Oscillation death

Oscillator death is a phenomena whereby two or more autonomously oscillating systems approach a stable rest state when coupled. Historically, the coupling is "diffusive" so that if the oscillators are identical and start with identical initial data, the coupling makes no contribution to the dynamics. However, oscillator death can occur, as will be shown below, even if the oscillators are identical. This article first describes the general concept of oscillator death via some experimental and theoretical examples from chemistry. The theory of oscillator death is discussed further in the second part of the article.

[edit] Examples and general thoughts

Let us look at a differential equation of the form: \[ \frac{dx}{dt}=f(x,k), \qquad x(0)=x_0 \] where \(x\) is the vector of the variables, \(k\) is the vector of the various rate constants and \(x_0\) is the vector of the initial values of the chemical species that enter the system. At certain values of the parameters, the system being in a steady state has a Hopf bifurcation and oscillations start. By Hopf bifurcation we mean that at least one of the real parts of the eigenvalues of the Jacobian becomes positive, while the imaginary part is nonzero. In this case, small deviations from the steady state (SS) will grow, and the nonlinearities of the system will take over causing the oscillations.

The system may describe a chemical, mechanical, biological, or electrical oscillator. Here, the focus is placed on chemical systems. Examples of such oscillators include the Oregonator (Field and Noyes 1974), Brusselator (Prigogine and Lefever 1968), the NFT (Noyes-Field-Thompson) mechanism (Noyes, Field, and Thompson 1971), the FKN (Field-Koros-Noyes) mechanism (Field, Koros, and Noyes 1972). When \(N\) such oscillators are coupled so that material can flow from one oscillator to its next door neighbors, the following system of equations is obtained: \[ \frac{dx_i}{dt} = f(x_i,k_i)+ k_c(x_{i-1}-2x_i+x_{i+1}), \quad i=1,\ldots, N \] In particular, for two oscillators coupled in this way, we obtain \[ \frac{dx_1}{dt} = f(x_1,k_1)+ k_c(x_2-x_1), \] \[ \frac{dx_2}{dt} = f(x_2,k_2)+ k_c(x_1-x_2), \] that is, there is a flow of material from cell 1 to cell 2 in proportion to the concentration difference between the two oscillators and the rate being \(k_c\ .\) The values of the parameters (or some of them) \(k_{1}\) and \(k_{2}\) may or may not be the same, so the oscillators may be identical or not.

It is fairly easy to physically construct such coupling: taking two CSTRs (Continuous Stirred Tank Reactors) in each of which an oscillating reaction occurs and connect them in such a way as to allow the chemical species to move from one CSTR to the other in a controlled manner, for instance with the help of a suitable pump. Assuming the single oscillator has \(n\) species, the coupled system will have \(N=2n\) species. It is easy to imagine what happens at very low or very high coupling rates i.e. very low or very high \(k_c\ ;\) in the first case, each subsystem will behave as if it were alone and the oscillations in each of them will resemble those of the disconnected CSTRs; on the other extreme, the two CSTRs will behave as one large CSTR with average properties i.e. with average \(k\)'s; In the case of intermediate \(k_c\) there is no way to guess the behavior of the large system and we need to calculate the outcome.

As an example let us look at the behavior of two CSTRs containing each an NFT oscillator (K.Bar-Eli 1984) with the following initial values: [BrO₃^-]₀=0.06M, [Ce⁺³]₀=1.5x10^-4M, [H⁺]₀=1.5M, and [Br^-]₀₁=0.032mM while [Br^-]₀₂=0.028mM and \(k_0\)=0.004^-1sec where \(k_0\) is the flow rate of chemical species into and out of the CSTR. Note that the two CSTR differ only in the concentrations of the bromide ion that are used. Note also that we are using a chemical system that is not the famous BZ (Belousov–Zhabotinsky) but a system that does not contain malonic acid and therefore oscillates in a smaller range of parameters. Detailed mechanism and specification of the rate constants are given in the references.

