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Oberst S, Niven RK, Lester DR, Ord A, Hobbs B, Hoffmann N. Detection of unstable periodic orbits in mineralising geological systems. Chaos 2018; 28:085711. [PMID: 30180652 DOI: 10.1063/1.5024134] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/30/2018] [Accepted: 07/24/2018] [Indexed: 06/08/2023]
Abstract
Worldwide, mineral exploration is suffering from rising capital costs, due to the depletion of readily recoverable reserves and the need to discover and assess more inaccessible or geologically complex deposits. For gold exploration, this problem is particularly acute. We propose an innovative approach to mineral exploration and orebody characterisation, based on the analysis of geological core data as a spatial dynamical system, using the mathematical tools of dynamical system analysis. This approach is highly relevant for orogenic gold deposits, which-in contrast to systems formed at chemical equilibrium-exhibit many features of nonlinear dynamical systems, including episodic fluctuations on various length and time scales. Feedback relationships between thermo-chemical and deformation processes produce recurrent fluid temperatures and pressures and the deposition of vein-filling minerals such as pyrite and gold. We therefore relax the typical assumption of chemical equilibrium and analyse the underlying processes as aseismic, non-adiabatic, and inherent to a hydrothermal, nonlinear dynamical open-flow chemical reactor. These processes are approximated using the Gray-Scott model of reaction-diffusion as a complex toy system, which captures some of the features of the underlying mineralisation processes, including the spatiotemporal Turing patterns of unsteady chemical reactions. By use of this analysis, we demonstrate the capability of recurrence plots, recurrence power spectra, and recurrence time probabilities to detect underlying unstable periodic orbits as one sign of deterministic dynamics and their robustness for the analysis of data contaminated by noise. Recurrence plot based quantification is then applied to three mineral concentrations in the core data from the Sunrise Dam gold deposit in the Yilgarn region of Western Australia. Using a moving window, we reveal the episodic recurring low-dimensional dynamic structures and the period doubling route to instability with depth, embedded in and originating from higher-dimensional processes of the complex mineralisation system.
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Affiliation(s)
- S Oberst
- Centre for Audio, Acoustics and Vibration, University of Technology Sydney, Broadway, Sydney, New South Wales 2007, Australia
| | - R K Niven
- School of Engineering and Information Technology, The University of New South Wales, Canberra, Northcott Drive, Campbell, Australian Capital Territory 2600, Australia
| | - D R Lester
- School of Engineering, Royal Melbourne Institute of Technology, GPO Box 2476, Melbourne, Victoria 3001, Australia
| | - A Ord
- Centre for Exploration Targeting, The University of Western Australia, 35 Stirling Highway Crawley, Perth, Western Australia 6009, Australia
| | - B Hobbs
- Commonwealth Scientific and Industrial Research Organisation, 26 Dick Perry Ave., Kensington, Western Australia 6152, Australia
| | - N Hoffmann
- Dynamics Group, Complex Systems, Mechanical Engineering, Technical University Hamburg, Schlossmühlendamm 30, 21073 Hamburg, Germany
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Abstract
Many ecosystems show both self-organized spatial patterns and multistability of possible states. The combination of these two phenomena in different forms has a significant impact on the behavior of ecosystems in changing environments. One notable case is connected to tristability of two distinct uniform states together with patterned states, which has recently been found in model studies of dryland ecosystems. Using a simple model, we determine the extent of tristability in parameter space, explore its effects on the system dynamics, and consider its implications for state transitions or regime shifts. We analyze the bifurcation structure of model solutions that describe uniform states, periodic patterns, and hybrid states between the former two. We map out the parameter space where these states exist, and note how the different states interact with each other. We further focus on two special implications with ecological significance, breakdown of the snaking range and complex fronts. We find that the organization of the hybrid states within a homoclinic snaking structure breaks down as it meets a Maxwell point where simple fronts are stationary. We also discover a new series of complex fronts between the uniform states, each with its own velocity. We conclude with a brief discussion of the significance of these findings for the dynamics of regime shifts and their potential control.
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Affiliation(s)
- Yuval R Zelnik
- Centre for Biodiversity Theory and Modelling, Theoretical and Experimental Ecology Station, CNRS and Paul Sabatier University, 09200 Moulis, France
| | - Punit Gandhi
- Mathematical Biosciences Institute, Ohio State University, Columbus, Ohio 43210, USA
| | - Edgar Knobloch
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Ehud Meron
- Department of Solar Energy and Environmental Physics, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 8499000, Israel
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Abstract
We compare spot patterns generated by Turing mechanisms with those generated by replication cascades, in a model one-dimensional reaction-diffusion system. We determine the stability region of spot solutions in parameter space as a function of a natural control parameter (feed-rate) where degenerate patterns with different numbers of spots coexist for a fixed feed-rate. While it is possible to generate identical patterns via both mechanisms, we show that replication cascades lead to a wider choice of pattern profiles that can be selected through a tuning of the feed-rate, exploiting hysteresis and directionality effects of the different pattern pathways.
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Affiliation(s)
- Michael Stich
- Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts, United States of America
| | - Gourab Ghoshal
- Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts, United States of America
| | - Juan Pérez-Mercader
- Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts, United States of America
- The Santa Fe Institute, Santa Fe, New Mexico, United States of America
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