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Tyloo M. Resilience of the slow component in timescale-separated synchronized oscillators. FRONTIERS IN NETWORK PHYSIOLOGY 2024; 4:1399352. [PMID: 38962160 PMCID: PMC11220911 DOI: 10.3389/fnetp.2024.1399352] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 03/11/2024] [Accepted: 05/06/2024] [Indexed: 07/05/2024]
Abstract
Physiological networks are usually made of a large number of biological oscillators evolving on a multitude of different timescales. Phase oscillators are particularly useful in the modelling of the synchronization dynamics of such systems. If the coupling is strong enough compared to the heterogeneity of the internal parameters, synchronized states might emerge where phase oscillators start to behave coherently. Here, we focus on the case where synchronized oscillators are divided into a fast and a slow component so that the two subsets evolve on separated timescales. We assess the resilience of the slow component by, first, reducing the dynamics of the fast one using Mori-Zwanzig formalism. Second, we evaluate the variance of the phase deviations when the oscillators in the two components are subject to noise with possibly distinct correlation times. From the general expression for the variance, we consider specific network structures and show how the noise transmission between the fast and slow components is affected. Interestingly, we find that oscillators that are among the most robust when there is only a single timescale, might become the most vulnerable when the system undergoes a timescale separation. We also find that layered networks seem to be insensitive to such timescale separations.
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Affiliation(s)
- Melvyn Tyloo
- Theoretical Division and Center for Nonlinear Studies (CNLS), Los Alamos National Laboratory, Los Alamos, NM, United States
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2
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Khramenkov VA, Dmitrichev AS, Nekorkin VI. Bistability of operating modes and their switching in a three-machine power grid. CHAOS (WOODBURY, N.Y.) 2023; 33:103129. [PMID: 37850866 DOI: 10.1063/5.0165779] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2023] [Accepted: 09/26/2023] [Indexed: 10/19/2023]
Abstract
We consider a power grid consisting of three synchronous generators supplying a common static load, in which one of the generators is located electrically much closer to the load than the others, due to a shorter transmission line with longitudinal inductance compensation. A reduced model is derived in the form of an ensemble with a star (hub) topology without parameter interdependence. We show that stable symmetric and asymmetric synchronous modes can be realized in the grid, which differ, in particular, in the ratio of currents through the second and third power supply paths. The modes of different types are not observed simultaneously, but the asymmetric modes always exist in pairs. A partition of the parameter space into regions with different dynamical regimes of the grid are obtained. Regions are highlighted where only synchronous operating modes can be established. It is shown that the grid can be highly multistable and, along with synchronous operating modes, have simultaneously various types of non-synchronous modes. We study non-local stability of the asymmetric synchronous modes and switchings between them under the influence one-time disturbances and additive noise fluctuations in the mechanical powers of the generators' turbines. The characteristics of one-time disturbances are obtained leading to either return the grid back to the initial synchronous mode or switching the grid to another synchronous mode or some non-synchronous mode. The characteristics of noise fluctuations are obtained, which provide either a more probable finding of the grid in the desirable quasi-synchronous mode, or switching to an undesirable one.
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Affiliation(s)
- V A Khramenkov
- Department of Nonlinear Dynamics, Institute of Applied Physics of RAS, 46 Ulyanov Str., 603950 Nizhny Novgorod, Russia
| | - A S Dmitrichev
- Department of Nonlinear Dynamics, Institute of Applied Physics of RAS, 46 Ulyanov Str., 603950 Nizhny Novgorod, Russia
| | - V I Nekorkin
- Department of Nonlinear Dynamics, Institute of Applied Physics of RAS, 46 Ulyanov Str., 603950 Nizhny Novgorod, Russia
- Department of Oscillation Theory and Automatic Regulation, Nizhny Novgorod State University, 23 Prospekt Gagarina, 603950 Nizhny Novgorod, Russia
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Bröhl T, von Wrede R, Lehnertz K. Impact of biological rhythms on the importance hierarchy of constituents in time-dependent functional brain networks. FRONTIERS IN NETWORK PHYSIOLOGY 2023; 3:1237004. [PMID: 37705698 PMCID: PMC10497113 DOI: 10.3389/fnetp.2023.1237004] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/08/2023] [Accepted: 08/09/2023] [Indexed: 09/15/2023]
Abstract
Biological rhythms are natural, endogenous cycles with period lengths ranging from less than 24 h (ultradian rhythms) to more than 24 h (infradian rhythms). The impact of the circadian rhythm (approximately 24 h) and ultradian rhythms on spectral characteristics of electroencephalographic (EEG) signals has been investigated for more than half a century. Yet, only little is known on how biological rhythms influence the properties of EEG-derived evolving functional brain networks. Here, we derive such networks from multiday, multichannel EEG recordings and use different centrality concepts to assess the time-varying importance hierarchy of the networks' vertices and edges as well as the various aspects of their structural integration in the network. We observe strong circadian and ultradian influences that highlight distinct subnetworks in the evolving functional brain networks. Our findings indicate the existence of a vital and fundamental subnetwork that is rather generally involved in ongoing brain activities during wakefulness and sleep.
