Stationary Probability Density Research Articles

Modeling correlated uncertainties in stochastic compartmental models

We consider compartmental models of communicable disease with uncertain contact rates. Stochastic fluctuations are often added to the contact rate to account for uncertainties. White noise, which is the typical choice for the fluctuations, leads to significant underestimation of the disease severity. Here, starting from reasonable assumptions on the social behavior of individuals, we model the contacts as a Markov process which takes into account the temporal correlations present in human social activities. Consequently, we show that the mean-reverting Ornstein–Uhlenbeck (OU) process is the correct model for the stochastic contact rate. We demonstrate the implication of our model on two examples: a Susceptibles–Infected–Susceptibles (SIS) model and a Susceptibles–Exposed–Infected–Removed (SEIR) model of the COVID-19 pandemic and compare the results to the available US data from the Johns Hopkins University database. In particular, we observe that both compartmental models with white noise uncertainties undergo transitions that lead to the systematic underestimation of the spread of the disease. In contrast, modeling the contact rate with the OU process significantly hinders such unrealistic noise-induced transitions. For the SIS model, we derive its stationary probability density analytically, for both white and correlated noise. This allows us to give a complete description of the model’s asymptotic behavior as a function of its bifurcation parameters, i.e., the basic reproduction number, noise intensity, and correlation time. For the SEIR model, where the probability density is not available in closed form, we study the transitions using Monte Carlo simulations. Our modeling approach can be used to quantify uncertain parameters in a broad range of biological systems.

Stochastic analysis and vibration suppression of a time-delayed system with nonlinear energy sink

Nonlinear energy sink (NES) is of great significance in attenuating structural vibration. Time-delay phenomenon in a mechanical system is inevitable. Different from existing works, the influences of time-delayed factors in a nonlinear oscillator attached with NES are considered in this paper under Gaussian white noise (GWN). Firstly, the time-delayed term is transformed into a linear combination of stiffness and damping terms, and an equivalent system without the time-delayed term is obtained. Stationary probability density function (SPDF) is deduced by leveraging stochastic averaging method. Secondly, good agreement is observed between analytical and numerical results, which verifies the effectiveness of the method. Furthermore, bifurcation conditions for the amplitude of SPDF of the primary system are obtained, indicating that delayed time, feedback gain and damping coefficients can cause stochastic P-bifurcation of the primary system. Finally, the vibration suppression performance of NES is analyzed via energy variation and the time–history diagram of displacement.

Topological detection of phenomenological bifurcations with unreliable kernel density estimates

Phenomenological (P-type) bifurcations are qualitative changes in stochastic dynamical systems whereby the stationary probability density function (PDF) changes its topology. The current state of the art for detecting these bifurcations requires reliable kernel density estimates computed from an ensemble of system realizations. However, in several real world signals such as Big Data, only a single system realization is available—making it impossible to estimate a reliable kernel density. This study presents an approach for detecting P-type bifurcations using unreliable density estimates. The approach creates an ensemble of objects from Topological Data Analysis (TDA) called persistence diagrams from the system’s sole realization and statistically analyzes the resulting set. We compare several methods for replicating the original persistence diagram including Gibbs point process modelling, Pairwise Interaction Point Modelling, and subsampling. We show that for the purpose of predicting a bifurcation, the simple method of subsampling exceeds the other two methods of point process modelling in performance.

Stochastic bifurcation and chaos study for nonlinear ship rolling motion with random excitation and delayed feedback controls

In this paper, we investigate the stochastic dynamics of a class of nonlinear ship rolling motion with multiplicative noise under both displacement and velocity delay feedback controls. The Itoˆ-stochastic differential equation for the amplitude and phase of the roll motion is derived using the stochastic center manifold method and stochastic average method. Subsequently, the stochastic stability and bifurcation behaviors of the system are analyzed. Furthermore, using the stationary probability density method, we derive the parameter conditions for the occurrence of stochastic D-bifurcation and stochastic P-bifurcation. We also analyze the properties and shape changes of the system’s probability density function under different parameters through numerical simulation. It has been determined that the system exhibits stochastic bifurcation behavior, specifically P-bifurcation and D-bifurcation. The validity of the method is verified by a numerical model. The theoretical chaos threshold of the system is determined using the random Melnikov method, and the impact of delayed feedback parameters on the chaotic motion of the system is analyzed by combining the bifurcation diagram, phase portrait, and time series.