Table 1 shows the coupling rate, the periods of the two oscillators under the above conditions, the ratio of [Br^-]_max/[Br^-]_min of the two oscillators and the number of the positive eigenvalues. At low coupling rates, the two oscillators behave nearly independently; at \(k_c\)=3.8x10^-4 sec^-1, the number of positive eigenvalues changes from 4 to 2 and the oscillations of the two CSTR's become nearly the same--the same period but not the same phase, nor the same amplitude. At \(k_c\) =10.97688x10^-4 there is a super-critical Hopf bifurcation and the SS become stable. In the region 10.97688x10^-4 < \( k_c \)< 103x10^-4 (where stable and unstable limit cycles coincide) the coupled system will evolve towards the SS in spite of the fact that each system alone would be oscillating- that is, oscillation death occurs. In the region 103x10^-4 < \(k_c\) < 285.36226x10^-4 (where a sub-critical Hopf bifurcation occurs) the coupled system will oscillate or goes towards the SS depending on the initial conditions. Since both a SS and a stable limit cycle coexist in this region, hysteresis may occur if the coupling rate is changed back and forth in this region. A sketch illustrating the transitions between different behaviors occurring in this system as described in Table 1 is shown in Figure 1.

Figure 1: Sketch illustrating different behaviors summarized in Table 1. Full (dashed) lines = stable (unstable) SS. Full (dashed) red lines = amplitude of stable (unstable) limit cycle. Green (blue) circle = super (sub) critical Hopf bifurcation. Purple circle = coincidence of stable and unstable limit cycle. Abscissa: coupling rate; ordinate: oscillation amplitude.

The oscillation death and the formation of a steady state can also occur with other oscillators as well such as the full FKN mechanism of the BZ reaction and with its simplified form the Oregonator (Field and Noyes 1974).

Experimentally, a BZ reaction has been carried out in two separate CSTRs with the coupling between them controlled by a pump transferring the solutions between them (Bar-Eli and Reuveni 1985). It is found that with small coupling rates, the oscillations continue in both CSTRs while when the coupling increases the oscillations stop in both CSTRs even when they are identical.

The plots in Figure 2 show typical time scans of two BZ oscillators. It is clearly seen how the oscillations of about 1 min period die when the coupling is on (upward arrow) and are renewed when the coupling goes off (downward arrow).

Figure 2: Typical time scans of two BZ oscillators.

It has been shown, (K. Bar-Eli 1985) that oscillator death can occur also in the case of identical oscillators. However, in this case, there remains a stable synchronous oscillation coincident with stable inhomogeneous equilibria.

The Brusselator model (Prigogine and Lefever 1968) \[ \frac{dx}{dt} = -(B+1)x+x^2 y+A, \] \[ \frac{dy}{dt} = Bx-x^2 y, \]

is a two dimensional kinetic scheme (X and Y being the concentrations of the chemical species) that depends on two parameters A and B. It can be shown that when \(B > A ^2 +1\ ,\) the steady state will be unstable and the system will oscillate. In the following, the values A=2 B=10 were used to give some examples of coupled identical oscillators that "die" and break the symmetry, i.e. unsymmetrical steady states appear.

With the parameters A=2 and B=10 having unstable SS at \(x(SS)=2 \) and \(y(SS)=5 \ ,\) the oscillations stop at coupling values of \( 0.3328 < k_c < 1.01157\ ,\) the symmetry breaks and two stable inhomogeneous solutions emerge seen in the following table for \(k_c=0.5\ :\)

(the problem being symmetric we can change the indices of the CSTRs). Symmetric steady states (similar to the above) can also be formed with 4 CSTRs:

and also with 6 CSTR's:

unsymmetric steady state solutions can also be formed. For example, for other such parameters as \(A=1\ ,\) \(B=10\ ,\) \(k_c=0.5\ :\)

Further calculations for two coupled identical Brusselators are described below. The plots of \(k_c\) vs. \(A\) at constant \(B \)where the non-homogeneous SS appear are shown in Figure 3. The non-homogeneous SS exist between the blue points (minimum \(k_c\)) and the red ones (minimum \(k_c\)). It is seen that as B decreases the region of the inhomogeneous SS decreases too.

The end points of the plots in Figure 3 (i.e. the coincidence of the red and blue points) are shown in the plots of Figure 4. The maxima (red) and minima (blue) \(A\)'s, from the above calculations (and similar ones) vs. \(B\ .\) The line is \(A=\sqrt{B-1}\ ,\) above which homogeneous SS exists, below which oscillations prevail. Between the red and blue dots the system can be either at the inhomogeneous SS or oscillating depending on the initial conditions. It is worthwhile to note that the stable inhomogeneous SS can coexist with the homogeneous one above the line and below the red dots.

