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Ghadami A, Epureanu BI. Deep learning for centre manifold reduction and stability analysis in nonlinear systems. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2022; 380:20210212. [PMID: 35719074 DOI: 10.1098/rsta.2021.0212] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/03/2021] [Accepted: 03/10/2022] [Indexed: 06/15/2023]
Abstract
Bifurcations cause large qualitative and quantitative changes in the dynamics of nonlinear systems with slowly varying parameters. These changes most often are due to modifications that occur in a low-dimensional subspace of the overall system dynamics. The key challenge is to determine what that low-dimensional subspace is, and construct a low-order model that governs the dynamics in that subspace. Centre manifold theory can provide a theoretical means to construct such low-order models for strongly nonlinear systems that undergo bifurcations. Performing a centre manifold analysis, however, is particularly challenging when the system dimensionality is high or impossible when an accurate model of the system is not available. This paper introduces a data-driven approach for identifying a reduced order model of the system based on centre manifold theory. The approach does not require a model of the full order system. Instead, a deep learning approach capable of identifying the centre manifold and the transformation to the centre space is created using measurements of the system dynamics from random perturbations. This approach unravels the characteristics of the system dynamics in the vicinity of bifurcations, providing critical information regarding the behaviour of the system. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.
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Affiliation(s)
- Amin Ghadami
- Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA
| | - Bogdan I Epureanu
- Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA
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Ghadami A, Epureanu BI. Data-driven prediction in dynamical systems: recent developments. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2022; 380:20210213. [PMID: 35719077 PMCID: PMC9207538 DOI: 10.1098/rsta.2021.0213] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/07/2023]
Abstract
In recent years, we have witnessed a significant shift toward ever-more complex and ever-larger-scale systems in the majority of the grand societal challenges tackled in applied sciences. The need to comprehend and predict the dynamics of complex systems have spurred developments in large-scale simulations and a multitude of methods across several disciplines. The goals of understanding and prediction in complex dynamical systems, however, have been hindered by high dimensionality, complexity and chaotic behaviours. Recent advances in data-driven techniques and machine-learning approaches have revolutionized how we model and analyse complex systems. The integration of these techniques with dynamical systems theory opens up opportunities to tackle previously unattainable challenges in modelling and prediction of dynamical systems. While data-driven prediction methods have made great strides in recent years, it is still necessary to develop new techniques to improve their applicability to a wider range of complex systems in science and engineering. This focus issue shares recent developments in the field of complex dynamical systems with emphasis on data-driven, data-assisted and artificial intelligence-based discovery of dynamical systems. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.
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Affiliation(s)
- Amin Ghadami
- Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA
| | - Bogdan I. Epureanu
- Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA
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Kuehn C, Lux K, Neamţu A. Warning signs for non-Markovian bifurcations: colour blindness and scaling laws. Proc Math Phys Eng Sci 2022. [DOI: 10.1098/rspa.2021.0740] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Warning signs for tipping points (or critical transitions) have been very actively studied. Although the theory has been applied successfully in models and in experiments for many complex systems such as for tipping in climate systems, there are ongoing debates as to when warning signs can be extracted from data. In this work, we shed light on this debate by considering different types of underlying noise. Thereby, we significantly advance the general theory of warning signs for nonlinear stochastic dynamics. A key scenario deals with stochastic systems approaching a bifurcation point dynamically upon slow parameter variation. The stochastic fluctuations are generically able to probe the dynamics near a deterministic attractor to reveal critical slowing down. Using scaling laws near bifurcations, one can then anticipate the distance to a bifurcation. Previous warning signs results assume that the noise is Markovian, most often even white. Here, we study warning signs for non-Markovian systems including coloured noise and
α
-regular Volterra processes (of which fractional Brownian motion and the Rosenblatt process are special cases). We prove that early warning scaling laws can disappear completely or drastically change their exponent based upon the parameters controlling the noise process. This provides a clear explanation as to why applying standard warning signs results to reduced models of complex systems may not agree with data-driven studies. We demonstrate our results numerically in the context of a box model of the Atlantic Meridional Overturning Circulation.