Bifurcation Regulations Induced by Joint Noise in a Tri-Rhythmic Van Der Pol System

Tri-rhythmical nature has attracted extensive attention from scholars in describing the dynamical behaviors of self-sustained systems. In this paper, we consider a tri-rhythmic van der Pol system and give a bifurcation analysis of a stochastic tri-rhythmic self-sustained system under joint noise perturbation. Based on an approximate approach, we give the stationary probability density function of amplitudes, and we find that the noise and time-delay feedback, regulating the velocity time-delay feedback strength parameter, may not cause transitions among unimodal, bimodal and trimodal in the tri-rhythmic system. More stochastic bifurcations appear by regulating the time delay in this system. The system, surprisingly, undergoes five times of stochastic bifurcations when the time delay is monotonically increased. It is shown that the time delay is more sensitive to the tri-rhythmic system and a much stronger dependence on the system. From a biological point of view, the reaction rate of biological molecules can be enhanced or diminished by the change of the noise intensity or correlation time of Gaussian colored noise. More surprisingly, an increase in displacement feedback will delay the reaction rate; however, the effect of an increase in velocity feedback on the reaction rate depends on the time delay. A detailed research on the parameter space indicates that time delay and colored noise parameters can effectively control the multirhythmicity of the system. Finally, this paper verifies the effectiveness of the theoretical solution through the numerical results of Monte Carlo simulation of the original system. These results may help to further explore forking bifurcations in real-world applications. This research provides new insight into the understanding of nontrivial effects of joint noise on the tri-rhythmic system.

Stochastic dynamics of a nonlinear vibration energy harvester subjected to a combined parametric and external random excitation: The distinct cases of Itô and Stratonovich stochastic integration

The Stratonovich and Itô interpretations of a stochastic integral are mathematically consistent, distinct representations that yield contrasting response characteristics for stochastic dynamical systems subjected to multiplicative noise, such as parametric random excitation. The contrasting dynamics from the two interpretations currently remain unexplored in vibration energy harvester dynamics. We analytically investigate the dynamics of a generic piezoelectric, nonlinear vibration energy harvester simultaneously subjected to stochastic damping and external random excitation, focusing on the contrasting dynamics engendered by the two interpretations. Numerically solving the Stratonovich and Itô stochastic differential equations in the nonequilibrium regime of harvester dynamics, we find that the Stratonovich version yields significantly higher root mean square values of the harvester voltage output, for lower values of the mechanical damping coefficient. Furthermore, we find positive values for the leading Lyapunov exponent indicating instabilities in the harvester response unique to the Stratonovich interpretation. Additionally, comparing the results for the harvester electrical output between the cases of external excitation represented by the white noise and the Ornstein–Uhlenbeck noise processes, we find lower root mean square values of the voltage output in the latter case. Moreover, the results indicate that the averaged voltage output becomes progressively lower with increasing correlation time of the Ornstein–Uhlenbeck process. Studying the equilibrium dynamics by deriving Fokker–Planck equations for the response amplitude of the harvester using the harmonic balance method and stochastic averaging techniques, we solve for the stationary probability densities to find that the harvester is more likely to attain a well-defined equilibrium state under the Itô interpretation. In summary, the results contribute to the theoretical understanding of the fundamental contrast between the Stratonovich and Itô interpretations in vibration energy harvester dynamics and are expected to motivate advances in harvester design for various applications.

Reduced probability densities of long-lived metastable states as those of distributed thermal systems: Possible experimental implications for supercooled fluids

When liquids are cooled sufficiently rapidly below their melting temperature, they may bypass crystalization and, instead, enter a long-lived metastable supercooled state that has long been the focus of intense research. Although they exhibit strikingly different properties, both the (i) long-lived supercooled liquid state and (ii) truly equilibrated (i.e., conventional equilibrium fluid or crystalline) phases of the same material share an identical Hamiltonian. This suggests a mapping between dynamical and other observables in these two different arenas. We formalize these notions via a simple theorem applicable to general systems and illustrate that given a Hamiltonian defining the dynamics: (1) the reduced probability densities of all possible stationary states are linear combinations of reduced probability densities associated with thermal equilibria at different temperatures, chemical potentials, etc. (2) Excusing special cases, amongst all of these stationary states, a clustering of correlations is only consistent with conventional thermal equilibrium states (associated with a sharp distribution of the above state variables). (3) Other stationary states may be modified so as to have local correlations. These deformations concomitantly lead to metastable (yet possibly very long-lived) states. Since the lifetime of the supercooled state is exceptionally long relative to the natural microscopic time scales, their reduced probability densities may be close to those that we find for exact stationary states (i.e., a weighted average of equilibrium probability densities at different state variables). This form suggests several new predictions such as the existence of dynamical heterogeneity stronger than probed for thus far and a relation between the specific heat peak and viscosities. Our theorem may further place constraints on the putative “ideal glass” phase.

