Experimental investigation on preconditioned rate induced tipping in a thermoacoustic system

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.


√ √ ⁄
We start with an initial parameter value in the bistable region, and with an initial amplitude such that the system is in the basin of attraction of the fixed point. Say, now we vary the parameter with a fixed rate ̇ ( ). Then, there is a possibility for the system to undergo preconditioned rate induced tipping. However, before going over to that, we would like to make some details clear. The points and no longer correspond to limit cycles (because the system cannot stay on any of these points as ̇ ). However, these points still demarcate the basins of attraction in the system at any time instant -trajectories move away from either towards or towards in the bistable zone. And since itself is varying, the points and change with time. We, nevertheless, use the terms stable and unstable limit cycle loosely, for the purpose of explanation. Now, if ̇ , the parameter must be beyond the Hopf point for the system to tip to the stable limit cycle. However, for ̇ , one can have tipping before the system reaches the Hopf point, just due to fast variation of . The following discussion explains the mechanism for such a tipping.
We can see from equation 2 that the unstable limit cycle amplitude decreases with increase in . Then, if there exists a rate ̇ large enough such that after time , the decrease in is greater than the decay of the system towards the fixed point, the system has crossed over from the basin of attraction of the fixed point to that of the stable limit cycle. If this situation occurs before reaches the Hopf point, we say that the system has undergone preconditioned rate induced tipping. To express this mathematically, we fix the final value of at (and ), and label the corresponding unstable limit cycle amplitude as . This condition can then be expressed as To show that tipping can occur for some rate ̇ , one would substitute in the above equation ∫ This, however, is of little use as it is difficult to integrate. We circumvent this problem by using the following reasoning. The minima of ̇ occurs at

̅ √ √
This represents the maximum possible velocity towards the fixed point ( ) at any instant of time. If we can find a rate such that, even if the system were to attain this maximum velocity at every time instant, the system tips, then for any , the system will always tip. In other words, we would have confirmed that the system can undergo preconditioned rate induced tipping for sufficiently large rates. We note here that the threshold rate is not the critical rate (critical rate is the smallest rate above which the system tips). The existence of a threshold rate just serves to show that large enough rates ( can tip the system. We proceed to find this threshold rate. Clearly the parameter at any time instant is given as Using equations 4, 5 and 6 in equation 3, in the limiting condition ( ), we have, This can now be integrated to obtain a ̅ where, which should be less than or equal to . We argue that we can always find a large enough so that ⁄ . This is because for , the final point the system attains in equation 12 tends to zero, and so (that is, the amplitude asymptotically reaches the origin). Therefore, we can choose a large enough (with ) such that the combination is large and the required condition of is satisfied. Now, because at time the amplitude has gone below the final unstable limit cycle amplitude and the parameter has not reached the end point , the system will not tip. So, we have shown that for any initial condition , we can always find a rate ( ⁄ ) such that the system does not tip. Note that we choose to be strictly less than the Hopf point, and the above analysis will not hold for the case when (since for ).
We discard the scenario , because in this case, the system might tip for any positive rate of change of control parameter. We explain the rationale behind this statement in the following discussion. Now, since the value inside exponential in equation (16) is negative (as ), we see that , while on the other hand, continuously approaches zero at an infinite rate (equation 15). Therefore, at some parameter value between and , the amplitude crosses and the system tips. In other words, irrespective of the rate , the system tips. To avoid such a case, we choose an end point strictly less than .
In summary, we have shown that for any initial condition , there exist large enough rates for which the system will cross-over to the basin of attraction of the stable limit cycle before reaching the Hopf point. Also, we have shown that for the same initial condition, there are sufficiently small rates for which the system does not exhibit this behavior. This implies that there exists a critical rate above which the system will tip. Therefore, we have shown that the rate at which the parameter is varied can determine whether the system tips, thus establishing the occurrence of preconditioned rate induced tipping.

Supplementary Note 2. Theoretical model of the thermoacoustic system
We consider a theoretical model 1 that incorporates the feedback that exists between the sound waves and the heat release rate fluctuations. The model is derived from the conservation equations of momentum and energy given as follows 2 .
where is the ratio of specific heats of air, ̃ is pressure, ̃ is density, ̃ is velocity and ̇ is the heat release rate per unit volume. The variables in equations (17) and (18)  neglected to obtain the following equations.
The heat release rate fluctuations per unit volume ( ̇ ) is given by 4 : Where is the heat conductivity of air, is the specific heat at constant volume, is the mean velocity of the flow, ̅ and ̅ are the mean density and temperature of the flow respectively, is the cross-sectional area of the duct, is the effective wire length, is the temperature of the wire, is the wire diameter and ̃ is the location of the heater. The response of the heat release rate to the sound waves (velocity fluctuations) is nonlinear even though the propagation of sound can be described using linear equations. Further, the term ̃ ̃ ̃ in equation (21) represents the time delayed response of the heat release rate to the fluctuating velocity. We express equations (19) and (20) in terms of the following non-dimensional quantities.
where is the length of the duct, is the speed of sound in the duct and ̅ is the mean pressure.
The non-dimensionalized equations of momentum and energy are as follows.
Then, we expand the velocity and pressure fluctuations in terms of the spatial modes of the duct.

∑ ∑ ̇
where and ̇ are time varying quantities. The above expansion for and can be substituted in equations (23) and (24) to convert the partial differential equations to the following set of ODEs.
We introduce a damping term 5 with damping coefficient where and are constant coefficients, in equation (26). The non-dimensional angular frequency of the j th mode is given by .

Supplementary Note 3. Identifying initial conditions for observing preconditioned rate induced tipping in experiments and in the model
In this section, we identify initial conditions that are required to observe preconditioned rate induced tipping in experiments and model. The initial conditions should be such that the system is within the basin of attraction of the fixed point in the bistable region. If the system decays to the fixed point, we can conclude that the initial condition is within the basin of attraction of the fixed point. Then, we use this initial condition to observe preconditioned rate induced tipping in the system.
In experiments, we maintained the normalized heater power at ̃ (in the bistable region) and provided a finite amplitude excitation using loudspeakers. We find that the perturbation decays as illustrated in Supplementary Fig. 3.1. Similarly, in the model, we identified initial conditions in ̃ and such that the system is in the basin of attraction of the fixed point in the bistable zone (see Supplementary Fig. 3.2). We maintained these initial conditions in experiments and in the model to identify preconditioned rate induced tipping.