|InterJournal Complex Systems, 549
|Manuscript Number: |
Submission Date: 20511
|Bios and Creatory Complexity|
Bios is a newly found form of organization that resembles chaos in its aperiodic pattern and its extreme sensitivity to initial conditions, but has additional properties (diversification, novelty, nonrandom complexity, episodic patterning, 1/f power spectrum) found in natural creative processes, and absent in chaos. New methods have been developed to measure the properties that differentiate bios from simpler chaos. Global diversification is the increase in variance with duration of the time series; local diversification is the increase in variance with increase in embedding [Sabelli and Abouzeid, Nonlinear Dynamics, in press]. Diversification differentiates three types of processes: (a) mechanical processes and random series conserve variance; (b) processes that converge to equilibrium or periodic or chaotic attractors initially decrease variance; (c) creative processes increase variance. Novelty is defined as the increase in recurrence isometry with shuffling of the data [Sabelli, Nonlinear Dynamics, 2001]. Heartbeat intervals, most economic series, meteorological data, colored noise, and mathematical bios display novelty. Periodic series are recurrent (higher isometry than their shuffled copy). Chaotic attractors are neither novel nor recurrent; they have the same number of isometries as their shuffled copies. Nonrandom complexity is characterized by the production of multiple episodic patterns (“complexes”), as contrasted to the uniformity of periodic, chaotic and random trajectories; arrangement (the ratio of consecutive to total recurrence) measures nonrandom complexity [Sabelli, J. Applied Systems Studies, in press]. The process equation At+1 = At + k * t * sin(At) generates convergence to p, a cascade of bifurcations (including a “unifurcation”), chaos (with prominent period 4), bios and infinitation, as the value of the feedback gain k * t increases [Kauffman and Sabelli, Cybernetics and Systems, 1998; Sabelli and Kauffman, Cybernetics and Systems, 1999]. When t is given a negative sign, biotic series show irreversibility while chaotic series show only hysteresis. Bios is composed of multiple chaotic complexes, and change their range and sequence with minor changes in initial conditions (global sensitivity); chaotic trajectories are bounded within one basin of attraction (local sensitivity to initial conditions). The generation of bios requires bipolar feedback. When the bipolar feedback provided by the trigonometric function is biased, the equation produces a time series that culminates in chaos with prominent period 3 similar to that observed with the logistic equation. Conservation is required for bios, which is replaced by chaos in At+1 = k * t * sin(At). The series At+1 - At of a biotic time series is chaotic; differentiating an empirical series prior to analysis may transform a biotic pattern into a chaotic one. Mathematical bios and heartbeats show 1/fN power spectra; the time series of differences shows a direct relation between frequency and power. Empirical and mathematical biotic series show asymmetry, positive autocorrelation, and extended partial autocorrelation. Random, chaotic and stochastic series show symmetric statistical distributions, and no partial autocorrelation.
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