Search

The problem of analyzing the bifurcation mechanisms of complex stochastic oscillations in population dynamics is considered. We study this problem on the basis of the modified Leslie–Gower prey–predator model with randomly forced Holling-II functional response. The paper focuses on the effects of noise in the Canard explosion zone. The phenomenon of noise-induced splitting of Canard cycles is discovered and studied in terms of stochastic P-bifurcations. In parametric analysis, we use the stochastic sensitivity technique with the apparatus of confidence domains to find the most noise-sensitive Canard. For the phenomenon of stochastic splitting, an underlying deterministic mechanism using critical curves near the cycle orbit and sub-/super-critical zones is revealed. [ABSTRACT FROM AUTHOR]

In this paper, we consider random iterations of polynomial maps z² + cn, where cn are complex-valued independent random variables following the uniform distribution on the closed disk with center c and radius r. The aim of this paper is twofold. First, we study the (dis)connectedness of random Julia sets. Here, we reveal the relationships between the bifurcation radius and connectedness of random Julia sets. Second, we investigate the bifurcation of our random iterations and give quantitative estimates of bifurcation parameters. In particular, we prove that for the central parameter c = -1, almost every random Julia set is totally disconnected with much smaller radial parameters r than expected. We also introduce several open questions worth discussing. [ABSTRACT FROM AUTHOR]

Copyright of Wuhan University Journal of Natural Sciences is the property of Wuhan University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The stability and Hopf bifurcation of a class of flywheel governor systems subject to stochastic excitation under time-delay feedback are studied. The model is simplified using center manifold reduction to obtain a paradigm for a two-dimensional central subsystem. The system converges asymptotically to an Itô stochastic differential equation based on polar transformations and stochastic averaging method. The stochastic stability of the system is analyzed by applying the maximum Lyapunov exponent and the singular boundary theory, and its probability density function is calculated. In addition, it is found through Monte Carlo numerical simulations that the effects of noise intensity and time-delay on the stochastic P-bifurcation of the system are opposite, and proper noise intensity makes the flywheel governor system converge to the equilibrium position, while a large time-delay leads to stochastic bifurcation and makes the flywheel governor system unstable. Finally, the two-parameter bifurcation diagram is used to further analyze the impact of time-delay and noise intensity on the bifurcation of the system under variable load torque. It is found that the influence of noise intensity and time-delay on the periodic oscillation state is the same, and the stability of the flywheel governor system will decrease with both of their increases. This also further demonstrates the duality of the impact of noise intensity on the same system under different conditions. The above research provides better theoretical references for the design and safe operation of speed control systems. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of nonlinear stochastic Duffing oscillators with multidelay feedback. We propose an effective reduction approach with the help of center manifold theory and stochastic averaging method. Taking the initial time-delay τ as the parameter, we reduce the original system to a one-dimensional averaged Itô equation. Our analysis reveals that the original system exhibits stochastic bifurcations, including stochastic D and P bifurcations. Once we have a clear understanding of the bifurcation structure, we can use this knowledge to choose appropriate system parameters and place the system in the desired state. For instance, by adjusting the initial time-delay τ of the control system, we can stabilize the system and achieve the desired outcome. Numerical simulations also verify the theoretical results. With appropriate parameter choices, multiple time delays can destabilize the equilibrium and promote chaotic behaviors, and can also lead to more stable dynamical behavior. Remarkably, we discovered that increasing the interval of time delays and feedback numbers can enhance system stability. It may potentially serve as a novel mechanism for stabilizing stochastic systems. The study provides a solid theoretical foundation for exploring stochastic systems subject to complex time-delay feedback control, and offers a valuable framework for related fields. [ABSTRACT FROM AUTHOR]

Phantom attractors in nonlinear systems under additive stochastic excitation have been recently discovered. This paper uncovers the existence of phantom attractors in a single-degree-of-freedom smooth nonlinear equation, which characterizes the vibration of an inextensible beam subjected to lateral stochastic excitation. It also elucidates that the stochastic averaging method, in this context, may lead to qualitatively erroneous probability density functions, identified as one of the reasons why these attractors were previously overlooked. The study then proceeds to analyze the formation process of the phantom attractor and the critical noise intensity associated with it. Subsequently, the key nonlinear term related to the emergence of phantom attractors is identified by observing whether the system still exhibits phantom attractors after the corresponding nonlinear terms are removed. It is revealed that in this system, the presence of phantom attractors is closely linked to the inertia nonlinearity of the hardening type. The system investigated in this paper is simpler compared to previously identified systems capable of generating phantom attractors. This simplicity aids in facilitating research focused on unraveling the general principles behind the formation of phantom attractors. [ABSTRACT FROM AUTHOR]

