and in

the process, becomes increasingly random or disordered. It is this increasing

disorder or entropy of a system that forces a spontaneous reaction to persist

but, once a system attains maximum entropy or equilibrium, the spontaneous

reaction ceases to continue.Â Non-equilibrium systems

are maintained in a state away from thermodynamic equilibrium by the steady

injection and transport of energy. Most interesting to us is systems displaying

regular or nearly regular spatial structures same ripple marks and river

pattern in geosystems.Â A particular

nonequilibrium system can be thought of as occupying a point in a

three-dimensional parameter space with axes labeled by three dimensionless

parameters R, Î, and N.Â The parameter R is some

dimensionless parameter like the Rayleigh number that measures the strength of

driving compared to dissipation. For many systems, driving a system further

from equilibrium by increasing R to larger values leads to chaos and then to

ever-more complicated spatiotemporal states for which there is ever finer

spatial structure and ever faster temporal dynamics.Â Â Â Lyapunov theory and regular patternÂ An n-dimensional (where the number of dimensions equals the number of

components) system has n Lyapunov exponents, which determine the rate of

convergence or divergence of initially similar system states in the system

phase space, and thus the sensitivity to perturbations or to variations in

initial conditions.Â The system is not, and

cannot be, chaotic unless there is at least one positive Lyapunov exponent.

Because an unstable system has at least one Î» > 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.

May 2014

Pages 51-64

**Receive Date:**14 June 2016**Revise Date:**14 April 2024**Accept Date:**14 June 2016