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Affiliation(s)
- Timo Bröhl
- Department of Epileptology, University of Bonn Medical Center, Bonn, Germany
- Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn, Bonn, Germany
| | - Randi von Wrede
- Department of Epileptology, University of Bonn Medical Center, Bonn, Germany
| | - Klaus Lehnertz
- Department of Epileptology, University of Bonn Medical Center, Bonn, Germany
- Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn, Bonn, Germany
- Interdisciplinary Center for Complex Systems, University of Bonn, Bonn, Germany
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Datseris G, Luiz Rossi K, Wagemakers A. Framework for global stability analysis of dynamical systems. CHAOS (WOODBURY, N.Y.) 2023; 33:073151. [PMID: 37499248 DOI: 10.1063/5.0159675] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/25/2023] [Accepted: 07/03/2023] [Indexed: 07/29/2023]
Abstract
Dynamical systems that are used to model power grids, the brain, and other physical systems can exhibit coexisting stable states known as attractors. A powerful tool to understand such systems, as well as to better predict when they may "tip" from one stable state to the other, is global stability analysis. It involves identifying the initial conditions that converge to each attractor, known as the basins of attraction, measuring the relative volume of these basins in state space, and quantifying how these fractions change as a system parameter evolves. By improving existing approaches, we present a comprehensive framework that allows for global stability analysis of dynamical systems. Notably, our framework enables the analysis to be made efficiently and conveniently over a parameter range. As such, it becomes an essential tool for stability analysis of dynamical systems that goes beyond local stability analysis offered by alternative frameworks. We demonstrate the effectiveness of our approach on a variety of models, including climate, power grids, ecosystems, and more. Our framework is available as simple-to-use open-source code as part of the DynamicalSystems.jl library.
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Affiliation(s)
- George Datseris
- Department of Mathematics and Statistics, University of Exeter, North Park Road, Exeter EX4 4QF, United Kingdom
| | - Kalel Luiz Rossi
- Theoretical Physics/Complex Systems, ICBM, Carl von Ossietzky University of Oldenburg, Carl-von-Ossietzky-Straße 9-11, 26111 Oldenburg, Germany
| | - Alexandre Wagemakers
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
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Bröhl T, Lehnertz K. A perturbation-based approach to identifying potentially superfluous network constituents. CHAOS (WOODBURY, N.Y.) 2023; 33:2894464. [PMID: 37276550 DOI: 10.1063/5.0152030] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/27/2023] [Accepted: 05/16/2023] [Indexed: 06/07/2023]
Abstract
Constructing networks from empirical time-series data is often faced with the as yet unsolved issue of how to avoid potentially superfluous network constituents. Such constituents can result, e.g., from spatial and temporal oversampling of the system's dynamics, and neglecting them can lead to severe misinterpretations of network characteristics ranging from global to local scale. We derive a perturbation-based method to identify potentially superfluous network constituents that makes use of vertex and edge centrality concepts. We investigate the suitability of our approach through analyses of weighted small-world, scale-free, random, and complete networks.
