Memory-dependent noise-induced resonance and diffusion in non-Markovian systems

S. S. Melnyk, O. V. Usatenko, and V. A. Yampol'skii

Phys. Rev. E 103, 032139 – Published 22 March 2021

Abstract

We study random processes with nonlocal memory and obtain solutions of the Mori-Zwanzig equation describing non-Markovian systems. We analyze the system dynamics depending on the amplitudes $ν$ and $μ_{0}$ of the local and nonlocal memory and pay attention to the line in the ( $ν, μ_{0}$ ) plane separating the regions with asymptotically stationary and nonstationary behavior. We obtain general equations for such boundaries and consider them for three examples of nonlocal memory functions. We show that there exist two types of boundaries with fundamentally different system dynamics. On the boundaries of the first type, diffusion with memory takes place, whereas on borderlines of the second type the phenomenon of noise-induced resonance can be observed. A distinctive feature of noise-induced resonance in the systems under consideration is that it occurs in the absence of an external regular periodic force. It takes place due to the presence of frequencies in the noise spectrum, which are close to the self-frequency of the system. We analyze also the variance of the process and compare its behavior for regions of asymptotic stationarity and nonstationarity, as well as for diffusive and noise-induced-resonance borderlines between them.

Received 4 January 2021
Accepted 2 March 2021

DOI:https://doi.org/10.1103/PhysRevE.103.032139

Physics Subject Headings (PhySH)

Diffusion Stochastic processes

Disordered systems Lattice models in statistical physics

Langevin equation Monte Carlo methods Non-Markovian processes Stochastic differential equations Time series analysis

Statistical Physics & Thermodynamics

Authors & Affiliations

S. S. Melnyk and O. V. Usatenko

O. Ya. Usikov Institute for Radiophysics and Electronics NASU, 61085 Kharkiv, Ukraine

V. A. Yampol'skii

O. Ya. Usikov Institute for Radiophysics and Electronics NASU, 61085 Kharkov, Ukraine and V. N. Karazin Kharkov National University, 61077 Kharkov, Ukraine

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Vol. 103, Iss. 3 — March 2021

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Images

Figure 1
The stationarity borderlines for the delta-functional memory $μ (t) = (μ_{0} / T) δ (t - T)$ with $T = 1$ in the plane $(ν, μ_{0})$ . The red solid curve at $μ_{0} > 1$ corresponds to the oscillatory borderline, and the black solid straight line corresponds to the diffusive one.
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Figure 2
The stationarity borderlines for the stepwise memory function $μ (t) = (μ_{0} / T^{2}) θ (T - t)$ with $T = 0.5$ in the plane $(ν, μ_{0})$ . The upper red solid curve is the oscillatory borderline, and the lower black solid straight line at $μ_{0} < 2$ is the diffusive one.
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Figure 3
The stationarity borderlines for the exponential memory function $μ (t) = (μ_{0} / T^{2}) exp (- t / T)$ with $T = 1$ in the plane $(ν, μ_{0})$ . The region of stationarity lies to the right of the solid line; the region of nonstationarity lies to the left of this line. The vertical red and oblique black solid lines correspond to the oscillatory and diffusive borderlines, respectively. A dashed parabola separates the areas where the correlation function decays exponentially without oscillations (below the parabola) and with oscillations (above this curve).
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Figure 4
The memory-dependent diffusion for the exponential memory function and zero prehistory $V (t ⩽ 0) = 0$ . The irregular black solid lines are the trajectories for different realizations of the stochastic process $V (t)$ on the diffusive borderline of stationarity. The green solid line is the analytical result for $\pm \sqrt{D (t)}$ where $D (t)$ is the variance, Eq. (42). The red symbols on this curve are the results of numerical simulation obtained by the ensemble averaging over $10^{3}$ realizations for each symbol. The dashed red line presents the $\pm \sqrt{D_{B} (t)} = \pm σ \sqrt{t}$ dependence for the Brownian diffusion. The dash-dotted curve is the dependence $\pm σ \sqrt{t} / (1 + ν T)$ which serves as the asymptote for $\pm \sqrt{D (t)}$ at $t ≫ T$ [see Eqs. (42) and (43)]. The parameters are $ν = - 0.4, μ_{0} = 0.4, T = 1$ , and $σ = 1$ .
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Figure 5
The noise-induced resonance in the $V (t)$ process with the exponential memory function and zero prehistory. The thin black solid line shows a realization of the stochastic process $V (t)$ on the oscillatory borderline of stationarity. The green solid line presents the analytical result for $\pm \sqrt{D (t)}$ , Eq. (45). The parameters are $ν = - 1 / T, μ_{0} = 1.01, T = 1$ , and $σ = 1$ .
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Figure 6
The variance $D (t)$ on the diffusive borderline for the exponential memory function and zero prehistory at different values of $μ_{0}$ : $μ_{0} = 0.4$ (the upper green solid curve), $μ_{0} = 0$ (the red straight dashed line), and $μ_{0} = - 0.4$ (the lower black dash-dotted curve). The black filled circles on these curves are the results of numerical simulations obtained by the ensemble averaging over $10^{3}$ realizations for each symbol. Other parameters: $ν = - μ_{0} / T, T = 1$ , and $σ = 1$ .
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