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Cantisán J, Yanchuk S, Seoane JM, Sanjuán MAF, Kurths J. Rate and memory effects in bifurcation-induced tipping. Phys Rev E 2023; 108:024203. [PMID: 37723724 DOI: 10.1103/physreve.108.024203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2023] [Accepted: 07/20/2023] [Indexed: 09/20/2023]
Abstract
A variation in the environment of a system, such as the temperature, the concentration of a chemical solution, or the appearance of a magnetic field, may lead to a drift in one of the parameters. If the parameter crosses a bifurcation point, the system can tip from one attractor to another (bifurcation-induced tipping). Typically, this stability exchange occurs at a parameter value beyond the bifurcation value. This is what we call, here, the shifted stability exchange. We perform a systematic study on how the shift is affected by the initial parameter value and its change rate. To that end, we present numerical simulations and partly analytical results for different types of bifurcations and different paradigmatic systems. We show that the nonautonomous dynamics can be split into two regimes. Depending on whether we exceed the numerical or experimental precision or not, the system may enter the nondeterministic or the deterministic regime. This is determined solely by the conditions of the drift. Finally, we deduce the scaling laws governing this phenomenon and we observe very similar behavior for different systems and different bifurcations in both regimes.
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Affiliation(s)
- Julia Cantisán
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos Tulipán s/n, 28933 Móstoles, Madrid, Spain
| | - Serhiy Yanchuk
- Department of Mathematics, Humboldt University Berlin, 12489 Berlin, Germany
- Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany
| | - Jesús M Seoane
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos Tulipán s/n, 28933 Móstoles, Madrid, Spain
| | - Miguel A F Sanjuán
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos Tulipán s/n, 28933 Móstoles, Madrid, Spain
- Department of Applied Informatics, Kaunas University of Technology Studentu 50-415, Kaunas LT-51368, Lithuania
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany
- Department of Physics, Humboldt University Berlin, 12489 Berlin, Germany
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K S S, E A G, Kurths J, Surovyatkina E. Emergency rate-driven control for rotor angle instability in power systems. CHAOS (WOODBURY, N.Y.) 2022; 32:061102. [PMID: 35778159 DOI: 10.1063/5.0093450] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/29/2022] [Accepted: 05/09/2022] [Indexed: 06/15/2023]
Abstract
Renewable energy sources in modern power systems pose a serious challenge to the power system stability in the presence of stochastic fluctuations. Many efforts have been made to assess power system stability from the viewpoint of the bifurcation theory. However, these studies have not covered the dynamic evolution of renewable energy integrated, non-autonomous power systems. Here, we numerically explore the transition phenomena exhibited by a non-autonomous stochastic bi-stable power system oscillator model. We use additive white Gaussian noise to model the stochasticity in power systems. We observe that the delay in the transition observed for the variation of mechanical power as a function of time shows significant variations in the presence of noise. We identify that if the angular velocity approaches the noise floor before crossing the unstable manifold, the rate at which the parameter evolves has no control over the transition characteristics. In such cases, the response of the system is purely controlled by the noise, and the system undergoes noise-induced transitions to limit-cycle oscillations. Furthermore, we employ an emergency control strategy to maintain the stable non-oscillatory state once the system has crossed the quasi-static bifurcation point. We demonstrate an effective control strategy that opens a possibility of maintaining the stability of electric utility that operates near the physical limits.