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Affiliation(s)
- Christian Kuehn
- Department of Mathematics, Technical University of Munich, Boltzmannstr. 3, Garching b. München 85748, Germany
- Complexity Science Hub Vienna, Josefstädter Str. 39, Vienna 1080, Austria
| | - Kerstin Lux
- Department of Mathematics, Technical University of Munich, Boltzmannstr. 3, Garching b. München 85748, Germany
| | - Alexandra Neamţu
- Department of Mathematics and Statistics, University of Konstanz, Universitätsstr. 10, Konstanz 78464, Germany
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Li K, Hajian-Forooshani Z, Su C, Perfecto I, Vandermeer J. Reduced rainfall and resistant varieties mediate a critical transition in the coffee rust disease. Sci Rep 2022; 12:1564. [PMID: 35091653 PMCID: PMC8799695 DOI: 10.1038/s41598-022-05362-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/12/2021] [Accepted: 12/30/2021] [Indexed: 11/13/2022] Open
Abstract
Critical transitions, sudden responses to slow changes in environmental drivers, are inherent in many dynamic processes, prompting a search for early warning signals. We apply this framework to understanding the coffee rust disease, which experienced an unprecedented outbreak in Mesoamerica in 2012-2013, likely a critical transition. Based on monthly infection data from 128 study quadrats in a 45-ha plot in southern Mexico from 2014 to 2020, we find that the persistent seasonal epidemic following the initial outbreak collapses in an evident subsequent critical transition. Characteristic signals of "critical slowing down" precede this collapse and are correlated with reduced rainfall, as expected from climate change, and planting of rust-resistant varieties, an ongoing management intervention. Recoveries from catastrophes may themselves be experienced as a critical transition and managers should consider the larger dynamical landscape for the possibility of subsequent transitions. Early warning signals could therefore be useful when evaluating mitigation effectiveness.
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Affiliation(s)
- Kevin Li
- School for Environment and Sustainability, University of Michigan, 440 Church St., Ann Arbor, MI, 48109, USA.
| | | | - Chenyang Su
- School for Environment and Sustainability, University of Michigan, 440 Church St., Ann Arbor, MI, 48109, USA
| | - Ivette Perfecto
- School for Environment and Sustainability, University of Michigan, 440 Church St., Ann Arbor, MI, 48109, USA
| | - John Vandermeer
- Department of Ecology and Evolutionary Biology, University of Michigan, Ann Arbor, MI, 48109, USA
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Füllsack M, Reisinger D. Transition prediction in the Ising-model. PLoS One 2021; 16:e0259177. [PMID: 34735514 PMCID: PMC8568180 DOI: 10.1371/journal.pone.0259177] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/08/2021] [Accepted: 10/15/2021] [Indexed: 11/18/2022] Open
Abstract
Dynamical systems can be subject to critical transitions where a system’s state abruptly shifts from one stable equilibrium to another. To a certain extent such transitions can be predicted with a set of methods known as early warning signals. These methods are often developed and tested on systems simulated with equation-based approaches that focus on the aggregate dynamics of a system. Many ecological phenomena however seem to necessitate the consideration of a system’s micro-level interactions since only there the actual reasons for sudden state transitions become apparent. Agent-based approaches that simulate systems from the bottom up by explicitly focusing on these micro-level interactions have only rarely been used in such investigations. This study compares the performance of a bifurcation estimation method for predicting state transitions when applied to data from an equation-based and an agent-based version of the Ising-model. The results show that the method can be applied to agent-based models and, despite its greater stochasticity, can provide useful predictions about state changes in complex systems.
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Affiliation(s)
- Manfred Füllsack
- Institute of Systems Sciences, Innovation and Sustainability Research, University of Graz, Graz, Austria
| | - Daniel Reisinger
- Institute of Systems Sciences, Innovation and Sustainability Research, University of Graz, Graz, Austria
- * E-mail:
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Bury TM, Bauch CT, Anand M. Detecting and distinguishing tipping points using spectral early warning signals. J R Soc Interface 2020; 17:20200482. [PMID: 32993435 PMCID: PMC7536046 DOI: 10.1098/rsif.2020.0482] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022] Open
Abstract
Theory and observation tell us that many complex systems exhibit tipping points—thresholds involving an abrupt and irreversible transition to a contrasting dynamical regime. Such events are commonly referred to as critical transitions. Current research seeks to develop early warning signals (EWS) of critical transitions that could help prevent undesirable events such as ecosystem collapse. However, conventional EWS do not indicate the type of transition, since they are based on the generic phenomena of critical slowing down. For instance, they may fail to distinguish the onset of oscillations (e.g. Hopf bifurcation) from a transition to a distant attractor (e.g. Fold bifurcation). Moreover, conventional EWS are less reliable in systems with density-dependent noise. Other EWS based on the power spectrum (spectral EWS) have been proposed, but they rely upon spectral reddening, which does not occur prior to critical transitions with an oscillatory component. Here, we use Ornstein–Uhlenbeck theory to derive analytic approximations for EWS prior to each type of local bifurcation, thereby creating new spectral EWS that provide greater sensitivity to transition proximity; higher robustness to density-dependent noise and bifurcation type; and clues to the type of approaching transition. We demonstrate the advantage of applying these spectral EWS in concert with conventional EWS using a population model, and show that they provide a characteristic signal prior to two different Hopf bifurcations in data from a predator–prey chemostat experiment. The ability to better infer and differentiate the nature of upcoming transitions in complex systems will help humanity manage critical transitions in the Anthropocene Era.
