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.
Highlights
Tipping[1,2,3,4,5,6,7,8] in a dynamical system can happen either due to a bifurcation in the system known as B-tipping or due to the influence of noise known as N-tipping[9]
The transition to an alternate stable state can be delayed when the control parameter is varied at a slow rate[11,12,13,14,15,16]
The system follows the gradually varying quasi-static attractor when the rate at which the parameter is varied is below a critical rate
Summary
Tipping[1,2,3,4,5,6,7,8] in a dynamical system can happen either due to a bifurcation in the system known as B-tipping or due to the influence of noise known as N-tipping[9]. The transition to an alternate stable state can be delayed when the control parameter is varied at a slow rate[11,12,13,14,15,16]. The scenario is different when the parameter is varied in a rapid manner, where rate depending tipping or R-tipping[9, 17] could occur. When the parameter is varied above the critical rate, the system will be driven outside the basin of attraction of the quasi-static attractor and will eventually be attracted to a new stable state. In order to show the differences between our study and that of Ashwin et al.[9], we introduce the concept of preconditioned rate induced tipping in the normal form equation of such a system. Increased, the stable and the unstable limit cycles are born at the fold point whereas the fixed point loses its stability at the Hopf point
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