Dichotomous Noise Research Articles

Stochastic resonance in a fractional linear oscillator subject to random viscous damping and signal-modulated noise

The stochastic resonance in a fractional linear system with random viscous damping driven by a dichotomous noise and signal-modulated noise is investigated. By the use of the properties of the noises and the Shapiro-Loginov formula and Laplace transform technology, the exact expression for the mean output-amplitude-gain (OAG) of the system is obtained. The non-monotonic influence of the viscous damping of the oscillator on the OAG is found. It is shown that the OAG is a non-monotonic function of the intensity and the correlation rate of the dichotomous noise. The OAG varies non-monotonically with the friction coefficient, with the frequency of the driving signal, with the frequency of the linear oscillator, as well as with the fractional exponent of the fractional oscillator.

Trichotomous noise induced stochastic resonance in a fractional oscillator with random damping and random frequency

.In this study, we investigate the stochastic resonance (SR) of a fractional linear oscillator subjected to multiplicative trichotomous noise and additive fractional Gaussian noise and driven by a periodic signal. Using the Shapiro-Loginov formula and the Laplace transformation technique, we acquire the exact expression of the first-order moment of the system’s steady response. Meanwhile, we discuss the evolutions of the output amplitude with the frequency of the periodic signal and noise parameters. We determine that bona fide SR, SR and reverse-resonance exist in this system. Specifically, the evolution of the output amplitude with the frequency of the periodic signal presents one-peak oscillation, double-peak oscillation and triple-peak oscillation. Moreover, the interplay of the trichotomous noise and memory can induce and diversify the stochastic multi-resonance (SMR) phenomena and give this linear system richer dynamic behavior. A hypersensitive response of the output amplitude to the noise intensity is demonstrated, which is not observed in systems driven by dichotomous noise.

Decreasing renewal dichotomous Markov noise shock model with hypothesis testing applications

In this paper, we investigate the probabilistic characteristics of systems driven by dichotomous Markov noise (DMN), described by an external random increasing and decreasing shocks. Based on the DMN, we define two new aging classes of the increasing/decreasing renewal dichotomous Markov noise (IRDMN/DRDMN), which are separated by an exponential steady state regime in the long time behavior. In addition, a moment inequality is derived for the system whose life belongs to DRDMN class. This inequality to devise a nonparametric testing procedure for exponentiality against an alternative DRDMN life distributions.

Heteroclinic chaos in a Josephson-junction system perturbed by dichotomous noise excitation

The chaotic behavior in a Josephson-junction system perturbed by dichotomous noise excitation is discussed in detail. Conditions for the onsets of chaos are derived by virtue of the random Melnikov method together with the mean-square criterion. It is shown that with the increase of the noise transition rate, the threshold of the dichotomous noise amplitude for the onset of chaos in the system increases. The effects of dichotomous noise on the Josephson-junction system are also determined by numerical simulations via the mean largest Lyapunov exponents, which verifies that the injection of the dichotomous noise can cause the change of the sign of the largest Lyapunov exponent and lead to noise-induced chaos. Phase portraits and time histories are further used to verify these results. It can be concluded that by changing the internal parameters of the dichotomous noise, we can adjust the threshold for the onset of the chaos and then control dynamical behaviors in the Josephson-junction system subjected to dichotomous noise excitation.

Dichotomous noise models of gene switches.

Molecular noise in gene regulatory networks has two intrinsic components, one part being due to fluctuations caused by the birth and death of protein or mRNA molecules which are often present in small numbers and the other part arising from gene state switching, a single molecule event. Stochastic dynamics of gene regulatory circuits appears to be largely responsible for bifurcations into a set of multi-attractor states that encode different cell phenotypes. The interplay of dichotomous single molecule gene noise with the nonlinear architecture of genetic networks generates rich and complex phenomena. In this paper, we elaborate on an approximate framework that leads to simple hybrid multi-scale schemes well suited for the quantitative exploration of the steady state properties of large-scale cellular genetic circuits. Through a path sum based analysis of trajectory statistics, we elucidate the connection of these hybrid schemes to the underlying master equation and provide a rigorous justification for using dichotomous noise based models to study genetic networks. Numerical simulations of circuit models reveal that the contribution of the genetic noise of single molecule origin to the total noise is significant for a wide range of kinetic regimes.

