Bistability generated via a noise-induced phase transition is reexamined from the view of macroscopic dynamical systems, which clarifies the role of fluctuation better than the conventional Fokker-Plank or Langevin equation approach. Using this approach, we investigated the spatially-extended systems with two degrees of freedom per site. The model systems undergo a noise-induced phase transition through a Hopf bifurcation, leading to a macroscopic limit cycle motion similar to the deterministic relaxation oscillation.
KEYWORDS: Stochastic processes, Oscillators, Signal to noise ratio, Complex systems, Interference (communication), Bistability, Solids, Statistical analysis, Neurons, Chemical elements
We study nonlinear systems under two noisy sources to demonstrate the concept of doubly stochastic effects. In such effects noise plays a twofold role: first it induces a special feature in the system, and second it interplays with this feature leading to noise-induced order. For this effect one needs to optimize both noisy sources, hence we call these phenomena doubly stochastic effects. To show the generality of this approach we apply this concept to several basic noise-induced phenomena: stochastic resonance, noise-induced propagation and coherence resonance. Additionally, we discuss an application of this concept to noise-induced transitions and ratchets. In all these noise-induced effects ordering occurs due to the joint action of two noisy sources.
Classical notion of synchronization, introduced originally for periodical self-sustained oscillators, can be extended to stochastic systems. This can be done even in the case when the characteristic times of a system are fully controlled by noise. Stochastic synchronization is then defined by imposing certain conditions to various statistical measures of the process. We review various approaches to stochastic synchronization and apply them to study synchronization in the electrosensory system of paddlefish.
We propose a model for a walker moving on an asymmetric periodic ratchet potential. The walker has two 'feet' represented as two finite-size particles coupled nonlinearly through a double-well potential. In contrast to linear coupling, the bistable potential admits a richer dynamics where the ordering of the particles can alternate. The transitions between the two stable points on the bistable potential, correspond to a walking with alternating particles. In our model, each particle is acted upon by independent white noises, modeling thermal noise, and additionally we have an external time-dependent force that drives the system out of equilibrium, allowing directed transport. This force can be common colored noise, periodic deterministic driving or fluctuations on the bistable potential. In the equilibrium case, where only white noise is present, we perform a bifurcation analysis which reveals different walking patterns available for various parameter settings. Numerical simulations showed the existence of current reversals and significant changes in the effective diffusion constant and in the synchronization index. We obtained an optimal coherent transport, characterized by a maximum dimensionless ratio of the current and the effective diffusion (Peclet number), when the periodicity of the ratchet potential coincides with the equilibrium distance between the two particles.
The problems studied here are relevant for an understanding of the functioning of hydrolytic enzyme molecules. These enzymes work like molecular machines breaking off the valence peptid bonds of substrates. In particular the role of Fermi resonance which is evident from a spectral lines of valence oscillations is studied. The influence of this resonance on valence splitting is discussed. It is shown that the breaking of these bonds has a higher probability, if the stochastic oscillations of atoms in catalytic groups at the active site have a large quality coefficient. We show that the corresponding low damping is essential for the Fermi resonance modes of these oscillations.
The issues about the behavior of variant test particles (TP) in 2D potential field with complex relief are studied. The cases with time-dependent parameters of potential field are considered. In particular the mathematical models that are discussed in this paper can describe either the process of penetration of TP (ligand) in active site of the enzyme or proton's transition from one potential pit to another.
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