Singularity of information flow at the Hopf bifurcation point

Abstract

We investigate the singular behavior of information flow near the Hopf bifurcation point by analyzing the learning rate, a key quantity in stochastic thermodynamics. As a model system exhibiting the Hopf bifurcation, we study the Brusselator. We first numerically compute the learning rate in the stationary regime and find that it remains finite even in the deterministic limit, suggesting that information flow can be quantified in deterministic dynamics through probabilistic descriptions. Linear analysis accurately reproduces the numerical results in the stationary regime but fails near the bifurcation point. To overcome this limitation, we employ the singular perturbation method, well known in deterministic bifurcation theory, and carry out the corresponding calculation explicitly for a stochastic system described by a Langevin equation. This allows us to evaluate the learning rate near the bifurcation point. We then theoretically derive its non-smooth behavior in the deterministic limit. Our results demonstrate that changes in dynamical behavior are reflected in the information flow and provide a basis for analyzing information processing in biochamical oscillations.

Figures (6)

• Figure 1: Learning rates in the steady state. The bifurcation point is $b_c = 2$, and the system size is $V = 10^3$.
• Figure 2: Learning rate in the steady-state regime. $b_c = 2$ and $V = 2 \times10^3$. Blue dots indicate the results of numerical simulations, and the orange line shows the result of the linear analysis given by Eq. (\ref{['Brusselator_linear_learning']}).
• Figure 3: Learning rate near the Hopf bifurcation in the Brusselator. $b_c = 2$ and $V=2\times10^3$. Blue dots represent the numerical simulation results, the orange line denotes the linear analysis result Eq. (\ref{['Brusselator_linear_learning']}), and the green line shows the result from singular perturbation method Eq. (\ref{['transformation_learning']}).
• Figure 4: Deterministic limit of the learning rate $l^1_{\mathrm{st}}$ near the bifurcation point in the Brusselator with $b_c = 2$. The purple line $l_{\infty}$ represents the asymptotic value in the stationary regime, namely the result of the linear analysis extended into the oscillatory regime. The yellow line represents the learning rate in the deterministic limit gevin by Eq. (\ref{['lst_piecewise']}).
• Figure 5: System-size dependence of the difference between the learning rate $l^1_{\mathrm{st}}$ and its asymptotic value in the stationary regime $l_\infty$.