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Affiliation(s)
- Suchithra K S
- Department of Electrical & Electronics Engineering, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Coimbatore 641112, India
| | - Gopalakrishnan E A
- Center for Computational Engineering and Networking (CEN), Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Coimbatore 641112, India
| | - Jürgen Kurths
- Department of Physics, Humboldt University of Berlin, Newtonstrasse 15, Berlin 12489, Germany
| | - E Surovyatkina
- Potsdam Institute for Climate Impact Research, Potsdam 14412, Germany
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Pavithran I, Sujith RI. Effect of rate of change of parameter on early warning signals for critical transitions. CHAOS (WOODBURY, N.Y.) 2021; 31:013116. [PMID: 33754769 DOI: 10.1063/5.0025533] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/16/2020] [Accepted: 12/09/2020] [Indexed: 06/12/2023]
Abstract
Many dynamical systems exhibit abrupt transitions or tipping as the control parameter is varied. In scenarios where the parameter is varied continuously, the rate of change of the control parameter greatly affects the performance of early warning signals (EWS) for such critical transitions. We study the impact of variation of the control parameter with a finite rate on the performance of EWS for critical transitions in a thermoacoustic system (a horizontal Rijke tube) exhibiting subcritical Hopf bifurcation. There is a growing interest in developing early warning signals for tipping in real systems. First, we explore the efficacy of early warning signals based on critical slowing down and fractal characteristics. From this study, lag-1 autocorrelation (AC) and Hurst exponent (H) are found to be good measures to predict the transition well before the tipping point. The warning time, obtained using AC and H, reduces with an increase in the rate of change of the control parameter following an inverse power law relation. Hence, for very fast rates, the warning time may be too short to perform any control action. Furthermore, we report the observation of a hyperexponential scaling relation between the AC and the variance of fluctuations during such a dynamic Hopf bifurcation. We construct a theoretical model for noisy Hopf bifurcation wherein the control parameter is continuously varied at different rates to study the effect of rate of change of the parameter on EWS. Similar results, including the hyperexponential scaling, are observed in the model as well.
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Affiliation(s)
| | - R I Sujith
- Department of Aerospace Engineering, IIT Madras, Chennai 600036, India
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Suchithra KS, Gopalakrishnan EA, Surovyatkina E, Kurths J. Rate-induced transitions and advanced takeoff in power systems. CHAOS (WOODBURY, N.Y.) 2020; 30:061103. [PMID: 32611081 DOI: 10.1063/5.0002456] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/25/2020] [Accepted: 05/22/2020] [Indexed: 06/11/2023]
Abstract
One of the most common causes of failures in complex systems in nature or engineering is an abrupt transition from a stable to an alternate stable state. Such transitions cause failures in the dynamic power systems. We focus on this transition from a stable to an unstable manifold for a rate-dependent mechanical power input via a numerical investigation in a theoretical power system model. Our studies uncover early transitions that depend on the rate of variation of mechanical input. Furthermore, we determine the dependency of a critical rate on initial conditions of the system. Accordingly, this knowledge of the critical rate can be used in devising an effective control strategy based on artificial intelligence (AI).
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Affiliation(s)
- K S Suchithra
- Center for Computational Engineering & Networking (CEN), Amrita School of Engineering, Coimbatore, Amrita Vishwa Vidyapeetham, India
| | - E A Gopalakrishnan
- Center for Computational Engineering & Networking (CEN), Amrita School of Engineering, Coimbatore, Amrita Vishwa Vidyapeetham, India
| | | | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, Potsdam 14473, Germany
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Ritchie P, Karabacak Ö, Sieber J. Inverse-square law between time and amplitude for crossing tipping thresholds. Proc Math Phys Eng Sci 2019; 475:20180504. [PMID: 30853839 DOI: 10.1098/rspa.2018.0504] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/01/2018] [Accepted: 01/22/2019] [Indexed: 11/12/2022] Open
Abstract
A classical scenario for tipping is that a dynamical system experiences a slow parameter drift across a fold tipping point, caused by a run-away positive feedback loop. We study what happens if one turns around after one has crossed the threshold. We derive a simple criterion that relates how far the parameter exceeds the tipping threshold maximally and how long the parameter stays above the threshold to avoid tipping in an inverse-square law to observable properties of the dynamical system near the fold. For the case when the dynamical system is subject to stochastic forcing we give an approximation to the probability of tipping if a parameter changing in time reverses near the tipping point. The derived approximations are valid if the parameter change in time is sufficiently slow. We demonstrate for a higher-dimensional system, a model for the Indian summer monsoon, how numerically observed escape from the equilibrium converge to our asymptotic expressions. The inverse-square law between peak of the parameter forcing and the time the parameter spends above a given threshold is also visible in the level curves of equal probability when the system is subject to random disturbances.