Stochastic oscillations in individual cells are usually characterized by a nonmonotonic power spectrum with an oscillatory autocorrelation function. Here we develop an analytical approach to stochastic oscillations in a minimal hybrid model of stochastic gene expression including promoter state switching, protein synthesis and degradation, as well as a genetic feedback loop. The oscillations observed in our model are noise-induced since the deterministic theory predicts stable fixed points. The autocorrelated function, power spectrum, and steady-state distribution of protein concentration fluctuations are computed in closed form. Using the exactly solvable model, we illustrate sustained oscillations as a circular motion along a stochastic hysteresis loop induced by gene state switching. A triphasic stochastic bifurcation upon the increasing strength of negative feedback is observed, which reveals how stochastic bursts evolve into stochastic oscillations. In our model, oscillations tend to occur when the protein is relatively stable and when gene switching is relatively slow. Translational bursting is found to enhance the robustness and broaden the region of stochastic oscillations. These results provide deeper insights into R. Thomas's two conjectures for single-cell gene expression kinetics. [ABSTRACT FROM AUTHOR]

The purpose of this manuscript is to analyze the bifurcation of the tri-rhythmic model with three stable limit cycles under the excitation of Gaussian colored noise, which aims to reveal the extremely complex nonlinear dynamics in biology. In the deterministic bifurcation of the tri-rhythmic model with time-delay, an interesting tri-rhythmic collapse and recovery phenomenon is found. By tuning the time-delay feedback, the model also exhibits a switch between tri-rhythmic and bi-rhythmic behavior. Furthermore, the tri-rhythmic model under colored noise excitation is discussed theoretically based on the stationary probability density function obtained by the stochastic averaging method. The variation in the number of peak values of the stationary probability density function is observed, thereupon enabling the stochastic bifurcation of the tri-rhythmic oscillator. Therefore, by adjusting the time-delay feedback and colored noise, many abundant bifurcation phenomena can be acquired. The theoretical analysis is consistent with the results of the dynamic numerical simulation. Our study may provide a new perspective for further exploring the bifurcation phenomenon of the tri-rhythmic model, which has important practical significance. [ABSTRACT FROM AUTHOR]

Abstract Considering the dynamic behavior of doubly‐fed induction generators (DFIGs) under the influence of random factors, this paper not only analyzes the phenomenon of stochastic instability and bifurcation of a DFIG dynamic variable in its random space when they are affected by environmental noise, but also proposes a method based on the Tkagi–Sugneo (T–S) fuzzy control strategy to control its stochastic bifurcation. First, a four‐dimensional stochastic dynamic DFIG model is established by using multiplicative white noise to simulate the influence of environmental noise on electrical variables, and stochastic central manifold theory is used to reduce the dimensionality of a planar model in the bifurcation neighborhood. Then, the stochastic stability of the model is investigated based on singular boundary theory, while the steady‐state probability density of the stochastic DFIG is determined using the Fokker–Plank–Kolmogorov equation to obtain the location and probability density of its stochastic P‐bifurcation. Finally, the influence of stochastic bifurcation behavior is eliminated by H∞‐fuzzy output feedback control. The numerical simulation results indicate that the location and probability of stochastic bifurcation in a DFIG will vary with the change in noise intensity, and the bifurcation parameter values and stochastic stability domain are obtained. The harm caused by random factors can be solved based on H∞‐fuzzy output feedback, which provides a theoretical basis for the stable operation of the DFIG system.

Considering the dynamic behavior of doubly‐fed induction generators (DFIGs) under the influence of random factors, this paper not only analyzes the phenomenon of stochastic instability and bifurcation of a DFIG dynamic variable in its random space when they are affected by environmental noise, but also proposes a method based on the Tkagi–Sugneo (T–S) fuzzy control strategy to control its stochastic bifurcation. First, a four‐dimensional stochastic dynamic DFIG model is established by using multiplicative white noise to simulate the influence of environmental noise on electrical variables, and stochastic central manifold theory is used to reduce the dimensionality of a planar model in the bifurcation neighborhood. Then, the stochastic stability of the model is investigated based on singular boundary theory, while the steady‐state probability density of the stochastic DFIG is determined using the Fokker–Plank–Kolmogorov equation to obtain the location and probability density of its stochastic P‐bifurcation. Finally, the influence of stochastic bifurcation behavior is eliminated by H∞‐fuzzy output feedback control. The numerical simulation results indicate that the location and probability of stochastic bifurcation in a DFIG will vary with the change in noise intensity, and the bifurcation parameter values and stochastic stability domain are obtained. The harm caused by random factors can be solved based on H∞‐fuzzy output feedback, which provides a theoretical basis for the stable operation of the DFIG system. [ABSTRACT FROM AUTHOR]

