Edge-of-chaos

One often hears the phraise edge-of-chaos in discussions of AL, as in such and such a system appears poised at the edge-of-chaos. As this important concept is still a topic of some debate, we make a few comments regarding it. 

Langton wais able to provide a tentative answer to his question by examining the behavior of the entire rule space of elementary one-dimensional cellular automata rules (see discussion in section 3.2) as parameterized by a single parameter A. He found that as A is increased from its minimal to maximal values, a path is effectively traced in the rule space that progresses from fixed point behavior to simple period- 

Langton s tentative answer to the question above is therefore We expect that information processing can emerge spontaneously and come to dominate the dynamics of a physical system in the vicinity of a critical phase transition. Langton speculates that the dynamics of phase transitions is fundamentally equivalent to the dynamics of information processing. 

Effective computation, such as that required by life processes and the maintenance of evolvability and adaptability in complex systems, requires both the storage and transmission of information. If correlations between separated sites (or agents) of a system are too small - as they are in the ordered regime shown in figure 11.3 -the sites evolve essentially independently of one another and little or no transmission takes place. On the other hand, if the correlations are too strong - as they are in the chaotic regime - distant sites may cooperate so strongly so as to effectively mimic each other s behavior, or worse yet, whatever ordered behavior is present may be 

However intuitive the edge-of-chaos idea appears to be, one shoidd be aware that it has received a fair amount of criticism in recent years. It is not clear, for example, how to even define complexity in more complicated systems like coevolutionary systems, much less imagine a phase transition between diffen ent complexity regimes. Even Langton s sugge.stion that effective computation within the limited domain of cellular automata can take place only in the transition region has been challenged.  

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Mitchell, M., P.T.IIrabor and J.P.Crutchfield, Revisiting the edge of chaos evolving cellular automata to perform computations, Complex Systems, Volume 7, 1993, 89-130. 

Iang90] Langton, C.G., Computation at the edge of chaos phase transitions and emergent computation , Physica, 42D (1990) 12-37. 

Iewin92] Lewin, R., Complexity Life at the Edge of Chaos, Macmillan Publishing Company (1992). 

It must be emphasized that the complex systems at all levels in our hierarchy are dynamical (interactive). There is a constant motion of the whole and of the parts. Therefore relationships, interactions or transactions that take place do so as stochastic events. This is precisely what Poincare described, who was perhaps the major founding father of post-Newtonian science. The fact that purposive encounters do occur as a part of biologically significant processes may be viewed either as a result of purely random events, or it may be characteristic of the behaviour of complex systems poised at the edge of chaos. 

Lewin, R. (1993). Complexity. Life at the Edge of Chaos , p. 177. Macmillan, New York. Lyman, W. (1982). Handbook of Chemical Property Estimation. Academic Press, New... 

When a system has long-time correlation, for which we expect fractional power scaling of excess heat, our assumption of the Boltzmann equilibrium distribution may always not be valid. Actually some power distributions such as the Tsallis distribution [14] have been reported at the edge of chaos [15]. A superstatistical equilibrium distribution is written as a superposition of Boltzmann distributions with different temperatures. Beck and Cohen [13] considered many types of distributions for the inverse of temperature. For example, they chose Gaussian, uniform, gamma, log-normal, and others. In particular, the Tsallis distribution is realized for gamma distribution. We will show that excess heat can be written as a superposition of correlation functions... 

Kauffman, S.A. and S. Johnsen. 1991. Coevolution to the edge of chaos coupled fitness landscapes, poised states, and coevolutionary avalanches.. Theor. Biol. 149 467-505. 

M. Bertram and A. Mikhailov. Pattern formation on the edge of chaos mathematical modeling of CO oxidation on a Pt(llO) surface under global delayed feedback. Phys. Rev. E, 67 1-9, 2003. 

The evolutionary field a) R. Lewin, Complexity - Life at the Edge of Chaos, Macmillan, New York 1992 Die Komplexitdtstheorie, Hoffmann Campe, Hamburg 1993 b) F. Cramer Chaos and Order, VCH, Weinheim 1993 Der Zeit-baum, Insel, Franfurt a. M. 1993 Symphonie des Lebendigen, Insel, Frankfurt a.M. 1996. 

The threshold between order and chaos seems to be an essential requisite of complex adaptive self-organising systems (order at the edge of chaos). As these systems are dissipative, an order through fluctuations is effective in working between the above mentioned conditions. 

Following the ideas of AndersonP l, Langton , who is one of the founders of artificial life research, coined the phrase life at the edge of chaos . To identify the condition of life for a cellular automaton system, he defined a control parameter A. For a cellular automaton with K states and N neighbors the A parameter is defined as follows. An arbitrary state Sq is chosen as a quiescent state, as for instance, the solution zero, with for the logistic map. If there are n, transitions defined to state Sq, the remaining... 

C. G. Langton, Life at the Edge of Chaos, in C.G. Langton, C.Taylor, J.D. Farmer, S. Rasmussen (eds.) Artificial Life II, Addison-Wesley, Redwood City, p. 41 (1992)... 

Kuhn, L. 2009. Adventures in Complexity - For organisations near the edge of chaos. Triarchy Press. Devon, UK. 

Conversety, for complexity theorists it is static equilibrium that puts complex adaptive systems at risk. Bounded instability, i.e. a state between stability and instability, is for them more conducive to evolution than either stable equilibrium or explosive instabilily. As the edge of chaos provides opportunities for coDective evolution, there is no (p. 528) use in apptying prejudicial terms such as burden to it (Pascale 2001). Rather problems and solutions should be formulated in terms of the whole system instead of individual components. 

Pascale, R., MUleman, M., and Goja, L. (2001) Surfing the Edge of Chaos. New York Three Rivers Press. 

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