Critical transitions in spatial systems induced by Ornstein–Uhlenbeck noise: spatial mutual information as a precursor

Complex dynamical systems are subject to perturbations across space and time, which can induce a critical transition or tipping in the state of the system. External perturbations are often correlated in time and can interplay with the underlying nonlinearity of the spatial system, affecting the occurrence of critical transitions. Theoretical analysis of the spatial system perturbed by the Ornstein–Uhlenbeck (OU) correlated noise poses challenges beyond the white noise assumptions and is yet to be done. Here, we resort to the mean-field approximation of a spatially extended system perturbed with OU noise and obtain the stationary probability density function deriving the Fokker–Planck equation for the same. This allows us to determine the role of diffusion and noise on the resilience of the spatial system. While the theoretical analysis guides us on the landscape of tipping thresholds of the system, critical transitions customary to a variety of systems, require a priori prediction. Here, we propose a probabilistic information-based indicator—spatial mutual information—that can successfully forecast tippings, complementing the previously developed spatial indicators. Further, validating its reliability on empirical data, we show that spatial mutual information serves as a robust indicator capturing information characteristic to an imminent tipping reaching peaks in its vicinity.

Thin film Weyl semimetals with turning number of Fermi surface greater than unity

The development of Weyl semimetals (WSMs) as a new type of quantum material for a variety of novel phenomena is underway. In this context WSM with broken time reversal symmetry and inversion symmetry confined to a thin film which exhibits topologically distinct Fermi surface in the low energy regime is presented. The equi-energy contours possess a topological invariant called turning number ν. We extend the numerical investigation of the magnetotransport in the WSM to include systems with the turning number of Fermi surface greater than unity, ν=2, which is due to the self intersection of equi-energy contours. This results in exotic Weyl orbits which are a hybrid of bulk and surface states and are crucial for magnetotransport. This is in contrast to the conventional electron gas in which orbits are closed curves without self intersection. The contour of Weyl orbits in real space is connected to the shape of Fermi surface. In the presence of a perpendicular magnetic field on a slab geometry, the probability density of edge states are wide and narrow on each edge rather than symmetric on both edges (the hall mark of conventional quantum Hall effect). We explore the effect of disorder on these asymmetric edge states and find that the conductance remains quantized at 2e2/h up to a critical disorder strength. Our findings show that the transport characteristics of WSM are more robust to disorder in a particular direction for these systems with higher turning number. These systems also show unusual conductance by a slanted magnetic field as the slant can modify the Landau levels at the Fermi level.

Bifurcation analysis of a tristable system with fractional derivative under colored noise excitation

Stochastic bifurcation has received much attention recently and is of great significance in the research on the dynamics of nonlinear systems. In this paper, we study the bifurcation of a self-sustained tristable system containing fractional derivative under the excitation of two colored noises, where the self-sustained tristable system consists of stable limit cycles and a stable state. The multiple scale method and generalized harmonic function are used to convert the original system into an equal system without obvious time delay and fractional derivative. Then the stationary probability density function (SPDF) of the system is obtained by using the stochastic averaging method to discuss stochastic bifurcation. Based on singularity theory, the equation satisfying amplitude is obtained to derive the corresponding bifurcation diagram. From the bifurcation analysis of the system, it is revealed that fractional order, fractional coefficient, and the intensity and correlation time of colored noise can be utilized as bifurcation parameters, causing peculiar stochastic bifurcation phenomena and regulating the output of the system, as well as the relatively big colored noise intensity and relatively small correlation time facilitate the realization of large magnitude limit cycle. Monte Carlo numerical results verify the validity of the theoretical approach. In addition, appropriately changing the delayed feedback parameter helps to govern the bifurcation range and causes richer bifurcation phenomena. The results of this paper contribute to a better understanding of self-sustained tristable systems with fractional derivative and may provide reference value in resolving the problems of chattering in high-speed aircraft and applications to viscoelastic materials.

Analysis of the stationary probability density of a generalized and bistable van der Pol system excited by colored noise

The stochastic P-bifurcation behavior of bi-stability in a generalized van der Pol oscillator with the fractional damping under colored noise and thermal excitation is investigated. Firstly, using the principle of minimal mean square error and lin-earization method, the non-linear stiffness terms can be equivalent to a linear stiffness which is a function of the system amplitude, and the original system is simplified to an equivalent integer order van der Pol system. Secondly, the system amplitude stationary probability density function is obtained by the stochastic averaging, and then based on the singularity theory, the critical parametric con-ditions for the system amplitude stochastic P-bifurcation are found. Finally, the types of the stationary probability density function of the system amplitude are qualitatively analyzed in each area divided by the transition set curves. The con-sistency between the analytical results and the numerical results acquired from Monte-Carlo simulation also testifies the theoretical analysis in this paper and the method used in this paper can directly guide the design of the fractional order controller to adjust the response of the system.