In this paper, a thermokinetic model forced by colored noise is studied. We analyze the mechanisms of stochastic excitement of equilibrium modes under variation of correlation time and noise intensity. It is shown that the phenomenon of colored-noise-induced excitement is accompanied by stochastic P-bifurcations. The region of the correlation parameter in which resonance occurs is localized. To study the phenomenon of colored-noise-induced excitement, we develop the probabilistic analysis based on the confidence domains method. [ABSTRACT FROM AUTHOR]

This article investigates the stochastic bifurcation of a generalized Duffing oscillator with fractional derivative term under multiplicative colored noise excitation. Firstly, a linear combination of restoring force and damping force is used to equivalently replace the fractional order derivative term in the system. Secondly, the cubic term in the system is linearized, the system is transformed into an integer order system by using the principle of minimum mean square error, the steadystate probability density function of the system is obtained by the stochastic averaging method. Finally, the maximum Lyapunov exponent of the invariant measure of the system is obtained by the stochastic averaging method of quasi- non-integrable Hamilton system, and the stochastic D-bifurcation and P-bifurcation of the system are analyzed. It is found that the changes of fractional derivative order, autocorrelation time of noise and noise intensity can induce stochastic P-bifurcation of the system. [ABSTRACT FROM AUTHOR]

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 oc‐ currence of bifurcation phenomenon are verified. [ABSTRACT FROM AUTHOR]

This article concerns the dynamical transitions of the stochastic Swift-Hohenberg equation with multiplicative noise on a two-dimensional domain (-L, L) × (-L, L). With α and L regarded as parameters, we show that the approximate reduced system corresponding to the invariant manifold undergoes a stochastic pitchfork bifurcation near the critical points, and the impact of noise on stochastic bifurcation of the Swift-Hohenberg equation. We find the approximation representation of the manifold and the corresponding reduced systems for stochastic Swift-Hohenberg equation when L2 and √ 2L1 are close together. [ABSTRACT FROM AUTHOR]

• Considering the randomness of the equivalent conicity of the wheelset system. • The analysis of wheelset system under non-smooth condition is more appropriate. • Considering the time delay displacement feedback control in the primary suspension. • Stochastic D(P)-bifurcation occur in the probabilistic sense in the wheelset system. • The hunting stability of the wheelset system during operation were analyzed. Considering not only the stochastic track irregularity and the possible effect of stochastic parameter excitation, but also the time delay of spring, that is, its response is in place, but the force generated is not in place, a stochastic delay wheelset system is established. The infinite dimensional system is reduced to the finite dimensional stochastic differential equation by using the center manifold, and further reduced to a one-dimensional diffusion process by using the stochastic averaging method. The stability of the wheelset system is obtained by analyzing the singular boundary theory and calculating the maximum Lyapunov exponent. The conditions and types of stochastic bifurcation in wheelset system are obtained by combining probability density function. The numerical simulation verifies the correctness of the theoretical analysis and shows that the time delay affects the critical hunting instability speed of wheelset. The stochastic term affects the lateral displacement of the stochastic delay wheelset system. [ABSTRACT FROM AUTHOR]

The Gaussian white noise was introduced into the permanent magnet synchronous motor (PMSM) model, to obtain the system Itô stochastic differential equation through the polar transformation and with the stochastic average method. Hence, the probability density function of the system was calculated, and the mechanism of the P-bifurcation of the system was revealed through numerical simulation. In addition, the complex dynamics of the system in the 2-parameter space was discussed. The simulation results show that, there are lots of "fish-shaped" periodic regions in the parameter space. The regions become unstable under effects of system noises. It is worth noting that, under noises of a certain intensity, the system dynamics will switch from periodic motion to convergence, indicating the dual nature of the effects of noises on the PMSM system dynamics. [ABSTRACT FROM AUTHOR]

In this paper, a wind turbine generator drive system with stochastic excitation under both displacement and velocity delayed feedback is considered. Firstly, the center manifold method is used to approximate the delay term of the system, so that the Itô-stochastic differential equation can be obtained by random average method. Through the maximal Lyapunov exponential method, the local stochastic stability and random D-bifurcation conditions of the system are obtained. Secondly, it is verified that the increase of noise intensity and delay value induces the occurrence of random P-bifurcation of the system through Monte Carlo numerical simulations. In addition, the theoretical chaos threshold of the system is derived by the random Melnikov method. The results show that the chaos threshold decreases as the noise intensity increases, and the increase in time delay leads to a delay in the chaotic behavior of the system. Finally, the correctness and effectiveness of the chaos-theoretic analysis are verified based on the one-parameter bifurcation diagrams and the two-parameter bifurcation diagrams. [ABSTRACT FROM AUTHOR]

