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The paper treats stationary response of a stochastically excited multi-degree-of-freedom (multi-DOF) vibro-impact system undergoing Markovian jump. The vibro-impact system with sudden abrupt changes in substructures or external excitations is modeled as a continuous-discrete Markovian jump system, which is essentially different from the traditional vibro-impact model. It is demonstrated that the random jump factors switch between a finite number of modes. This salient feature allows us to identify this type of dynamic behaviors as response of hybrid vibro-impact systems undergoing Markovian jump. Utilizing a two-step approximate technique, we can reduce the considered multi-DOF hybrid system to one-dimensional averaged Ito equation of the form of system’s total energy. The approximate analytical solution of the associated Fokker–Planck–Kolmogorov (FPK) equation of system’s energy is derived to predict the stationary response of original hybrid systems.

Highly energetic electrons are trapped in the magnetic field of Earth’s radiation belts. The physical mechanisms driving the dynamics of the Van Allen belts can be understood from the electron’s energy spectrum, which is believed to be steeply falling with increasing energy. This view has been prevalent for the past 60 years since the energy spectra were first measured. Here, we report the observation of a reversed energy spectrum with abundant high-energy and fewer low-energy electrons spanning from hundreds of keV to around two MeV in electron energy in data collected with NASA’s Van Allen Probes. We find that this spectrum dominates inside the plasmasphere—a dense cold plasma region co-rotating with the Earth. Using two-dimensional Fokker–Planck diffusion simulations with a time-dependent, data-driven model of hiss waves in the plasmasphere, we demonstrate that the formation of the reversed spectrum is explained by the scattering of hiss waves. The results have important implications for understanding the distributions of charged particles and wave–particle interactions in magnetized plasmas throughout the solar system and beyond. Observations reveal that electrons in Earth’s outer radiation belt possess a spectrum that partially rises with increasing energy, contrary to common beliefs. Plasma hiss waves scattered off electrons are found to be the origin of this phenomenon.

In this paper, the modeling and dynamical response of a vibro-impact capsule system with Hertzian contact and random environmental perturbation is studied. The capsule system is modeled as a two-degree-of-freedom dynamical system composed of a rigid capsule and a movable internal mass. The random environmental perturbation is incorporated in the models of the capsule system, which is more realistic working condition of capsule robotic system. The impact between the rigid capsule and the internal mass will occur when the relative displacement is larger than the gap between them. The impact interaction is described by using the Hertzian contact model. The resistance force between the capsule and the sliding face is described by Coulomb friction model. The response of the vibro-impact capsule system is obtained through Monte Carlo simulation. A kind of random stick–slip phenomena can be found in the response of the proposed model, which is consistent with the actual behavior of capsule system. At last, it is worthy noting that stochastic P-bifurcation occurs when the parameters of the system vary.

A new procedure for designing optimal bounded control of stochastically excited multi-degree-of-freedom (MDOF) nonlinear viscoelastic systems is proposed based on the stochastic averaging method and the stochastic maximum principle. First, the system is formulated as a quasi-integrable Hamiltonian system with viscoelastic terms and each viscoelastic term is replaced approximately by an elastically restoring force and a visco-damping force based on the randomly periodic behavior of the motion of quasi-integrable Hamiltonian system. Thus, a stochastically excited MDOF nonlinear viscoelastic system is converted to an equivalent quasi-integrable Hamiltonian system without viscoelastic terms. Then, by applying stochastic averaging, the system is further reduced to a partially averaged system of less dimension. The adjoint equation and maximum condition for the optimal control problem of the partially averaged system are derived by using the stochastic maximum principle, and the optimal bounded control force is determined from the maximum condition. Finally, the probability and statistics of the stationary response of optimally controlled system are obtained by solving the Fokker–Plank–Kolmogorov equation (FPK) associated with the fully averaged Ito equation of the controlled system. An example is worked out to illustrate the proposed procedure and its effectiveness.

