Figure 15

Phase computation methods on the Brusselator. The approximate schemes are almost a perfect match for the golden reference PhCompBF. The equations are very fast as indicated by the speed-up figures given in the text. Results of the phase equations are quite close to each other. In the interval 400-600 s, PhEqnQL comes closer to the true value. The following observations and facts are iterated for this first figure of the results. For convenience, these comments are not going to be repeated for every figure that follows. Note that the phase equations are in reality differential equations, solving one by one for the instantaneous phase of points in an oscillator sample path. Therefore, due to the approximations involved in their design and, furthermore, due to the imperfect discretizations (of the differential equations that they are represented by) for their numerical solutions, the phase equations are doomed to suffer from accumulating truncation errors. This is why, in many results figures for the oscillators in this article, we observe the results of phase equations tending to deviate from the golden reference PhCompBF as time progresses. However, computational complexity-wise the phase equations are indeed very fast. This makes the phase equations a feasible and accurate choice for the phase computations of less noisy oscillators, possibly with a dense grid of timepoints in an SSA sample path and high molecule numbers for every species in the system (especially in a container of large volume), deviating not much from their limit cycles. The phase computation schemes, on the other hand, do not employ as many approximations as the phase equations do in their design. Furthermore, these schemes are in the form of algebraic equations, again solving one by one for the instantaneous phase of points in an oscillator sample path. Therefore, the schemes, for their numerical solution, do not involve time discretizations as the phase equations do. This means that the schemes do not suffer from truncation error accumulation. The schemes are subject to errors originating from the approximations committed in their theoretical development, and once again, these approximations are not on the same scale as those employed in the derivation of phase equations, i.e., the schemes are much more accurate than the equations. However, the numerical procedures associated with the schemes render them more costly in computational complexity with respect to the equations. Therefore, one may rightfully contend that the phase computation schemes are tailored to fit phase computations for moderately noisy oscillators in small volume, with low molecule numbers for each species and possibly a sparse grid of timepoints in an SSA sample path.