Extensive engineering applications have confirmed the necessity of introducing stochastic disturbance into the permanent magnet synchronous motor (PMSM) system. Therefore the stochastic stability and bifurcation of the system under Gaussian white noise are investigated in this study. The stochastic average method transforms the system equation into Itô stochastic differential equation. Further, the stability of the model under white noise excitation is analysed based on the theoretical calculation process. Notably, the mechanism of P-bifurcation of the system is revealed by simulating the evolution process of probability density function with the change in noise intensity. Moreover, the complex dynamics of the system in two-parameter space are explored using multiple numerical tools, in which extensive fish-shaped periodic regions appear. It is particularly interesting that the noise inevitably erodes the boundaries of these fish-shaped periodic regions. Besides, it is noteworthy that a new phenomenon is found from the numerical simulation results that noise intensity can induce convergence behaviour in the periodic oscillation region, which also shows the two-sidedness of noise effect on the system. [ABSTRACT FROM AUTHOR]

The fractional stochastic vibration system is quite different from the traditional one, and its application potential is enormous if the noise can be deployed correctly and the connection between the fractional order and the noise property is unlocked. This article uses a fractional modification of the well-known van der Pol oscillator with multiplicative and additive recycling noises as an example to study its stationary response and its stochastic bifurcation. First, based on the principle of the minimum mean square error, 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. Second, the Itô differential equations and One-dimensional Markov process are obtained according to the stochastic averaging method, using Oseledec multiplicative ergodic theorem and maximal Lyapunov exponent to judge local stability, and judging global stability is done by using the singularity theory. Lastly, the stochastic D-bifurcation behavior of the model is analyzed by using the Lyapunov exponent of the dynamical system invariant measure, and the stationary probability density function of the system is solved according to the FPK equation. The results show that the fractional order and noise property can greatly affect the system’s dynamical properties. This paper offers a profound, original, and challenging window for investigating fractional stochastic vibration systems.

Since the stochastic bifurcation theory is still in its infancy, we try to analyze some stochastic bifurcation phenomenon from a simple mathematical model. Thus, this paper mainly focuses on studying the stochastic bifurcation of a stochastic logistic model with distributed delay in the strong kernel case, which is affected by noise. Therefore, we use the intrinsic growth rate as a bifurcation parameter. First, we study the stochastic D-bifurcation and stochastic P-bifurcation for stochastic logistic model. Furthermore, by deriving the corresponding Fokker–Planck equation, we obtain the expression of the joint density function of the stochastic logistic system near the positive equilibrium point. Finally, some conclusions are given. [ABSTRACT FROM AUTHOR]

The dynamics of a new modified Van der Pol (VDP) self-sustained oscillator, driven by fractional time-delay feedback under correlated noise, is addressed in this paper. The studied system presents a tristability mode with the coexistence of three stable limit cycles in the deterministic case. Under the generalized harmonic balance technique, the fractional derivative simultaneously includes an equivalent quasi-linear dissipative force and quasi-linear restoring force, which reduces the whole problem to an equivalent VDP equation without a fractional derivative. The stochastic averaging method investigates analytical solutions for the equivalent stochastic equation. The critical parametric conditions for stochastic P -bifurcation of amplitude are obtained via the singularity theory for the system switch among the three steady states. The analytical solutions are confronted with direct numerical simulations, in a process where the dynamical features of the system are characterized using the stationary probability density function (PDF) of amplitude and joint PDF of displacement and velocity. A satisfactory agreement is obtained between both approaches, therefore confirming the accuracy of the theoretical predictions. Changing the fractional order, the fractional coefficient, the time delay parameter, and the correlation time also appears to induce the occurrence of the stochastic P -bifurcation. [ABSTRACT FROM AUTHOR]

Energy harvesting has become a popular topic in recent years. A number of studies have been conducted in the field of vibration energy harvesting system (VEHS). However, few studies have concentrated on viscoelastic energy harvesters driven by bounded noise excitation. In this paper, the stochastic response of a viscoelastic energy harvester subjected to bounded noise is discussed. Approximate solutions of the system were derived by utilizing the method of multiple scales, and the expressions of the mean square voltage (MSV) and mean output power (MOP) were obtained. The relation between the detuning frequency and first-order steady moment was first revealed. The effectiveness of the approach was verified by a good agreement between theoretical results and numerical results. Furthermore, the variations in the detuning frequency can result in the stochastic jump phenomenon, and stochastic bifurcation is induced with the changes in the viscoelastic parameter and detuning frequency. Finally, the impacts of system parameters on the MSV and the MOP were also analyzed. [ABSTRACT FROM AUTHOR]

We study the dynamical transitions of the stochastic Swift-Hohenberg equation with multiplicative noise on a one-dimensional domain (0 , L). With α and the length of the domain L regarded as parameters, we show that the approximate reduced system corresponding to the invariant manifold undergoes a stochastic pitchfork bifurcation near the critical points, and analyze the case when the bifurcation points nearly coincide. We show the impact of noise on stochastic bifurcation of the Swift-Hohenberg equation. [ABSTRACT FROM AUTHOR]