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Affiliation(s)
- Timo Bröhl
- Department of Epileptology, University of Bonn Medical Centre, Venusberg Campus 1, 53127 Bonn, Germany
- Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn, Nussallee 14-16, 53115 Bonn, Germany
| | - Klaus Lehnertz
- Department of Epileptology, University of Bonn Medical Centre, Venusberg Campus 1, 53127 Bonn, Germany
- Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn, Nussallee 14-16, 53115 Bonn, Germany
- Interdisciplinary Center for Complex Systems, University of Bonn, Brühler Straße 7, 53175 Bonn, Germany
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Abstract
AbstractThe transformation of ecosystems proceeds at unprecedented rates. Recent studies suggest that high rates of environmental change can cause rate-induced tipping. In ecological models, the associated rate-induced critical transition manifests during transient dynamics in which populations drop to dangerously low densities. In this work, we study how indirect evolutionary rescue—due to the rapid evolution of a predator’s trait—can save a prey population from the rate-induced collapse. Therefore, we explicitly include the time-dependent dynamics of environmental change and evolutionary adaptation in an eco-evolutionary system. We then examine how fast the evolutionary adaptation needs to be to counteract the response to environmental degradation and express this relationship by means of a critical rate. Based on this critical rate, we conclude that indirect evolutionary rescue is more probable if the predator population possesses a high genetic variation and, simultaneously, the environmental change is slow. Hence, our results strongly emphasize that the maintenance of biodiversity requires a deceleration of the anthropogenic degradation of natural habitats.
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Alkhayuon H, Tyson RC, Wieczorek S. Phase tipping: how cyclic ecosystems respond to contemporary climate. Proc Math Phys Eng Sci 2021; 477:20210059. [PMID: 35153584 PMCID: PMC8511773 DOI: 10.1098/rspa.2021.0059] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/22/2021] [Accepted: 09/06/2021] [Indexed: 11/30/2022] Open
Abstract
We identify the phase of a cycle as a new critical factor for tipping points (critical transitions) in cyclic systems subject to time-varying external conditions. As an example, we consider how contemporary climate variability induces tipping from a predator–prey cycle to extinction in two paradigmatic predator–prey models with an Allee effect. Our analysis of these examples uncovers a counterintuitive behaviour, which we call phase tipping or P-tipping, where tipping to extinction occurs only from certain phases of the cycle. To explain this behaviour, we combine global dynamics with set theory and introduce the concept of partial basin instability for attracting limit cycles. This concept provides a general framework to analyse and identify easily testable criteria for the occurrence of phase tipping in externally forced systems, and can be extended to more complicated attractors.
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Affiliation(s)
- Hassan Alkhayuon
- University College Cork, School of Mathematical Sciences, Western Road, Cork T12 XF62, Ireland
| | - Rebecca C Tyson
- CMPS Department (Mathematics), University of British Columbia Okanagan, Kelowna, British Columbia, Canada
| | - Sebastian Wieczorek
- University College Cork, School of Mathematical Sciences, Western Road, Cork T12 XF62, Ireland
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Schoenmakers S, Feudel U. A resilience concept based on system functioning: A dynamical systems perspective. CHAOS (WOODBURY, N.Y.) 2021; 31:053126. [PMID: 34240958 DOI: 10.1063/5.0042755] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/03/2021] [Accepted: 04/21/2021] [Indexed: 06/13/2023]
Abstract
We introduce a new framework for resilience, which is traditionally understood as the ability of a system to absorb disturbances and maintain its state, by proposing a shift from a state-based to a system functioning-based approach to resilience, which takes into account that several different coexisting stable states could fulfill the same functioning. As a consequence, not every regime shift, i.e., transition from one stable state to another, is associated with a lack or loss of resilience. We emphasize the importance of flexibility-the ability of a system to shift between different stable states while still maintaining system functioning. Furthermore, we provide a classification of system responses based on the phenomenological properties of possible disturbances, including the role of their timescales. Therefore, we discern fluctuations, shocks, press disturbances, and trends as possible disturbances. We distinguish between two types of mechanisms of resilience: (i) tolerance and flexibility, which are properties of the system, and (ii) adaptation and transformation, which are processes that alter the system's tolerance and flexibility. Furthermore, we discuss quantitative methods to investigate resilience in model systems based on approaches developed in dynamical systems theory.
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Affiliation(s)
- Sarah Schoenmakers
- Theoretical Physics/Complex Systems, ICBM, Carl von Ossietzky University of Oldenburg, 26111 Oldenburg, Germany
| | - Ulrike Feudel
- Theoretical Physics/Complex Systems, ICBM, Carl von Ossietzky University of Oldenburg, 26111 Oldenburg, Germany
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