It has been reported that traveling waves propagate periodically and stably in sub-excitable systems driven by noise [Phys. Rev. Lett. \textbf{88}, 138301 (2002)]. As a further investigation, here we observe different types of traveling waves under different noises and periodic forces, using a simplified Oregonator model. Depending on different noises and periodic forces, we have observed different types of wave propagation (or their disappearance). Moreover, the reversal phenomena are observed in this system based on the numerical experiments in the one-dimensional space. As an explanation, we regard it as the effect of periodic forces. Thus, we give qualitative explanations to how reversal phenomena stably appear, which seem to arise from the mixing function of the periodic force and the noise. And the output period and three velocities (the normal, the positive and the negative) of the travelling waves are defined and their relationship with the periodic forces, along with the types of waves, are also studied in sub-excitable system under a fixed noise intensity.
Deep Dive into Propagation of travelling waves in sub-excitable systems driven by noise and periodic forcing.
It has been reported that traveling waves propagate periodically and stably in sub-excitable systems driven by noise [Phys. Rev. Lett. \textbf{88}, 138301 (2002)]. As a further investigation, here we observe different types of traveling waves under different noises and periodic forces, using a simplified Oregonator model. Depending on different noises and periodic forces, we have observed different types of wave propagation (or their disappearance). Moreover, the reversal phenomena are observed in this system based on the numerical experiments in the one-dimensional space. As an explanation, we regard it as the effect of periodic forces. Thus, we give qualitative explanations to how reversal phenomena stably appear, which seem to arise from the mixing function of the periodic force and the noise. And the output period and three velocities (the normal, the positive and the negative) of the travelling waves are defined and their relationship with the periodic forces, along with the types
The effects of noise on nonlinear systems are the subject of intense experimental and theoretical investigations. Noise can induce transition [1,2], bifurcations [3], and stochastic resonance [4,5,6,7]. Especially, in Ref. [7] the synchronization of spatiotemporal patterns were observed in an excitable medium by the numerical evidence. Moreover noise can enhance propagation in arrays of coupled bistable oscillators [8,9,10,11]. In an excitable system, an external periodic forcing can dramatically change its behavior. As reported by previous documents, the phase locking, quasi-periodicity, period doubling, and chaos were observed [12]. The temporal evolution of the concentration patterns has been modeled by partial differential reactiondiffusion equations. Such models include oscillatory, excitable or bistable systems with either none, one or two linearly stable homogeneous states [13,14]. It is also well known that in sub-excitable systems noise also can induce travelling waves [15], drive avalanche behavior [16], and sustain pulsating patterns and global oscillations [17]. Sub-excitable systems under noises and periodic forcing are able to send out travelling and spiral waves. Belousov-Zhabotinsky (BZ) reaction [18,19] is a popular symbol in Send offprint requests to: sifenni@163.com (Fen-Ni Si) nonlinear dynamical realm to study excitable and subexcitable system. It has been widely agreed that the noise and periodic forcing play a very important role in wave propagation and stability.
Recently, it was observed that noise can support wave propagations in sub-excitable [20,17,15] systems due to a noise-induced transition [21,22]. In these studied, the medias are static, and transports are governed by diffusion [23]. However in many situations, the medias are not static but subject to a motion. For example, stirred by a flow, or by the periodic forcing, the convective-like phenomena were observed due to electric field in Ref. [24], which occurs especially in chemical reactions in a fluid environment. In such cases, diffusive transport usually dominates only at small spatial scales while mixing due to the flow in much faster at large scale. In Refs. [25,26,27], the authors shown that in an inhomogeneous self-sustained oscillatory media, an increasing rate of mixing can lead to a transition to a global synchronization of the whole media. Especially, Ref. [28] showed that the interplay among excitability, noise, diffusion and mixing can generate various pattern formation in a 2D FigzHugh-Nagumo (FHN) model subject to the advection by a chaotic flow. Here, we research the effect of noise and periodic forcing on sub-excitable systems using the Oregonator model in one dimension, which advances from the BZ reaction. The re-versal phenomenon is not observed in the previous documents about the propagation of travelling waves. It is founded in this paper, which relates to a new concept. In Ref. [30], Richard A. Albanese refers to reversal concept in wave propagation concern extraction of information about distant structural features from the measurements of scattered waves, but it is irrelevant the excitable system.
In our paper, we define that the general traveling waves propagate forward in one constant direction and vice versa. Under this definition, we found the reversal phenomena in our numerical simulations. However in our simulation we have discovered that after some time, the waves change its propagation direction and turn backward to travel in the opposite direction. The traveling waves propagate forward and backward alternately and periodically. That is called the reversal phenomenon (see the supplementary material on-line movie for this phenomenon, Movie-0 ). In the mathematical language, the definition of definition of an “reversal phenomenon” as at time t, the spatial position of a traveling wave front is at point x. After some time ∆t > 0, the wave front reaches point x again with reversal direction. In fact, this phenomenon was observed in the FHN model by numerical simulations [28]. In this article, our focuses are the effect of the periodic forcing and noise on the propagation of traveling waves in subexcitable systems.
Most of the systems we are interested in reside in a ddimensional world. This means that our variables (fields or concentrations) depend on time and space. In present paper, the starting deterministic model [20] is based on partial differential equations, and when the randomness is introduced we transform them into stochastic partial differential equation. A representative example is the deterministic reaction-diffusion equation,
where φ(x, t) represent the density of a physical observable, f (φ(x, t), µ) is a nonlinear function of the field φ and µ denotes the relevant control parameter. The above equation can be made more complicated when considering vector fields, higher-order derivatives, or nonlocal operators. The effect of fluctuations is introduced through a stochastic proce
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