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Affiliation(s)
- T M Bury
- Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada ON N2L 3G1.,School of Environmental Sciences, University of Guelph, Guelph, Ontario, Canada ON N1G 2W1
| | - C T Bauch
- Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada ON N2L 3G1
| | - M Anand
- School of Environmental Sciences, University of Guelph, Guelph, Ontario, Canada ON N1G 2W1
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Chen S, O'Dea EB, Drake JM, Epureanu BI. Eigenvalues of the covariance matrix as early warning signals for critical transitions in ecological systems. Sci Rep 2019; 9:2572. [PMID: 30796264 PMCID: PMC6385210 DOI: 10.1038/s41598-019-38961-5] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2017] [Accepted: 12/27/2018] [Indexed: 11/23/2022] Open
Abstract
Many ecological systems are subject critical transitions, which are abrupt changes to contrasting states triggered by small changes in some key component of the system. Temporal early warning signals such as the variance of a time series, and spatial early warning signals such as the spatial correlation in a snapshot of the system's state, have been proposed to forecast critical transitions. However, temporal early warning signals do not take the spatial pattern into account, and past spatial indicators only examine one snapshot at a time. In this study, we propose the use of eigenvalues of the covariance matrix of multiple time series as early warning signals. We first show theoretically why these indicators may increase as the system moves closer to the critical transition. Then, we apply the method to simulated data from several spatial ecological models to demonstrate the method's applicability. This method has the advantage that it takes into account only the fluctuations of the system about its equilibrium, thus eliminating the effects of any change in equilibrium values. The eigenvector associated with the largest eigenvalue of the covariance matrix is helpful for identifying the regions that are most vulnerable to the critical transition.
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Affiliation(s)
- Shiyang Chen
- Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan, USA
| | - Eamon B O'Dea
- Odum School of Ecology, University of Georgia, Athens, Georgia, USA
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, Georgia, USA
| | - John M Drake
- Odum School of Ecology, University of Georgia, Athens, Georgia, USA
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, Georgia, USA
| | - Bogdan I Epureanu
- Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan, USA.
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Ghadami A, Gourgou E, Epureanu BI. Rate of recovery from perturbations as a means to forecast future stability of living systems. Sci Rep 2018; 8:9271. [PMID: 29915262 PMCID: PMC6006279 DOI: 10.1038/s41598-018-27573-0] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2018] [Accepted: 05/25/2018] [Indexed: 12/19/2022] Open
Abstract
Anticipating critical transitions in complex ecological and living systems is an important need because it is often difficult to restore a system to its pre-transition state once the transition occurs. Recent studies demonstrate that several indicators based on changes in ecological time series can indicate that the system is approaching an impending transition. An exciting question is, however, whether we can predict more characteristics of the future system stability using measurements taken away from the transition. We address this question by introducing a model-less forecasting method to forecast catastrophic transition of an experimental ecological system. The experiment is based on the dynamics of a yeast population, which is known to exhibit a catastrophic transition as the environment deteriorates. By measuring the system's response to perturbations prior to transition, we forecast the distance to the upcoming transition, the type of the transition (i.e., catastrophic/non-catastrophic) and the future equilibrium points within a range near the transition. Experimental results suggest a strong potential for practical applicability of this approach for ecological systems which are at risk of catastrophic transitions, where there is a pressing need for information about upcoming thresholds.
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Affiliation(s)
- Amin Ghadami
- Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan, 48109, USA
| | - Eleni Gourgou
- Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan, 48109, USA
- Department of Internal Medicine, Medical School, University of Michigan, Ann Arbor, Michigan, 48109, USA
- Mechanical Engineering, University of Michigan, Ann Arbor, MI, United States
| | - Bogdan I Epureanu
- Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan, 48109, USA.
