Oscillator death

Ermentrout, G. B., and Kopell, N. (1990) Oscillator death i n systems of coupled neural oscillators. SIAM J. Appl. Math. 50,125. 

Figure 8.4 The order parameter R vs. the stirring rate shows the synchronization transition for different values of the non-uniformity parameter 6 (top). Phase diagram as a function of the stirring rate v and inhomogeneity S (bottom). The symbols represent synchronization, oscillator death and no synchronization (from Neufeld et al. (2003)).

Thus in the oscillator death state (Mirollo and Strogatz, 1990), the stirring completely inhibits the chemical oscillations, transforming the unstable steady state of the local dynamics into a stable attractor of the reaction-advection-diffusion system. This arises from the competition between the non-uniform frequencies that lead to the dispersion of the phase of the oscillations, while mixing tends to homogenize the system and bring it back to the unstable steady state. Therefore this behavior is only possible in oscillatory media with spatially non-uniform frequencies. The phase diagram of the non-uniform oscillatory system in the plane of the two main control parameters (v, 5) is shown in Fig. 8.4. 

Behavior similar to the oscillator death described above has been observed as the suppression of oscillations due to stirring in some experiments with oscillatory chemical reactions, and was also found in a few other systems without mixing, like globally coupled phase-amplitude oscillators or oscillators with delayed coupling (Mirollo and Strogatz, 1990 Ramana Reddy et al., 1998). 

Z. Neufeld, I.Z. Kiss, C. Zhou, and J. Kurths. Synchronization and oscillator death in oscillatory media with stirring. Phys. Rev. Lett., 91 084101, 2003. 

Points A and B are the steady states at which the oscillator can come to rest when oscillator death occurs. (Adapted from Crowley and Epstein, 1989.)... 

A novel interplay between entanglement as a QIP resource and entanglement as the source of decoherence was detailed [116]. Two entangled qubits were analyzed, each coupled to a bath via common modes. The non-Markovian timescale was considered, as well as dynamical modulations. It was shown how the entanglement of the qubits could vanish after a finite time (entanglement sudden death, ESD), but later restored by non-Markovian modulation-induced oscillations of the system-bath coherence. 

Fig. 3.8. Representation of the onset, growth, and death of oscillations in the isothermal autocatalytic model as /z varies for reaction with the uncatalysed step included, showing emergence of the stable limit cycle at and its disappearance at n. ...

We have now seen how local stability analysis can give us useful information about any given state in terms of the experimental conditions (i.e. in terms of the parameters p and ku for the present isothermal autocatalytic model). The methods are powerful and for low-dimensional systems their application is not difficult. In particular we can recognize the range of conditions over which damped oscillatory behaviour or even sustained oscillations might be observed. The Hopf bifurcation condition, in terms of the eigenvalues k2 and k2, enabled us to locate the onset or death of oscillatory behaviour. Some comments have been made about the stability and growth of the oscillations, but the details of this part of the analysis will have to wait until the next chapter. 

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

Low-entropy Planck-power (or other [21-25,33-35]) input such as hydrogen in nonoscillating cosmologies, or two-time low-entropy boundary conditions in oscillating ones [61,62,101-105], would enable our Universe — and likewise any Universe in the Multiverse — to forever thwart the heat death predicted by the Second Law of Thermodynamics. It should be noted that there also are other ways that the heat death can be thwarted see, for example, Ref. [127], Hopefully, one way or another, the heat death is thwarted in the real Universe, whether within an inflationary Multiverse [89-94,105] or otherwise [88,89,101-105,127]. 

These processes correspond, respectively, to spontaneous reproduction of the prey, reproduction of the predator mediated by prey consumption, and predator death. In fact one obtains Eqs. (3.68)-(3.69) with a2 = oi, but this is just a consequence of measuring the amount of chemicals in moles or molecules, and it expresses the fact that in (3.70) the consumption of each molecule of P leads to a new molecule of Z. Changing to other units such as mass makes the coefficients of the nonlinear terms to become different as in (3.68)-(3.69). In biological settings, common units for either P and Z are either mass or number of individuals, and in these units it is no longer true that ingestion of one unit of P produces one unit of new Z. Lotka presented his model as a chemical scheme that would produce persistent chemical oscillations. 

R.E. Mirollo and S.H. Strogatz. Amplitude death in an array of limit-cycle oscillators. J. Stat. Phys., 60 245-262, 1990. 

D. V. Ramana Reddy, A. Sen, and G. L. Johnston. Time delay induced death in coupled limit cycle oscillators. Phys. Rev. Lett., 80 5109-5112, 1998. 

Cohen, P.I., Petrich, G.S., Pukite, P.R., Whaley, G.J., and Arrott, A.S., Birth-death models of epitaxy I. Diffraction oscillations from low index surfaces. Surf. ScL, 216, 222, 1989. 

I. Ozden, S. Venkataramani, M. A. Long, B. W. Connors, and A. V. Nur-mikko. Strong coupling of nonlinear electronic and biological oscillators Reaching the amplitude death regime. Phys. Rev. Lett, 93 158102, 2004. 

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