In this paper, an improved mathematical model is proposed to analyze a special internal relationship between rainfall intensity and gas pollutants in urban atmospheric system, where the nonlinear relationship between precipitation and the concentration of pollutants in the air is considered, as well as the influence of random disturbances from the outside world. After the approximate solution of the system is obtained, the stability and bifurcation characteristics of the system with and without random disturbance are both analyzed. Theoretical analysis and numerical simulation results show that there is a close covariant relationship between rainfall and gaseous pollutants; the system has abundant bifurcation characteristics, which, in the physical sense, represents sudden changes of the gaseous pollutants in the urban atmospheric system; the urban atmospheric system can always be kept at a lower pollution level by applying slight disturbances to offset external disturbances. [ABSTRACT FROM AUTHOR]

In the present work, we establish the approximation via modulation or amplitude equations of the nonlinear stochastic partial differential equation (SPDE) driven by cylindrical α-stable Lévy processes. We study SPDEs with a cubic nonlinearity, where the deterministic equation is close to a change of stability of the trivial solution. The natural separation of time scales close to this bifurcation allows us to obtain an amplitude equation describing the essential dynamics of the bifurcating pattern, thus reducing the original infinite dimensional dynamics to simpler finite dimensional effective dynamics. In the presence of a multiplicative stable Lévy noise that preserves the constant trivial solution we study the impact of noise on the approximation. In contrast to Gaussian noise, where due to averaging effects nondominant patterns are uniformly small in time, large jumps in the Lévy noise might lead to large error terms, and thus new estimates are needed to take this into account. [ABSTRACT FROM AUTHOR]

An eco-epidemiological model of dynamical interaction of susceptible prey, infected prey, and predator is studied in the presence of the Allee effect and random disturbances. A bifurcation analysis of the deterministic variant of the model versus Allee parameter is carried out. Various regimes of tri-rhythmicity with coexistence of three cycles, or two cycles and one chaotic attractor are revealed. The complication of population dynamics due to stochastic transitions between coexisting oscillatory attractors and order-chaos transformations is studied both numerically and analytically, by the method of confidence ellipsoids and tori. The phenomenon of noise-induced extinction of predator with increasing amplitudes of oscillations for susceptible or infected prey is discussed. [ABSTRACT FROM AUTHOR]

The phenomenon of stochastic bifurcation driven by the correlated non-Gaussian colored noise and the Gaussian white noise is investigated by the qualitative changes of steady states with the most probable phase portraits. To arrive at the Markovian approximation of the original non-Markovian stochastic process and derive the general approximate Fokker-Planck equation (FPE), we deal with the non-Gaussian colored noise and then adopt the uni¯ed colored noise approximation (UCNA). Subsequently, the theoretical equation concerning the most probable steady states is obtained by the maximum of the stationary probability density function (SPDF). The parameter of the uncorrelated additive noise intensity does enter the governing equation as a non-Markovian effect, which is in contrast to that of the uncorrelated Gaussian white noise case, where the parameter is absent from the governing equation, i.e., the most probable steady states are mainly controlled by the uncorrelated multiplicative noise. Additionally, in comparison with the deterministic counterpart, some peculiar bifurcation behaviors with regard to the most probable steady states induced by the correlation time of non-Gaussian colored noise, the noise intensity, and the non-Gaussian noise deviation parameter are discussed. Moreover, the symmetry of the stochastic bifurcation diagrams is destroyed when the correlation between noises is concerned. Furthermore, the feasibility and accuracy of the analytical predictions are verified compared with those of the Monte Carlo (MC) simulations of the original system. [ABSTRACT FROM AUTHOR]

Vibration energy harvester, which can convert mechanical energy to electrical energy so as to achieve self-powered micro-electromechanical systems (MEMS), has received extensive attention. In order to improve the efficiency of vibration energy harvesters, many approaches, including the use of advanced materials and stochastic loading, have been adopted. As the viscoelastic property of advanced materials can be well described by fractional calculus, it is necessary to further discuss the dynamical behavior of the fractional-order vibration energy harvester. In this paper, the stochastic P-bifurcation of a fractional-order vibration energy harvester subjected to colored noise is investigated. Variable transformation is utilized to obtain the approximate equivalent system. Probability density function for the amplitude of the system response is derived via the stochastic averaging method. Numerical results are presented to verify the proposed method. Critical conditions for stochastic P-bifurcation are provided according to the change of the peak number for the probability density function. Then bifurcation diagrams in the parameter planes are analyzed. The influences of parameters in the system on the mean harvested power are discussed. It is found that the mean harvested power increases with the enhancement of the noise intensity, while it decreases with the increase of the fractional order and the correlation time. [ABSTRACT FROM AUTHOR]