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Affiliation(s)
- Paul Ritchie
- Earth System Science, College of Life and Environmental Sciences, Harrison Building, University of Exeter, Exeter EX4 4QF, UK
| | - Özkan Karabacak
- Department of Electronic Systems, Automation and Control, Aalborg University, Fredrik Bajers Vej 7 C, Aalborg East 9220, Denmark
| | - Jan Sieber
- Centre for Systems,Dynamics and Control, College of Engineering, Mathematics and Physical Sciences, Harrison Building, University of Exeter, Exeter EX4 4QF, UK
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Unni VR, Gopalakrishnan EA, Syamkumar KS, Sujith RI, Surovyatkina E, Kurths J. Interplay between random fluctuations and rate dependent phenomena at slow passage to limit-cycle oscillations in a bistable thermoacoustic system. CHAOS (WOODBURY, N.Y.) 2019; 29:031102. [PMID: 30927835 DOI: 10.1063/1.5088943] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/10/2019] [Accepted: 03/11/2019] [Indexed: 06/09/2023]
Abstract
We study the impact of noise on the rate dependent transitions in a noisy bistable oscillator using a thermoacoustic system as an example. As the parameter-the heater power-is increased in a quasi-steady manner, beyond a critical value, the thermoacoustic system undergoes a subcritical Hopf bifurcation and exhibits periodic oscillations. We observe that the transition to this oscillatory state is often delayed when the control parameter is varied as a function of time. However, the presence of inherent noise in the system introduces high variability in the characteristics of this critical transition. As a result, if the value of the system variable-the acoustic pressure-approaches the noise floor before the system crosses the unstable manifold, the effect of rate on the critical transition becomes irrelevant in determining the transition characteristics, and the system undergoes a noise-induced tipping to limit-cycle oscillations. The presence of noise-induced tipping makes it difficult to identify the stability regimes in such systems by using stability maps for the corresponding deterministic system.
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Affiliation(s)
- Vishnu R Unni
- Department of Mechanical and Aerospace Engineering, University of California San Diego, San Diego, California 92093, USA
| | - E A Gopalakrishnan
- Center for Computational Engineering and Networking, Amrita School of Engineering, Coimbatore 641112, India
| | - K S Syamkumar
- Indian Institute of Technology Madras, Chennai 600036, India
| | - R I Sujith
- Indian Institute of Technology Madras, Chennai 600036, India
| | | | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, Berlin 14412, Germany
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Tony J, Subarna S, Syamkumar KS, Sudha G, Akshay S, Gopalakrishnan EA, Surovyatkina E, Sujith RI. Experimental investigation on preconditioned rate induced tipping in a thermoacoustic system. Sci Rep 2017; 7:5414. [PMID: 28710448 PMCID: PMC5511272 DOI: 10.1038/s41598-017-05814-y] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2017] [Accepted: 06/05/2017] [Indexed: 11/10/2022] Open
Abstract
Many systems found in nature are susceptible to tipping, where they can shift from one stable dynamical state to another. This shift in dynamics can be unfavorable in systems found in various fields ranging from ecology to finance. Hence, it is important to identify the factors that can lead to tipping in a physical system. Tipping can mainly be brought about by a change in parameter or due to the influence of external fluctuations. Further, the rate at which the parameter is varied also determines the final state that the system attains. Here, we show preconditioned rate induced tipping in experiments and in a theoretical model of a thermoacoustic system. We provide a specific initial condition (preconditioning) and vary the parameter at a rate higher than a critical rate to observe tipping. We find that the critical rate is a function of the initial condition. Our study is highly relevant because the parameters that dictate the asymptotic behavior of many physical systems are temporally dynamic.
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Affiliation(s)
- J Tony
- Indian Institute of Technology Madras, Chennai, India
| | - S Subarna
- Indian Institute of Technology Madras, Chennai, India
| | - K S Syamkumar
- Indian Institute of Technology Madras, Chennai, India
| | - G Sudha
- Indian Institute of Technology Madras, Chennai, India
| | - S Akshay
- Indian Institute of Technology Madras, Chennai, India
| | - E A Gopalakrishnan
- Center for Computational Engineering and Networking (CEN), Amrita Vishwa Vidyapeetham, Amrita University, Coimbatore, India
| | - E Surovyatkina
- Space Research Institute of Russian Academy of Sciences, Moscow, Russia
- Potsdam Institute for Climate Impact Research, Potsdam, Germany
| | - R I Sujith
- Indian Institute of Technology Madras, Chennai, India.
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