Dichotomous-noise-induced chaos in a generalized Duffing-type oscillator with fractional-order deflection

The chaotic behavior of a generalized Duffing-type oscillator with fractional-order deflection under dichotomous noise excitation is discussed in detail. Using the Melnikov method together with mean-square criterion, necessary conditions for the onsets of chaos are derived. It is shown that with the increase of noise transition rate, the threshold of noise amplitude for chaos firstly decreases to a minimum, and then increases. The effects of dichotomous noise on the Duffing-type oscillator are also obtained by vanishing the mean largest Lyapunov exponent of the oscillator. This is further verified by phase portraits and time histories with numerical simulations.

Noise-induced effects in nonlinear relaxation of condensed matter systems

Noise-induced phenomena characterise the nonlinear relaxation of nonequilibrium physical systems towards equilibrium states. Often, this relaxation process proceeds through metastable states and the noise can give rise to resonant phenomena with an enhancement of lifetime of these states or some coherent state of the condensed matter system considered. In this paper three noise induced phenomena, namely the noise enhanced stability, the stochastic resonant activation and the noise-induced coherence of electron spin, are reviewed in the nonlinear relaxation dynamics of three different systems of condensed matter: (i) a long-overlap Josephson junction (JJ) subject to thermal fluctuations and non-Gaussian, Lévy distributed, noise sources; (ii) a graphene-based Josephson junction subject to thermal fluctuations; (iii) electrons in a n-type GaAs crystal driven by a fluctuating electric field. In the first system, we focus on the switching events from the superconducting metastable state to the resistive state, by solving the perturbed stochastic sine-Gordon equation. Nonmonotonic behaviours of the mean switching time versus the noise intensity, frequency of the external driving, and length of the junction are obtained. Moreover, the influence of the noise induced solitons on the mean switching time behaviour is shown. In the second system, noise induced phenomena are observed, such as noise enhanced stability (NES) and stochastic resonant activation (SRA). In the third system, the spin polarised transport in GaAs is explored in two different scenarios, i.e. in the presence of Gaussian correlated fluctuations or symmetric dichotomous noise. Numerical results indicate an increase of the electron spin lifetime by rising the strength of the random fluctuating component. Furthermore, our findings for the electron spin depolarization time as a function of the noise correlation time point out (i) a non-monotonic behaviour with a maximum in the case of Gaussian correlated fluctuations, (ii) an increase up to a plateau in the case of dichotomous noise. The noise enhances the coherence of the spin relaxation process.

Rising relative fluctuation as a warning indicator of discontinuous transitions in symbiotic metapopulations

The long-time limit behavior of the stochastic Lotka–Volterra model of a symbiotic metapopulation subjected to generalized Verhulst self-regulation is considered. The influence of a time-variable environment on the carrying capacities of subpopulations is modeled as a periodic deterministic part and a symmetric dichotomous noise. Relying on the mean-field approach it is established that at certain parameter regimes the mean field (average subpopulations size) exhibits hysteresis in respect to the noise correlation time, manifested in the appearance of colored-noise-induced discontinuous transitions. Especially, it is shown that the relative fluctuation of the subpopulation sizes exhibits accelerated increase prior to abrupt transitions of the metapopulation state. Moreover, in certain cases the autocorrelation function of the population sizes demonstrates anticorrelation at some values of the lag time.

Stabilization Effects of Dichotomous Noise on the Lifetime of the Superconducting State in a Long Josephson Junction

We investigate the superconducting lifetime of a long overdamped current-biased Josephson junction, in the presence of telegraph noise sources. The analysis is performed by randomly choosing the initial condition for the noise source. However, in order to investigate how the initial value of the dichotomous noise affects the phase dynamics, we extend our analysis using two different fixed initial values for the source of random fluctuations. In our study, the phase dynamics of the Josephson junction is analyzed as a function of the noise signal intensity, for different values of the parameters of the system and external driving currents. We find that the mean lifetime of the superconductive metastable state as a function of the noise intensity is characterized by nonmonotonic behavior, strongly related to the soliton dynamics during the switching towards the resistive state. The role of the correlation time of the noise source is also taken into account. Noise-enhanced stability is observed in the investigated system.