Energy harvesting has become a popular topic in recent years. A number of studies have been conducted in the field of vibration energy harvesting system (VEHS). However, few studies have concentrated on viscoelastic energy harvesters driven by bounded noise excitation. In this paper, the stochastic response of a viscoelastic energy harvester subjected to bounded noise is discussed. Approximate solutions of the system were derived by utilizing the method of multiple scales, and the expressions of the mean square voltage (MSV) and mean output power (MOP) were obtained. The relation between the detuning frequency and first-order steady moment was first revealed. The effectiveness of the approach was verified by a good agreement between theoretical results and numerical results. Furthermore, the variations in the detuning frequency can result in the stochastic jump phenomenon, and stochastic bifurcation is induced with the changes in the viscoelastic parameter and detuning frequency. Finally, the impacts of system parameters on the MSV and the MOP were also analyzed.

A stochastic vibro-impact system has triggered a consistent body of research work aimed at understanding its complex dynamics involving noise and nonsmoothness. Among these works, most focus is on integer-order systems with Gaussian white noise. There is no report yet on response analysis for fractional-order vibro-impact systems subject to colored noise, which is presented in this paper. The biggest challenge for analyzing such systems is how to deal with the fractional derivative of absolute value functions after applying nonsmooth transformation. This problem is solved by introducing the Fourier transformation and deriving the approximate probabilistic solution of the fractional-order vibro-impact oscillator subject to colored noise. The reliability of the developed technique is assessed by numerical solutions. Based on the theoretical result, we also present the critical conditions of stochastic bifurcation induced by system parameters and show bifurcation diagrams in two-parameter planes. In addition, we provide a stochastic bifurcation with respect to joint probability density functions. We find that fractional order, coefficient of restitution factor and correlation time of colored noise excitation can induce stochastic bifurcations. [ABSTRACT FROM AUTHOR]

Impact together with friction can be widely found in the mechanical engineering. Although some scholars have investigated the stochastic systems with impact and friction, they only involve the integer-order systems and do not consider the fractional-order cases. In fact, for the system with viscoelastic material, the damping term depends not only on the current time and position but also on the previous states. The memory property of viscoelastic material is characterized by a power-law kernel function which is associated with the fractional derivative. Based on this viewpoint, in this article, we focus on the friction-damped system with fractional derivative damping under Gaussian white noise excitation. We propose an approximate approach to investigate the stochastic response and bifurcation of a fractional-order friction-damped system with the help of variable transformations and stochastic averaging method. One example is employed to verify the effectiveness of the proposed approach. We also explore the stochastic bifurcation phenomenon induced by the fractional order, fractional coefficient and other system parameters through the critical conditions. At last, the difference of bifurcation regions for the fractional-order model and the integer-order model are presented. [ABSTRACT FROM AUTHOR]

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 I t o ˆ -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. • 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 Ito-stochastic differential equation for the amplitude and phase of the roll motion is derived using the stochastic center manifold method and stochastic average method. • Using the stationary probability density method, we derive the parameter conditions for the occurrence of stochastic D-bifurcation and stochastic P-bifurcation. • The theoretical chaos threshold is determined using the random Melnikov method. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we mainly study the stochastic stability and stochastic bifurcation of Brusselator system with multiplicative white noise. Firstly, by a polar coordinate transformation and a stochastic averaging method, the original system is transformed into an Itô averaging diffusion system. Secondly, we apply the largest Lyapunov exponent and the singular boundary theory to analyze the stochastic local and global stability. Thirdly, by means of the properties of invariant measures, the stochastic dynamical bifurcations of stochastic averaging Itô diffusion equation associated with the original system is considered. And we investigate the phenomenological bifurcation by analyzing the associated Fokker–Planck equation. We will show that, from the view point of random dynamical systems, the noise “destroys” the deterministic stability. Finally, an example is given to illustrate the effectiveness of our analyzing procedure.

The variability of stochastic dynamics for a three-dimensional dynamic model in a parametric zone with 2-tori is investigated. It is shown how weak Gaussian noise transforms deterministic quasiperiodic oscillations into noisy bursting. The phenomenon of stochastic generation of a phantom attractor and its shift with noise amplification is revealed. This phenomenon, accompanied by order-chaos transitions, is studied in terms of stochastic P - and D -bifurcations. [ABSTRACT FROM AUTHOR]