Longtime evolution and stationary response of a stochastic tumor-immune system with resting T cells.

In this paper, we take the resting T cells into account and interpret the progression and regression of tumors by a predator-prey like tumor-immune system. First, we construct an appropriate Lyapunov function to prove the existence and uniqueness of the global positive solution to the system. Then, by utilizing the stochastic comparison theorem, we prove the moment boundedness of tumor cells and two types of T cells. Furthermore, we analyze the impact of stochastic perturbations on the extinction and persistence of tumor cells and obtain the stationary probability density of the tumor cells in the persistent state. The results indicate that when the noise intensity of tumor perturbation is low, tumor cells remain in a persistent state. As this intensity gradually increases, the population of tumors moves towards a lower level, and the stochastic bifurcation phenomena occurs. When it reaches a certain threshold, instead the number of tumor cells eventually enter into an extinct state, and further increasing of the noise intensity will accelerate this process.

Periodicity-assist double delay-controlled stochastic resonance for the fault detection of bearings

This paper analyses the stochastic resonance of a double delay-controlled nonlinear system based on the high-order-degradation potential function. It provides a theoretical derivation of the stationary probability density, mean first-passage time (MFPT), and signal-to-noise ratio (SNR), while the marginal probability density is examined via Monte-Carlo simulation. In the field of fault detection for faulty bearings, the stochastic resonance of the nonlinear system encounters a challenge where the exact fault period must be known beforehand. Unfortunately, it is not available for the real engineering application. Consequently, this paper proposes the periodicity-assisted stochastic resonance method, which uses the blind deconvolution technique. The proposed method allows for adaptive determination of the fault frequency from the vibration signal and successfully enhances the power of the signal component corresponding to the determined fault frequency. Compared with the state-of-the-art methods, the proposed method shows better performance, which is validated by the simulated and experimental data. Overall, the proposed method is advantageous for enhancing the weak fault characteristic which is interfered by the strong noise and random impulses.

Analysis of electronic properties of hexagonal and triangular Bismuth quantum dot with different edges in the presence of magnetic field

We investigate energy spectra of Bi honeycomb quantum dots (BiQDs) considering buckled hexagonal and triangular structures with zigzag and armchair edges. We apply the four-orbital tight-binding approach. By using of the probability density of states, we distinguish some edge states in the regime of the band gap of Bi monolayer. We see that the number of edge states of zigzag (armchair) triangular BiQDs is more than the number of edge states of zigzag (armchair) hexagonal BiQDs. Also, we investigated energy spectra of hexagonal and triangular BiQDs with zigzag and armchair edges in the presence of the perpendicular magnetic fields. In the low and strong magnetic fields, there are relativistic Landau levels in the electronic spectra of BiQDs (in the hexagonal and triangular boundary conditions). Moreover, we show that edge atoms have more important roles in the formation of the line connectors between i and |i+1| Landau levels of hexagonal BiQDs than ones that are in the triangular BiQDs. In the middle numbers of the magnetic fields, more states collapse into Landau levels in which bulk atoms have a significant role to constitute them in both hexagonal and triangular BiQDs. Furthermore as the same as Bi nanoribbon spectrum, there are some helical edge modes in the band gap regions of Bi monolayer in which edge atoms are responsible to form them. These helical edge modes have topological properties.

A stochastic predator–prey model with distributed delay and Ornstein–Uhlenbeck process: Characterization of stationary distribution, extinction, and probability density function

As the evolution of species relies on not only the current state but also the past information, it is more reasonable and realistic to take delay into an ecological model. This paper deals with a stochastic predator–prey model that considers the distribution delay and assume that the intrinsic growth rate and the death rate in the model are governed by Ornstein–Uhlenbeck process to simulate the random factors in the environment. Based on the existence and uniqueness of the global solution to the model and the boundedness of the order moments of the solution, several conditions are established to analyze the survival of the species. Specifically, a criteria for the existence of the stationary distribution to the stochastic system is established by constructing some suitable Lyapunov functions. And the analytical expression of the probability density function of the model around the quasi‐equilibrium is obtained. Furthermore, the extinction of species in the model is also explored. Finally, numerical simulations are carried out to illustrate the theoretical results obtained in this paper.