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Liang J, Hu Y, Chen G, Zhou T. A universal indicator of critical state transitions in noisy complex networked systems. Sci Rep 2017; 7:42857. [PMID: 28230166 PMCID: PMC5322368 DOI: 10.1038/srep42857] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2016] [Accepted: 01/18/2017] [Indexed: 11/28/2022] Open
Abstract
Critical transition, a phenomenon that a system shifts suddenly from one state to another, occurs in many real-world complex networks. We propose an analytical framework for exactly predicting the critical transition in a complex networked system subjected to noise effects. Our prediction is based on the characteristic return time of a simple one-dimensional system derived from the original higher-dimensional system. This characteristic time, which can be easily calculated using network data, allows us to systematically separate the respective roles of dynamics, noise and topology of the underlying networked system. We find that the noise can either prevent or enhance critical transitions, playing a key role in compensating the network structural defect which suffers from either internal failures or environmental changes, or both. Our analysis of realistic or artificial examples reveals that the characteristic return time is an effective indicator for forecasting the sudden deterioration of complex networks.
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Affiliation(s)
- Junhao Liang
- School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P.R. China
| | - Yanqing Hu
- School of Data and Computer Sciences, Sun Yat-Sen University, Guangzhou 510275, P.R. China
| | - Guanrong Chen
- Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong SAR, P.R. China
| | - Tianshou Zhou
- School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P.R. China.,Key Laboratory of Computational Mathematics, Guangdong Province, Guangzhou 510275, P.R. China
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D’Souza K, Epureanu BI, Pascual M. Forecasting Bifurcations from Large Perturbation Recoveries in Feedback Ecosystems. PLoS One 2015; 10:e0137779. [PMID: 26356503 PMCID: PMC4565629 DOI: 10.1371/journal.pone.0137779] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2015] [Accepted: 08/21/2015] [Indexed: 11/18/2022] Open
Abstract
Forecasting bifurcations such as critical transitions is an active research area of relevance to the management and preservation of ecological systems. In particular, anticipating the distance to critical transitions remains a challenge, together with predicting the state of the system after these transitions are breached. In this work, a new model-less method is presented that addresses both these issues based on monitoring recoveries from large perturbations. The approach uses data from recoveries of the system from at least two separate parameter values before the critical point, to predict both the bifurcation and the post-bifurcation dynamics. The proposed method is demonstrated, and its performance evaluated under different levels of measurement noise, with two ecological models that have been used extensively in previous studies of tipping points and alternative steady states. The first one considers the dynamics of vegetation under grazing; the second, those of macrophyte and phytoplankton in shallow lakes. Applications of the method to more complex situations are discussed together with the kinds of empirical data needed for its implementation.
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Affiliation(s)
- Kiran D’Souza
- Mechanical and Aerospace Engineering Department, The Ohio State University, Columbus, Ohio, United States of America
- * E-mail:
| | - Bogdan I. Epureanu
- Mechanical Engineering Department, University of Michigan, Ann Arbor, Michigan, United States of America
| | - Mercedes Pascual
- Department of Ecology and Evolution, University of Chicago, Chicago, Illinois, United States of America
- Santa Fe Institute, Santa Fe, New Mexico, United States of America
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Scheffer M, Carpenter SR, Lenton TM, Bascompte J, Brock W, Dakos V, van de Koppel J, van de Leemput IA, Levin SA, van Nes EH, Pascual M, Vandermeer J. Anticipating critical transitions. Science 2012; 338:344-8. [PMID: 23087241 DOI: 10.1126/science.1225244] [Citation(s) in RCA: 742] [Impact Index Per Article: 61.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
Abstract
Tipping points in complex systems may imply risks of unwanted collapse, but also opportunities for positive change. Our capacity to navigate such risks and opportunities can be boosted by combining emerging insights from two unconnected fields of research. One line of work is revealing fundamental architectural features that may cause ecological networks, financial markets, and other complex systems to have tipping points. Another field of research is uncovering generic empirical indicators of the proximity to such critical thresholds. Although sudden shifts in complex systems will inevitably continue to surprise us, work at the crossroads of these emerging fields offers new approaches for anticipating critical transitions.
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Affiliation(s)
- Marten Scheffer
- Department of Environmental Sciences, Wageningen University, Post Office Box 47, NL-6700 AA Wageningen, Netherlands.
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Kéfi S, Dakos V, Scheffer M, Van Nes EH, Rietkerk M. Early warning signals also precede non-catastrophic transitions. OIKOS 2012. [DOI: 10.1111/j.1600-0706.2012.20838.x] [Citation(s) in RCA: 146] [Impact Index Per Article: 12.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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