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. © 2017 The Author(s).

R. I. Sujith

Department of Aerospace Engineering

J. Tony

S. Subarna

K. S. Syamkumar

G. Sudha

S. Akshay

behavior

experimental model

theoretical model

Fulltext

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IIT Madras

Scientific Reports

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. © 2019 Author(s).

Chaos

Interplay between random fluctuations and rate dependent phenomena at slow passage to limit-cycle oscillations in a bistable thermoacoustic system

Dynamical systems can undergo critical transitions where the system suddenly shifts from one stable state to another at a critical threshold called the tipping point. The decrease in recovery rate to equilibrium (critical slowing down) as the system approaches the tipping point can be used to identify the proximity to a critical transition. Several measures have been adopted to provide early indications of critical transitions that happen in a variety of complex systems. In this study, we use early warning indicators to predict subcritical Hopf bifurcation occurring in a thermoacoustic system by analyzing the observables from experiments and from a theoretical model. We find that the early warning measures perform as robust indicators in the presence and absence of external noise. Thus, we illustrate the applicability of these indicators in an engineering system depicting critical transitions. © 2016 The Author(s).

Early warning signals for critical transitions in a thermoacoustic system

The influence of system parameters such as heater power, heater location and mass flow rate on the hysteresis characteristics of a horizontal Rijke tube is presented in this paper. It is observed that a hysteresis zone is present for all the mass flow rates considered in the present study. A power law relation is established between the non-dimensional hysteresis width and the Strouhal number, defined as the ratio between convective time scale and acoustic time scale. The transition to instability in a horizontal Rijke tube is found to be subcritical in all the experiments performed in this study. When heater location is chosen as the control parameter, period-2 oscillations are found for specific values of mass flow rate and heater power.

International Journal of Spray and Combustion Dynamics

Influence of system parameters on the hysteresis characteristics of a horizontal Rijke tube

The role of non-normality and nonlinearity in thermoacoustic interaction in a Rijke tube is investigated in this paper. The heat release rate of the heating element is modeled by a modified form of King's law. This fluctuating heat release from the heating element is treated as a compact source in the one-dimensional linear model of the acoustic field. The temporal evolution of the acoustic perturbations is studied using the Galerkin technique. It is shown that any thermoacoustic system is non-normal. Non-normality can cause algebraic growth of oscillations for a short time even though the eigenvectors of the system could be decaying exponentially with time. This feature of non-normality combined with the effect of nonlinearity causes the occurrence of triggering, i.e., the thermoacoustic oscillations decay for some initial conditions whereas they grow for some other initial conditions. If a system is non-normal, then there can be large amplification of oscillations even if the excited frequency is far from the natural frequency of the system. The dependence of transient growth on time lag and heater positions is studied. Such amplifications (pseudoresonance) can be studied using pseudospectra, as eigenvalues alone are not sufficient to predict the behavior of the system. The geometry of pseudospectra can be used to obtain upper and lower bounds on the growth factor, which provide both necessary and sufficient conditions for the stability of a thermoacoustic system. © 2008 American Institute of Physics.

Physics of Fluids

Thermoacoustic instability in a Rijke tube: Non-normality and nonlinearity

Physical Review E

Transitions through critical temperatures in nematic liquid crystals

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system

Experimental investigation on preconditioned rate induced tipping in a thermoacoustic system

Journal	Scientific Reports
Publisher	Nature Publishing Group
ISSN	20452322
Open Access	No