Figure 3: The plots of \(k_c\) vs. \(A\) at constant \(B \)where the non-homogeneous SS. The plots are (from right to left)\[B=11\ ,\] \(B=10\ ,\) \(B=9.5\ ,\) and \(B=9\ .\)

Figure 4: The end points of the plots in Figure 3. The maxima (red) and minima (blue) \(A\)'s, from the above calculations (and similar ones) vs. \(B\ .\) The line is \(A=\sqrt{B-1}\ ,\) above which homogeneous SS exists, below which oscillations prevail.

Thus at \(B=35\ ,\) \(A=4 \ ,\) and \(k_c=0.5\) the initial point \(x[1,0]=0.2\ ,\) \(x[2,0]=7.7\ ,\) \(y[1,0]=19.4\ ,\) \(y[2,0]=4\ ,\) the system goes to the inhomogeneous SS at \(x[1]=0.249\ ,\) \(x[2]=7.751\ ,\) \(y[1]=19.644\ ,\) \(y[2]=4.6405\) while at the near by initial point namely, \(x[1,0]=0.19\ ,\) \(x[2,0]=7.6\ ,\) \(y[1,0]=18.7\ ,\) \(y[2,0]=3.9\ ,\) the system start oscillating.

In a similar fashion, above the line in the figure, the system can go either to an homogeneous steady state or to an inhomogeneous one. Thus at \(A=6.5\ ,\) \(B=35\ ,\) \(k_c=0.5\) starting at \(x[1,0]=0.5\ ,\) \(x[2,0]=9\ ,\) \(y[1,0]=22\ ,\) \(y[2,0]=2\) the system ends at an inhomogeneous SS at \(x[1]=0.66640964\ ,\) \(x[2]=12.33359\ ,\) \( y[1]=26.2488\ ,\) \(y[2]=2.9145\) while starting at \(x[1,0]=0.5\ ,\) \(x[2,0]=8\ ,\) \( y[1,0]=20\ ,\) \(y[2,0]=1\) the homogeneous SS \(x[1]=x[2]=6.5\) and \(y[1]=y[2]=5.385\) is reached.

[edit] Theory

Figure 5: Two independent oscillators going around a limit cycle at different frequencies. Diffusive coupling between them attempts to pull them toward their average which if the oscillators differ widely in frequency may be the near the equilibrium point.

The intuition behind oscillator death is rather straightforward. Figure 5 illustrates two independent oscillators going around a limit cycle at different frequencies. Diffusive coupling between them attempts to pull them toward their average which if the oscillators differ widely in frequency may be the near the equilibrium point. The analysis of oscillator death makes this point precise. General models of oscillators exhibiting death are difficult to analyze, so instead, a typical technique is to use a normal form near some bifurcation. Most analysis of oscillator death uses the normal form near a Hopf bifurcation: \[ z_j' = (1+i\omega_j)z_j -(1 + i q) z_j|z_j|^2 + (c+i d) (z_k-z_j), \quad j=1,2, \quad k=3-j. \] In absence of any coupling each oscillator has a frequency of \(\omega_j-q\ .\) In addition to the limit cycle, there is an unstable equilibrium point, \(z_j=0\) which remains in the presence of coupling. In the simplest form, oscillator death is the stabilization of the unstable rest state due to coupling. In Aronson et al. (1990), it was proven that the rest state was asymptotically stable for \(d=0\) when the following conditions held. Let \(\Delta=|\omega_2-\omega_1|\ .\) Then \(\Delta>2\) and \(1< c < (1+\Delta^2/4)/2\ .\) This says, that if the coupling is neither too strong nor too weak, and if there is enough of a frequency dispersion, the rest state will be stable. Note that in this case, it can also be shown that there are no stable limit cycles, so unlike the case of identical coupling in the above section, here, there is no bistability between the rest state and the oscillation. The oscillator is dead.