Stochastic Resonance in a Fractional Oscillator with Random Mass and Random Frequency

For a fractional linear oscillator subjected to two multiplicative dichotomous noises and a additive fractional Gaussian noise and driven by a periodic signal, we study the stochastic resonance (SR) in this paper. Using (fractional) Shapiro–Loginov formula and the Laplace transformation technique, we acquire the exact expression of the first-order moment of the system’s steady response. Meanwhile, we discuss the evolutions of the output amplitude with frequency of the periodic signal, noise parameters, fractional order, and friction coefficient. We find that SR in the wide sense existing in this system. Specially, the evolution of the output amplitude with frequency of the periodic signal presents one-peak oscillation and two-peak oscillation. Moreover, the friction coefficient can induce stochastic multi-resonance.

Moment stability for a predator–prey model with parametric dichotomous noises**Project supported by the National Natural Science Foundation of China (Grant No. 11272051).

In this paper, we investigate the solution moment stability for a Harrison-type predator–prey model with parametric dichotomous noises. Using the Shapiro–Loginov formula, the equations for the first-order and second-order moments are obtained and the corresponding stable conditions are given. It is found that the solution moment stability depends on the noise intensity and correlation time of noise. The first-order and second-order moments become unstable with the decrease of correlation time. That is, the dichotomous noise can improve the solution moment stability with respect to Gaussian white noise. Finally, some numerical results are presented to verify the theoretical analyses.

Statistics of a neuron model driven by asymmetric colored noise.

Irregular firing of neurons can be modeled as a stochastic process. Here we study the perfect integrate-and-fire neuron driven by dichotomous noise, a Markovian process that jumps between two states (i.e., possesses a non-Gaussian statistics) and exhibits nonvanishing temporal correlations (i.e., represents a colored noise). Specifically, we consider asymmetric dichotomous noise with two different transition rates. Using a first-passage-time formulation, we derive exact expressions for the probability density and the serial correlation coefficient of the interspike interval (time interval between two subsequent neural action potentials) and the power spectrum of the spike train. Furthermore, we extend the model by including additional Gaussian white noise, and we give approximations for the interspike interval (ISI) statistics in this case. Numerical simulations are used to validate the exact analytical results for pure dichotomous noise, and to test the approximations of the ISI statistics when Gaussian white noise is included. The results may help to understand how correlations and asymmetry of noise and signals in nerve cells shape neuronal firing statistics.

Stochastic resonance in periodic potential driven by dichotomous noise

Periodic potential system is widely used in a lot of areas such as biological ratchet model of motor, Josephson junction in the field of physics, engineering mechanics of the damping pendulum model, etc. Meanwhile, in the study of stochastic resonance, noise is crucial for dynamical system evolution. There are mostly colored Gaussian noises with nonzero correlation times in practical problems. Dichotomous noises belong to the color noises, and they have some simple statistical properties. In this paper, we study the motion of a Brownian particle in a periodic potential, driven by both a periodic signal and a dichotomous noise. The periodic potential system is different from the bistable system, so we use multiple indexes to explain the stochastic resonance. We calculate the average input energy of the system and the average output signal amplitude and phase difference by using stochastic energetics. Then we discuss the influences of the dichotomous noise intensity, noise correlation time and asymmetric coefficient of potential energy on the stochastic resonance. The results show that with the increase of the noise correlation time, a minimum value and a maximum value occur on the curve of the average input energy, meanwhile, the phenomenon of resonance appears in the system. With the increase of the noise intensity, the value of noise correlation time becomes greater when the phenomenon of stochastic resonance appears. Therefore, the region of stochastic resonance becomes bigger as the noise intensity or the asymmetry coefficient increases. Moreover, with the increase of the noise intensity, a mono peak is found for the signal-to-noise ratio (SNR) of the system and the stochastic resonance appears in this system. With the increase of the noise intensity, we compare the change of the SNR, the average input energy, and the average output signal amplitude. We find that the values of the amplitudes of the average output signal and SNR are basically the same, while the values of the amplitude of the average input energy of the system are a little different. This is because during the process of periodic signal doing work to the system, noise does work and passive dissipation energy of the system occures. In addition, when the curves of the amplitude of the average output signal and SNR reach their corresponding minimum values, the phase difference between the output signal and input signal is minimal.

Path Attractors and the Origins of Stochastic Bifrucation and Dephasing in Genetic Networks

Gene regulatory networks are driven stochastic systems with the noise having two distinct components due to the to birth and death of metabolite molecules and dichotomous nature of gene state switching. Presence of dichotomous gene noise alone has the capacity to significantly perturb the optimal transition paths and steady state probability distributions compared to the macroscopic models and their weak noise approximations. Most importantly dichotomous gene noise can also lead to multimodal distributions due to stochastic bifurcation of the underlying nonlinear dynamical system, which underlies the mechanism of formation of population heterogeneity. In this note we derive approximate path based expression of the time dependent probability of gene circuits which enables deeper exploration of the role of gene noise in formation of epigenetic states and dephasing-like phenomena.