Van der Pol–Duffing oscillator, which can be used a model for many dynamical system, has been widely concerned. However, most of the systems by scholars are either stable steady states or limit cycles. Here, the self-sustained oscillator with the coexistence of steady state and limit cycles, which is famous for describing the flutter of airfoils with large span ratio in low-speed wind tunnels, is treated in this paper. Using the energy balance method, the deterministic bifurcation of the tristable system with time-delay feedback is investigated. The presence of time-delay feedback expands the bifurcation range of the parameters, making the bifurcation phenomenon more abundant. In addition, according to the stationary probability density function obtained by the stochastic averaging method, stochastic bifurcation of the system with time-delay feedback and noise is explored theoretically. The numerical results confirm the correctness of the theoretical analysis. Transition between the unimodal structure, the bimodal structure and the trimodal structure is found. Many rich bifurcations are available by adjusting the time-delay and noise intensity, which may be conductive to achieve the desired phenomenon in the real-world application. [ABSTRACT FROM AUTHOR]

We investigate the bifurcation phenomena for stochastic systems with multiplicative Gaussian noise, by examining qualitative changes in mean phase portraits. Starting from the Fokker–Planck equation for the probability density function of solution processes, we compute the mean orbits and mean equilibrium states. A change in the number or stability type, when a parameter varies, indicates a stochastic bifurcation. Specifically, we study stochastic bifurcation for three prototypical dynamical systems (i.e. saddle-node, transcritical, and pitchfork systems) under multiplicative Gaussian noise, and have found some interesting phenomena in contrast to the corresponding deterministic counterparts. [ABSTRACT FROM AUTHOR]

The spectrum of the generator (Kolmogorov operator) of a diffusion process, referred to as the Ruelle-Pollicott (RP) spectrum, provides a detailed characterization of correlation functions and power spectra of stochastic systems via decomposition formulas in terms of RP resonances; see Part I of this contribution (Chekroun et al. in Theory J Stat. 10.1007/s10955-020-02535-x, 2020). Stochastic analysis techniques relying on the theory of Markov semigroups for the study of the RP spectrum and a rigorous reduction method is presented in Part I Chekroun et al. (2020). This framework is here applied to study a stochastic Hopf bifurcation in view of characterizing the statistical properties of nonlinear oscillators perturbed by noise, depending on their stability. In light of the Hörmander theorem, it is first shown that the geometry of the unperturbed limit cycle, in particular its isochrons, i.e., the leaves of the stable manifold of the limit cycle generalizing the notion of phase, is essential to understand the effect of the noise and the phenomenon of phase diffusion. In addition, it is shown that the RP spectrum has a spectral gap, even at the bifurcation point, and that correlations decay exponentially fast. Explicit small-noise expansions of the RP eigenvalues and eigenfunctions are then obtained, away from the bifurcation point, based on the knowledge of the linearized deterministic dynamics and the characteristics of the noise. These formulas allow one to understand how the interaction of the noise with the deterministic dynamics affect the decay of correlations. Numerical results complement the study of the RP spectrum at the bifurcation point, revealing useful scaling laws. The analysis of the Markov semigroup for stochastic bifurcations is thus promising in providing a complementary approach to the more geometric random dynamical system (RDS) approach. This approach is not limited to low-dimensional systems and the reduction method presented in Chekroun et al. (2020) is applied to a stochastic model relevant to climate dynamics in the third part of this contribution (Tantet et al. in J Stat Phys. 10.1007/s10955-019-02444-8, 2019). [ABSTRACT FROM AUTHOR]

The response of a low-frequency mode of climate variability, El Niño–Southern Oscillation, to stochastic forcing is studied in a high-dimensional model of intermediate complexity, the fully-coupled Cane–Zebiak model (Zebiak and Cane 1987), from the spectral analysis of Markov operators governing the decay of correlations and resonances in the power spectrum. Noise-induced oscillations excited before a supercritical Hopf bifurcation are examined by means of complex resonances, the reduced Ruelle–Pollicott (RP) resonances, via a numerical application of the reduction approach of the first part of this contribution (Chekroun et al. 2019) to model simulations. The oscillations manifest themselves as peaks in the power spectrum which are associated with RP resonances organized along parabolas, as the bifurcation is neared. These resonances and the associated eigenvectors are furthermore well described by the small-noise expansion formulas obtained by Gaspard (2002) and made explicit in the second part of this contribution (Tantet et al. 2019). Beyond the bifurcation, the spectral gap between the imaginary axis and the real part of the leading resonances quantifies the diffusion of phase of the noise-induced oscillations and can be computed from the linearization of the model and from the diffusion matrix of the noise. In this model, the phase diffusion coefficient thus gives a measure of the predictability of oscillatory events representing ENSO. ENSO events being known to be locked to the seasonal cycle, these results should be extended to the non-autonomous case. More generally, the reduction approach theorized in Chekroun et al. (2019), complemented by our understanding of the spectral properties of reference systems such as the stochastic Hopf bifurcation, provides a promising methodology for the analysis of low-frequency variability in high-dimensional stochastic systems. [ABSTRACT FROM AUTHOR]

