Synchronization of weakly perturbed Markov chain oscillators

Rate processes are simple and analytically tractable models for many dynamical systems that switch stochastically between a discrete set of quasistationary states; however, they may also approximate continuous processes by coarse-grained, symbolic dynamics. In contrast to limit-cycle oscillators that are weakly perturbed by noise, in such systems, stochasticity may be strong, and topologies more complicated than a circle can be considered. Here we apply a second-order time-dependent perturbation theory to derive expressions for the mean frequency and phase diffusion constant of discrete-state oscillators coupled or driven through weakly time-dependent transition rates. We also describe a method of global control to optimize the response of the mean frequency in complex transition networks.

Figure 1

Schematics of (a) a stochastic oscillator with three metastable states in a tilted harmonic potential. The transitions are Kramers rate processes and can be perturbed through a change in the activation energy or noise strength. In this example the forward transitions are perturbed in form of a traveling wave. (b) Nonsequential stochastic oscillator with four states. The diagrams on the right show possible lifts of the Markov chain to periodic lattices and associated Poincaré sections. The red (dotted lines in lower chain) Poincaré section disregards subthreshold oscillations $1\to 2\to 4\to 1$.

Figure 2

(a) Transition rates for a biased jump process on a one-dimensional periodic lattice. The transition rate is divided into a time-independent forward bias, diffusion part, and perturbation depending on time and state. (b) Transition rates for a forward jump process ($\gamma =0$) on a two-dimensional periodic lattice with attractive coupling in two directions. The speed of the transitions is indicated by different emphasis of the arrows.

Figure 3

Relative frequency change (left column) and change in the Péclet number (right column) of a stochastic biased jump process on a ring in (a) and (b) for the case of harmonic driving of the forward transition rates with $\Lambda =i{\omega }_1$, in (c) and (d) for driving with another stochastic oscillator of the same length $L$ and ${\Lambda }^{*}={\lambda }^{df}+{\lambda }^{fl}{\omega }_1/2\pi$, and in (e) and (f) for mutually coupled oscillators of same length $L$, ${\mathbf{k}}^0=(2\pi /L,-2\pi /L)$ and $\beta =\pi /2$. Numerical simulations were performed using a generalized Gillespie algorithm for time-dependent transition rates. Averages were taken over 1000 runs for 1000 time units each with the parameters $\gamma =0.1$, $\varepsilon =0.5$. The inset in subfigure (a) applies to all subfigures.

Figure 4

Plot of tilted potential (bold, blue line) in the stochastic Adler equation (36). The local minimum at ${\varphi }_0$ also determines the two adjacent local maxima at $-{\varphi }_0$ and $2\pi -{\varphi }_0$. The potential difference over the left barrier is $-2U\left({\varphi }_0\right)$ and $2\pi \Delta -2U\left({\varphi }_0\right)$ over the right barrier.

Figure 5

Rescaled frequency shift $(\Omega -{\omega }_0)/D_0$ of the periodically driven drift and diffusion process on a circle (a) as a function of $x=\Delta /D_0$ and $y=2\pi \varepsilon /D_0$. Values $x<0$ and $x>0$ correspond to negative (blue) and positive (red) frequency shifts, respectively (color bar aligned on top). The dashed lines mark the border of the Arnold tongue region of phase locking in the limit $D_0\to 0$. The Kramers rate theory is valid between the yellow (light, solid) curves that indicate the condition $(U_{\max }-U_{\min })/D_0=1$, whereas below the blue (dark, solid) line for $y<\sqrt{2}$, our perturbation theory will give physically sensible results for all frequency differences, in particular, at resonance point $\Delta =0$. Above the blue (dark, solid) line, the perturbation theory may be still give good results sufficiently far away from the resonance point. (b) Frequency shift as a function of $y$ with $x=1.0$ fixed (solid curve). The two dashed curves indicate the second-order perturbation approximation for low values of $y$ and the Kramers rate approximation for high $y$. The exact frequency shift was determined by numerical evaluation of the integrals in Ref. [14].

Figure 6

(a) Unidirectional ring network with $L=30$ states and $E_{sc}=15$ random shortcuts. (b) Rearranging the nodes on the unit circle according to the argument of the complex eigenvector to eigenvalue ${\lambda }_{\max }=-0.52-0.77i$ of maximal ratio $r_{\max }=1.48$ between imaginary and real part strong circular flows in the network can be detected. Forward transitions are shown in blue (solid arrows) and backward transitions in red (dashed arrows). (c) Convex hull of the complex eigenvalues excluding ${\lambda }_0=0$ for single realizations of directed ring networks of size $L=1000$ at various shortcut densities $\sigma =E_{s}c/L$. Here $E_{s}c$ is the number of additional random transition channels (including duplicates). The dashed lines originating at zero are tangents of slope $\left|r\right|=1.09$ to the convex hull for $\sigma =1.0$. The dots indicate nonzero complex eigenvalues for a network realization at $\sigma =2.0$. The spectral gap increases with $\sigma$ and the slope of the tangent decreases. (d) Double logarithmic plot of the average maximum value of $r=Im\left[\lambda \right]/Re\left[\lambda \right]$ in the spectrum versus shortcut density $\sigma$. Each data point and standard deviation was determined from a sample of 1000 random networks of size $L=500$. The dashed line in (c) is $r=1/\sqrt{\sigma }$.

Figure 7

Optimization of the driving protocol for the network shown in Fig. 6b. Assigning the same rate and perturbation cost to each transition in (a), we plot the eigenvalues of the derivative of the frequency shift operator (thin lines) with respect to frequency ${\omega }_1$ of deterministic harmonic driving. In (b) the eigenvalues of the second-order frequency-shift operator (thin lines) are shown as functions of the driving frequency. The red (bold, light gray) and blue (bold, dark gray) lines in (a) and (b) show the frequency shift and its derivative when the driving protocol is chosen to have the optimal frequency response at either of the two apparent resonant frequencies seen in (a).