Trichotomous Noise Induced Resonance Behavior for a Fractional Oscillator with Random Mass

We investigate the stochastic resonance (SR) phenomenon in a fractional oscillator with random mass under the external periodic force. The fluctuations of the mass are modeled as a trichotomous noise. Applying the Shapiro–Loginov formula and the Laplace transform technique, we obtain the exact expression of the first moment of the system. The non-monotonic behaviors of the spectral amplification (SPA) versus the driving frequency indicate that the bona fide SR appears. The necessary and sufficient conditions for the emergence of the generalized stochastic resonance phenomena on the noise flatness and on the noise intensity in the particular case of \(\Omega =\omega _0 ,v\rightarrow 0\) are established. Particularly, the hypersensitive response of the SPA to the noise intensity is found, which is previously reported and believed to be absent in the case of dichotomous noise.

Stability of an Oscillator with Random Mass

We consider an oscillator with a random mass for which the particles of the surrounding medium adhere to the oscillator for some random time after the collision, thereby changing its mass (Brownian motion with adhesion). For the case of dichotomous multiplicative noise, the first moment can diverge, indicating that the system is unable to reach a steady state.

Integrate-and-fire neurons driven by asymmetric dichotomous noise.

We consider a general integrate-and-fire (IF) neuron driven by asymmetric dichotomous noise. In contrast to the Gaussian white noise usually used in the so-called diffusion approximation, this noise is colored, i.e., it exhibits temporal correlations. We give an analytical expression for the stationary voltage distribution of a neuron receiving such noise and derive recursive relations for the moments of the first passage time density, which allow us to calculate the firing rate and the coefficient of variation of interspike intervals. We study how correlations in the input affect the rate and regularity of firing under variation of the model's parameters for leaky and quadratic IF neurons. Further, we consider the limit of small correlation times and find lowest order corrections to the first passage time moments to be proportional to the square root of the correlation time. We show analytically that to this lowest order, correlations always lead to a decrease in firing rate for a leaky IF neuron. All theoretical expressions are compared to simulations of leaky and quadratic IF neurons.

Response of a Superconducting Quantum Interference Device to Alternating Magnetic Field

We study the response of a superconducting quantum interference device (SQUID) to an alternating magnetic field. It is found that, for some suitably selected parameters' values, we can obtain stochastic resonance for the amplitude of the steady average voltage of the SQUID versus the dichotomous noise strength, and frequency (stochastic) resonance for the amplitude of the stationary average voltage of the SQUID as a function of the frequency of the alternating magnetic field. Our results can be useful for electrical power generation using the alternating magnetic field energy of a SQUID.

Transient and stationary characteristics of the Malthus-Verhulst-Bernoulli model with non-Gaussian fluctuating parameters

Two generalized Verhulst equations with non-Gaussian fluctuations of the reproduction rate and the volume of resources are under analytical investigation. For the first model, using the central limit theorem, we find the asymptotic behavior of the probability distribution of population density for an arbitrary non-Gaussian colored noise with nonzero power spectral density at zero frequency. Specifically, we confirm this result in the case of Markovian dichotomous noise and examine the evolution of mean population density. For fluctuating resources with one-sided stable distribution the transient dynamics of probability density function and statistical characteristics in the steady state are obtained. As shown, the scenario of the population's evolution depends on the parameter of nonlinearity in the original stochastic equation.

Disorder–order transitions in diluted and nondiluted direct transmission lines with asymmetric dichotomous noise

In this work we study the localization properties of diluted and nondiluted disordered direct transmission lines (TLs), when we distribute two values of inductances, LA and LB, according to an asymmetric dichotomous sequence. Using the scaling properties of the participation number D we study the localization properties in the thermodynamic limit. For certain τ and β parameter specific values, which characterize the dichotomous noise, we have found the following limit conditions: lim β→∞m(ω,τ,β) → 1.0 or lim τ→∞m(ω,τ,β) → 1.0, for the appearing of the disorder–order transitions. Here m(ω,τ,β) are the slopes of the linear relationships between ln (D) and ln (N). In addition, in each ωc resonance frequency generated by the dilution process we demonstrate the existence of extended intermediate states in the thermodynamic limit.