Vibro-impact system (VI-S) have demonstrated their potential to contribute to engineering practice throughout the last few years. Nonetheless, their scope of application are still very limited. One of a primary reason is that dynamic behaviors of VI-S under the combination of deterministic period excitations (deterministic period excitation) and random noises (random noises) are not entirely understood. Toward this, the radial basis function neural network (rbfnn) is used to analyze dynamical behaviors evolution of VI-S under deterministic period excitation combined with random noises. To illustrate the potential of the proposed scheme, two classical examples of Mathieu–Duffing oscillator and Duffing–Rayleigh oscillator are considered. The evolutionary of dynamics behaviors and stochastic P-bifurcation phenomenon are examined through a qualitative change of the probability distribution, which indicates that system parameters and noise intensity, respectively, can be treated as bifurcation-induced parameters. Besides, numerical results are also carried out to confirm the accuracy of semi-analytical prediction. • Transient response prediction for VI-S under combined excitation with RBFNN. • Two examples are presented for illustrating the applicable of the proposed method. • Stochastic bifurcation with respect to the system parameters were discussed. [ABSTRACT FROM AUTHOR]

A general jerky equation with random excitation is investigated in this paper. Before introducing the random excitation term, the equation is reduced to a two-dimensional model when undergoing a Hopf bifurcation. Then the model with the parametric excitation and external excitation is converted to a stochastic differential equation with singularity based on the stochastic average theory. For the equation, its dynamical behaviors are analyzed in different parameters' spaces, including the stability, stochastic bifurcation and stationary solution. Besides, numerical simulations are given to show the asymptotic behavior of the stationary solution. [ABSTRACT FROM AUTHOR]

In this manuscript, an investigation on bifurcations induced by two delays and additive and multiplicative colored noises in a self-sustained birhythmic oscillator is presented, both theoretically and numerically, which serves for the purpose of unveiling extremely complicated nonlinear dynamics in various spheres, especially in biology. By utilizing the multiple scale expansion approach and stochastic averaging technique, the stationary probability density function (SPDF) of the amplitude is obtained for discussing stochastic bifurcations. With time delays, intensities and correlation time of noises regarded as bifurcation parameters, rich bifurcation arises. In the case of additive noise, it is identified that the bifurcations induced by the two delays are entirely distinct and longer velocity delay can accelerate the conversion rate of excited enzyme molecules. A novel type of P-bifurcation emerges from the process in the case of multiplicative colored noise, with the SPDF qualitatively changing between crater-like and bimodal distributions, while it cannot be generated when the multiplicative colored noise is coupled with additive noise. The feasibility and effectiveness of analytical methods are confirmed by the good consistency between theoretical and numerical solutions. This investigation may have practical applications in governing dynamical behaviors of birhythmic systems. [ABSTRACT FROM AUTHOR]

This paper investigates a stochastic Internet congestion control model with distributed and constant delays driven by Gaussian white noise. We study the stochastic dynamics of the system at different delay feedback intensities. Firstly, by using center manifold theory and the stochastic averaging method, the system is reduced to the average Itô equation. Secondly, the stochastic bifurcation behavior of the system is investigated by using the Fokker–Planck–Kolmogorov equation corresponding to the average Itô equation. It is found that the stochastic P(D)-bifurcation occurs when the delay feedback intensity reaches the critical value. Finally, numerical simulations are given to validate the effectiveness of the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper aimed to investigate the stochastic P-bifurcation of Van der Pol oscillator with a fractional derivative damping term driven by Gaussian white noise excitation. Firstly, based on the method of stochastic averaging method and Stratonovich-Khasminskii theorem, the corresponding Fokker-Plank-Kolmogorov (FPK) equation is deduced. To describe the P-bifurcation of system, the stationary probability densities of amplitude can be obtained by solving the FPK equation. Then, the effects of the fractional order, the fractional coefficient, and the intensity of Gaussian white noise on the fractional systems are discussed in detail. The results show that increasing order α will change obviously the number and the height of peaks under certain parameter conditions. Finally, comparing the analytical and numerical results, a very satisfactory agreement can be found. [ABSTRACT FROM AUTHOR]

To the best of authors' knowledge, little work has been focused on the noisy vibro-impact systems with fractional derivative element. In this paper, stochastic bifurcation of a vibro-impact oscillator with fractional derivative element and a viscoelastic term under Gaussian white noise excitation is investigated. First, the viscoelastic force is approximately replaced by damping force and stiffness force. Thus the original oscillator is converted to an equivalent oscillator without a viscoelastic term. Second, the nonsmooth transformation is introduced to remove the discontinuity of the vibro-impact oscillator. Third, the stochastic averaging method is utilized to obtain analytical solutions of which the effectiveness can be verified by numerical solutions. We also find that the viscoelastic parameters, fractional coefficient and fractional derivative order can induce stochastic bifurcation. [ABSTRACT FROM AUTHOR]