Fractal evolution

Hierarchically Emergent Fractal Evolution (HEFE)

We have discovered a universal dynamical property in discrete systems (like QCA or CML) which we call fractal evolution [Fussy and Grössing 1994]. .
It is observed in lattice models if any additional temporal feedback operation is introduced (involving some memory of the system's states or some randomly generated mechanism) and if a normalization procedure after each time step is performed. Fractal evolution is characterized by a power-law behavior of a system's order parameter with regard to a resolution-like parameter which controls the deviation from an undisturbed (i.e., feedback-less) system's evolution, and it provides a dynamically invariant measure for the emerging spatiotemporal patterns. We call the latter the fractal evolution exponent to distinguish it from the fractal dimension of a static "object". As opposed to the scale invariance of the pattern of such a static object, our power law exponent characterizes the scale invariance of the pattern generation mechanism. Although our power law exhibits features similar to self-organized critical phenomena, it is not at all identical with them. It can thus be applied to systems showing power law behavior without the requirement of their overall state being critical.

fractal evolutio 2   fractal evolution 1

Figure: Pattern formation in a coupled map lattice for two different values of a resolution parameter entering a non-linear feedback process constitutive for the evolution of the whole array (from bottom to top). Plotting the resolution parameter versus the average lifetimes of the patterns on a log-log scale provides the power law typical for fractal evolution (for details see the text of the paper below).

 

Finally, if in a further step an intrinsic change of above-mentioned resolution-like parameter during the systems evolution is implemented, one observes a self-organized growth of the order parameter. This is characterized as hierarchically emergent fractal evolution with different levels of pattern formation intrinsically emerging during the systems evolution.

The evolution of evolution

We have applied the phenomenon of HEFE within a simple toy-model of biological macroevolution. (Similarly, applications may be investigated for economics.) Our model exhibits both the property of punctuated equilibrium and the dynamics of potentialities for different species to evolve towards increasingly higher complexity. With the introduction of a realistic background noise limiting the range of the feedback operation in a system showing HEFE, we obtain a pattern signature in fitness space with temporal boost/mutation distances according to a randomized devil's staircase function.

You can download the pdf-file of our paper: "A Simple Model for the Evolution of Evolution" by Siegfried Fussy, Gerhard Grössing, and Herbert Schwabl by clicking here.

You can also download the pdf-file of the paper: "Progressive Evolution and a Measure for its Noise-dependent Complexity" by Siegfried Fussy, Gerhard Grössing, and Herbert Schwabl by clicking here.