Stochastic Analysis of Predator–Prey Models under Combined Gaussian and Poisson White Noise via Stochastic Averaging Method

In the present paper, the statistical responses of two-special prey–predator type ecosystem models excited by combined Gaussian and Poisson white noise are investigated by generalizing the stochastic averaging method. First, we unify the deterministic models for the two cases where preys are abundant and the predator population is large, respectively. Then, under some natural assumptions of small perturbations and system parameters, the stochastic models are introduced. The stochastic averaging method is generalized to compute the statistical responses described by stationary probability density functions (PDFs) and moments for population densities in the ecosystems using a perturbation technique. Based on these statistical responses, the effects of ecosystem parameters and the noise parameters on the stationary PDFs and moments are discussed. Additionally, we also calculate the Gaussian approximate solution to illustrate the effectiveness of the perturbation results. The results show that the larger the mean arrival rate, the smaller the difference between the perturbation solution and Gaussian approximation solution. In addition, direct Monte Carlo simulation is performed to validate the above results.

On the modeling of amplitude-sensitive electron spin resonance (ESR) detection using voltage-controlled oscillator (VCO)-based ESR-on-a-chip detectors

In this paper, we present an in-depth analysis of a voltage-controlled oscillator (VCO)-based sensing method for electron spin resonance (ESR) spectroscopy, which greatly simplifies the experimental setup compared to conventional detection schemes. In contrast to our previous oscillator-based ESR detectors, where the ESR signal was encoded in the oscillation frequency, in the amplitude-sensitive method, the ESR signal is sensed as a change of the oscillation amplitude of the VCO. Therefore, using VCO architecture with a built-in amplitude demodulation scheme, the experimental setup reduces to a single permanent magnet in combination with a few inexpensive electronic components. We present a theoretical analysis of the achievable limit of detection, which uses perturbation-theory-based VCO modeling for the signal and applies a stochastic averaging approach to obtain a closed-form expression for the noise floor. Additionally, the paper also introduces a numerical model suitable for simulating oscillator-based ESR experiments in a conventional circuit simulator environment. This model can be used to optimize sensor performance early on in the design phase. Finally, all presented models are verified against measured results from a prototype VCO operating at 14 GHz inside a 0.5 T magnetic field.

Asymptotic Stability of Controlled Nonlinear Stochastic Systems Considering the Dynamics of Sensors and Actuators

A closed-loop controlled system usually consists of the main structure, sensors, and actuators. In this paper, asymptotic stability of trivial solutions of a controlled nonlinear stochastic system considering the dynamics of sensors and actuators is investigated. Considering the inherent and intentional nonlinearities and random loadings, the coupled dynamic equations of the controlled system with sensors and actuators are given, which are further formulated by a controlled, randomly excited, dissipated Hamiltonian system. The Hamiltonian of the controlled system is introduced, and, based on the stochastic averaging method, the original high-dimensional system is reduced to a one-dimensional averaged system. The analytical expression of Lyapunov exponent of the averaged system is derived, which gives the approximately necessary and sufficient condition of the asymptotic stability of trivial solutions of the original high-dimensional system. The validation of the proposed method is demonstrated by a four-degree-of-freedom controlled system under pure stochastically parametric excitations in detail. A comparative analysis, which is related to the stochastic asymptotic stability of the system with and without considering the dynamics of sensors and actuators, is carried out to investigate the effect of their dynamics on the motion of the controlled system. Results show that ignoring the dynamics of sensors and actuators will get a shrink stable region of the controlled system.

Stochastic Hopf Bifurcation of MDOF Quasi-Integrable Hamiltonian Systems with Delayed Feedback Fractional-Order PD Controller

Hopf bifurcation, as the most representative dynamic bifurcation, is closely related to the stability of many engineering structures. In this work, the stochastic Hopf bifurcation (SHB) of a controlled quasi-integrable Hamiltonian system (H.S.) of multi-degree-of-freedom (MDOF) is investigated, where the system is subjected to wide-band noise and controlled by a Fractional-order Proportional-Derivative (FOPD) controller with time delay. By decoupling FOPD control force and simplifying it without time delay, the averaged Itô differential equations of the approximated system are derived with the technique of stochastic averaging. Then, the average bifurcation parameter expression of system is obtained, which can determine the criterion of the SHB deduced by the FOPD control force. Last, an illustration of coupled Rayleigh oscillators is given to demonstrate the validity of the procedure. The influences of time delay, noise intensities and fractional order on the system SHB are discussed.

Stochastic Response of Nonlinear Viscoelastic Systems with Time-Delayed Feedback Control Force and Bounded Noise Excitation

In this paper, an approximate analytical procedure is proposed to derive the stochastic response of nonlinear viscoelastic systems with time-delayed feedback control force and bounded noise excitation. The viscoelastic force and the time-delayed control force depend on the past histories of the state variables, which will result in infinite-dimensional problem in theoretical analysis. To resolve these difficulties, the viscoelastic force and the time-delayed control force are approximated by the current state variable based on the quasi-periodic behavior of the systematic response. Then, by using the stochastic averaging method for strongly nonlinear systems subjected to bounded noise excitation, an averaged equation for the equivalent system is derived. The Fokker–Plank–Kolmogorov (FPK) equation of the associated averaged equation is solved to derive the stochastic response of the equivalent system. Finally, two typical nonlinear viscoelastic oscillators are worked out and the results demonstrated the effectiveness of the proposed procedure. By utilizing the quasi-periodic behavior and stochastic averaging method of the strongly nonlinear system, the time-delayed control force and the viscoelastic terms can be simplified with equivalent damping force and equivalent restoring force and the resonant response under bounded noise excitation can be obtained analytically. The numerical results showed the accuracy of the proposed method.