An underdamped and delayed tri-stable model-based stochastic resonance

Stochastic resonance (SR) is investigated in an underdamped tri-stable potential system driven by Gaussian colored noise and a periodic excitation, where both displacement and velocity time-delayed states feedback are considered. It is challenging to study SR in a second-order delayed multi-stable system analytically. In this paper, the improved energy envelope stochastic average method is developed to derive the analytical expressions of stationary probability density (SPD) and spectral amplification. The effects of noise intensity, damping coefficient, and time delay on SR are analyzed. The results show that the shapes of joint SPD can be adjusted to the desired structure by choosing the time delay and feedback gains. For fixed time delay, the SR peak is increased for negative displacement or velocity feedback gain. Meanwhile, the SR peak is decreased while the optimal noise intensity increases with increasing correlation time of noise. The Monte Carlo simulations (MCS) confirm the effectiveness of the theoretical results.

Impact of Non-Gaussian Noise and Time Delay on Stability and Stochastic Resonance for a FitzHugh-Nagumo Neural System Subjected to a Multiplicative Periodic Signal

In this paper, we focus on the investigations on the stochastic stability and the stochastic resonance (SR) phenomena for a FitzHugh-Nagumo system with time delay induced by a multiplicative non-Gaussian colored noise and an additive Gaussian colored noise. By use of the fast descent method, the unified colored noise approximation and the two-state theory for the SR, the stationary probability density function (SPDF) and the signal-to-noise ratio (SNR) caused by different noise terms and time delay are explored. The investigation results indicate that the two noise intensities, time delay and the departure parameter from the Gaussian noise can all reduce the probability density around the two stable states and destroy the stability of the neural system; while the two noise correlation times [Formula: see text] and [Formula: see text] can both improve the probability density around both stable states and reinforce the biological stability of the neural system. As regards the SNR, it is found that the two noise intensities and the departure coefficient can all weaken the SR effect, while time delay [Formula: see text] and the correlation time [Formula: see text] of the multiplicative noise will always magnify the SR phenomenon. It is worth to mention that the correlation time [Formula: see text] of the additive noise can stimulate the SR effect, but not alter the maximum of the SNR.

Interplay of spin–orbit and exchange interaction in a ferromagnet/heavy-metal hybrid system: Ni on W(110)

In a combined experimental and theoretical study, we investigate the interplay of spin–orbit interaction (SOI) and exchange interaction (XI) in the electronic structure of ultrathin Ni films on W(110). Using spin- and angle-resolved inverse photoemission, we observe that the size of the spin splitting of Ni-related exchange-split states differs for opposite magnetization directions. A quenched spin splitting for one of the magnetization directions reveals a contribution of SOI on an equal footing with XI. Using density-functional theory calculations, we explore the underlying mechanisms responsible for the experimentally observed coupling of SOI and XI. We find that a hybridization between adsorbate and substrate states, along with a high probability density of the respective states at the heavy W nuclei, cause the strong influence of SOI on the Ni-related exchange-split states.

Response and Bifurcation of Fractional Duffing Oscillator under Combined Recycling Noise and Time-Delayed Feedback Control

Response and bifurcation of fractional Duffing oscillator under recycling noise and time-delayed feedback control are investigated. Firstly, based on the principle of the minimum mean square error and small time-delayed approximation and linearize the cubic stiffness term, the fractional derivative is equivalent to a linear combination of damping and restoring forces, and the original system is simplified into an equivalent integer order system. Secondly, the Itô differential equations and one-dimensional Markov process are obtained according to stochastic averaging method, and the stochastic stability and stochastic bifurcation of the system are analyzed. Lastly, through joint probability density function diagram and the stationary probability density function diagram, the stochastic bifurcation behavior of system under the different time-delay, fractional order and noise intensity are discussed respectively, the validity of the theory and the occurrence of bifurcation phenomenon are verified.

An Improved Stochastic Averaging on the Piezoelectric Vibrational Harvester Model with Stiffness and Inertia Nonlinearities

In this paper, an improved stochastic averaging method is introduced to solve a monostable piezoelectric vibration energy harvester (VEH) model subject to basal Gaussian white and colored noises. First, the electric voltage is expressed as perturbations to the linear stiffness term and the damping term of the vibrational equation. Second, the original model is simplified into an Itô differential equation by the improved stochastic averaging method. Third, stationary solution of the Fokker–Plank–Kolmogorov function driven from the Itô equation is obtained, so that the stationary probability density functions (PDFs) of the displacement, velocity, and electrical voltage are obtained. With the help of the Monte Carlo, the effectiveness of these theoretical predictions is verified. Finally, the impacts of the nonlinear stiffness coefficient and nonlinear inertia coefficient on the responses are compared. It is found that the stiffness nonlinearity has stronger effect on the outputn electric voltage than the inertia nonlinearity.