It is possible to generalize this idea to models of the form: \[ x_1' = f(x_1) + d (x_2-x_1) \] \[ x_2' = r f(x_2) + d (x_1-x_2)\ .\] Note that in absence of coupling, if system \(x_1\) has a limit cycle with frequency \(\omega\) then system \(x_2\) has a limit cycle with frequency \(r \omega\ .\) If the oscillation \(x'=f(x)\) has emerged from a Hopf bifurcation, then there is a fixed point which has only one pair of eigenvalues with positive real parts and these have an imaginary component. Let \(x=0\) be the fixed point and let the lone unstable eigenvalues be \(\alpha\pm i\omega.\) Ermentrout and Troy (1989) showed that if \(\alpha/\omega\) is sufficiently small, then there are values of \(d\) neither too large, nor too small, such that \(x_1=x_2=0\) is asymptotically stable. This is a local result and does not necessarily imply the loss of any limit cycles, but in practice the limit cycle is lost when the fixed point stabilizes via a Hopf bifurcation as in the above normal-form example.

Oscillator death does not just happen for pairs of oscillators. It can happen in a spatially distributed medium. Ermentrout and Troy (1986) analyzed reaction diffusion equations with a linear gradient in frequency of the form: \[ z_t = z(1 + i \sigma (x-1/2) - |z|^2) + d z_{xx}, \quad 0 < x < 1. \] They find that for \(d>1\) (strong coupling), as \(\sigma\ ,\) the gradient in the frequencies, increases, the amplitude of the oscillations \(|z(x,t)|\) goes to zero uniformly in \(x\) and for \(\sigma\) large enough, \(z=0\) is asymptotically stable.

Mirollo and Strogatz (1990) and Ermentrout (1990) studied the phenomena of oscillator death in networks of globally coupled oscillators with random frequencies. There have been several recent experiments in a variety of systems which verify the general theory underlying oscillator death (Herrero et al. 1990, Ozden et al. 2004, Zhai et al. 2004).

[edit] References

• Aronson, D, Ermentrout, GB, Kopell, N, Physica D:403-449, 1990.
• Bar-Eli, K, J. Phys. Chem 88:3616-3622, 1984.
• Bar-Eli, K, Physica D 14:242-252, 1985.
• Bar-Eli,K and Reuveni S J.Phys.Chem. 89: 1329-1330, 1985
• Ermentrout, GB and Troy, WC, SIAM J. Appl. Math, 46:359-367, 1986.
• Ermentrout, GB and Troy, WC, SIAM J. Math. Anal., 20:1436-1446, 1989.
• Ermentrout, GB, Physica D, 41:219-231, 1990.
• Field, RJ , Noyes, RM J.Chem. Phys. 60: 1877-1700, 1974
• Field, RJ, Koros, E, and Noyes, RM, J. Am. Chem., Soc. 94, 8649-8664, 1972.
• Herrero, R, Figueras, M, Rius, J, Pi, F, and Orriols, G, Phys. Rev. Lett., 84:5312-5315, 2000.
• Mirollo, RE, Strogatz, SH, Journal of Statistical Physics, 245-262, 1990.
• Noyes, RM, Field, RJ, Thompson, RC, J. Am. Chem. Soc., 93, 7315-7316, 1971.
• Ozden, I, Venkataramani, S, Long, MA, Connors, BW, and Nurmikko, AV, Phys. Rev. Lett., 93:158102, 2004.
• Prigogine, I, Lefever, R, J. Chem. Phys. 48: 1695-1700,1968.
• Zhai, Y, Kiss, IZ, Hudson, JL, Phys. Rev. E 69:026208, 2004.

Internal references

• Yuri A. Kuznetsov (2006) Andronov-Hopf bifurcation. Scholarpedia, 1(10):1858.

• Anatol M. Zhabotinsky (2007) Belousov-Zhabotinsky reaction. Scholarpedia, 2(9):1435.

• John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.

• Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.

• James Murdock (2006) Normal forms. Scholarpedia, 1(10):1902.

• Richard J. Field (2007) Oregonator. Scholarpedia, 2(5):1386.

• Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.

• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

• Arkady Pikovsky and Michael Rosenblum (2007) Synchronization. Scholarpedia, 2(12):1459.

[edit] External Links

• Kedma Bar-Eli’s website
• Bard Ermentrout’s website

[edit] See Also

Andronov-Hopf bifurcation, Bifurcations, Brusselator, Oregonator, Oscillator