Mathematical Attractor Theory and Plutonic-Volcanic Episodicity

Abstract

The mathematical theory of attractors is reviewed descriptively. An attractor can be thought of as a map of the characteristic dynamic states of an evolving system, whether the form produced is continuous or discontinuous, regularly periodic or stochastic. Graphical examples of attractors showing dynamic bifurcations in the evolution of magma transport from the mantle to the surface are developed in terms of a model of nonlinearly pumped volumetric capacitors. Dynamic bifurcation means that the system alternates regularly or irregularly between two or more volumetric states. The potential for a volume domain (whether a plexus or a magma chamber) to store and discharge increments of magma is the basis for calling the system a volumetric capacitor. Pumping is represented by percolation and/or extensional fracture mechanisms of variable magma transport from a mantle source. Steady, oscillatory, and pulsatory flow regimes are predicted which qualitatively simulate the behavior of near-surface inflation/deflation cycles and volcanic eruptions. Buildup to catastrophic pyroclastic eruptions of silicic magma during the growth of caldera-forming ash-flow systems is described as attractor evolution produced by convergences of multidimensional rate processes. Such convergent behavior is analogous to the dynamics of slaved systems described by synergetic feedback, such as Haken’s (1979) theory of the optical laser. In the magmatic context, slaving refers to a condition where focused volumetric states of the system are outgrowths of cooperation among transport paths originating from multiple volumetric states (e. g., the net effects on heat flow and magma transport paths of basaltic fields surrounding an evolving silicic system). The multiple states, in turn, have arisen as bifurcations of magma transport from a source system of primitive magma in the mantle. In a metaphorical sense, culminating stages of silicic volcanism are laser-like. The metaphor is based on parallelism with the idea that energy (in the form of protons or magma) is pumped into the system in multi-mode states which ultimately feed a single state. In the optical analogy this is the distinction between the laser mode and multiple modes of ordinary light. In the magma system it is the distinction between a central evolved silicic chamber and the multimode field of basaltic injections of the crust that supports its evolution (the “basaltic shadow” of Smith and Shaw, 1973; 1975). Synergy is expressed by the cooperative effects which create the centralized chamber. Entirely new kinds of energetic states are thereby made possible (e. g., crustal melting, convection, chemical zonation, and large-scale eruptive pulses). The synergetic process is self-organizing in the sense that the composite system is slaved to an effectively single culminating mode of eruption leading to the caldera-forming stage of ash-flow magmatism. Other eruptive styles occurring both before and after such an episode represent detuning of the optimally slaved state. Attractor models show features that resemble Smith’s (1979) concepts of the evolution of silicic magmatism.