The recently developed high-quality WHU ELF/VLF receiver system has been deployed in Suizhou, China (geomagnetic latitude 21.81°N, longitude 174.44°E, L =1.16) to detect low latitude extremely-low-frequency (ELF: 0.3‒3 kHz) and very-low-frequency (VLF: 3‒30 kHz) emissions originating from either natural or artificial sources since February 2016. During the first-month operation of the receiver system, a total of 3039 clear whistlers have been recorded at this low latitude station with the majority (97.0%) occurring on 28 February and 1 March 2016. Observed whistlers manifest various types including single one-hop, echo train, multi-flash, and multi-path. They tend to intensify after local midnight, reach the peak around 04‒05 LT, and then weaken quickly. Both features of lower cutoff frequencies of most whistlers below ~1.6 kHz and almost uniform dispersion for many successive multi-flash whistlers suggest that these whistlers propagate along the geomagnetic field lines in the duct mode. The computed dispersion varies between ~15 s1/2 and 23 s1/2 for observed one-hop whistlers and is greater than 50 s1/2 for three-hop echo train whistlers, indicating that the whistlers observed at the Suizhou station are low latitude whistlers.

The stationary response analysis of stochastic vibro-impact (VI) dynamical system with a unilateral nonzero offset barrier and viscoelastic damping under additive and multiplicative random excitations is explored. First, the conversion of the original stochastic VI system coupled with viscoelastic damping into an equivalent stochastic VI system without viscoelastic damping terms is performed by adding the equivalent stiffness and damping. Then, the equivalent stochastic VI system is studied according to the level of system energy instead of non-smooth transformation of the previous literature; thus, description on effects of impacts draws support from the loss of the system energy. The energy loss at each impact is found to be related to the restitution factor and the energy level before impact. Furthermore, the probability density functions of the equivalent VI system based on assumption of the weak damping and random perturbation is produced by the application of the stochastic averaging of energy envelope. The effectiveness of the suggested method is verified by comparing the analytical results with those from Monte Carlo simulations.

A new control strategy for preventing the extinction of predator–prey ecosystem in a random environment is proposed. First, a controlled stochastic Lotka–Volterra system is introduced and its some properties are presented. The stochastic averaging method is applied to transfer the original controlled system to a partially averaged one. Then, the adjoint equation and maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The optimal control is determined from the maximum condition and solving the forward-backward stochastic differential equation (FBSDE). For infinite time-interval ergodic control, the adjoint process is stationary and the FBSDE is reduced to an ordinary differential equation. Finally, the stationary probability density of the prey of optimally controlled system is obtained to show that the controlled ecosystem is more stable than the uncontrolled one. The mean transition time of optimally controlled system is also calculated to show that the proposed control strategy is effective in preventing the possible extinction.

In the present paper, the random vibration problem of Rayleigh–Van der Pol oscillator is considered by means of a single-degree-of-freedom system with a unilateral rigid barrier at its equilibrium position. The random excitations are two parametric Gaussian white noises. The nonsmooth transformation method is applied to convert the vibroimpact system to an equivalent system without velocity jump. The modified stochastic averaging technique for energy envelope is used to deal with the transformed system, and the averaging Ito stochastic differential equation is obtained. The analytical solution of the response of the original vibroimpact system is derived by solving the corresponding Fokker–Planck equation and using the inverse transformation of nonsmooth transformation. The validity of the analytical results is verified by those from Monte Carlo simulations based on original system. Effects of different system parameters and parametric white noises on the response of the system are examined. In addition, the critical condition of stochastic bifurcations is also explored.

As an important plasma wave mode in the geospace, magnetosonic waves can interact with both radiation belt electrons and protons, thereby impacting the dynamics of magnetospheric particles. Based on the Doppler-shifted resonance condition and the cold plasma dispersion relation, we investigate the profiles of resonance zone and resonant frequency of the Landau resonance between radiation belt electrons and magnetosonic waves and the cyclotron resonances with protons. The results demonstrate that resonant interactions between magnetosonic waves and magnetospheric charged particles largely rely on L-shell, wave normal angle, and kinetic energy and equatorial pitch angle of particles. Resonance zones for the Landau resonance between magnetosonic waves and radiation belt electrons are confined to a very narrow (mostly less than 1°) extent of magnetic latitude, which tends to shift to lower latitudes with increasing equatorial pitch angle and decreasing electron energy. Landau resonance frequencies also increase with magnetosonic wave normal angle. In contrast, higher order cyclotron resonances of magnetosonic waves with protons are much easier to occur in a broad range of magnetic latitude. As the resonance order increases, the coverage of the resonance zone shrinks overall and occupies the geomagnetic equatorial region. In addition, resonant frequencies increase with resonance order. Corresponding to higher order cyclotron resonances, protons are more likely to interact with magnetosonic waves at intermediate to high frequencies. Our study can be useful to elaborate the resonant interaction processes between magnetosonic waves and radiation belt electrons and protons and improve the current understanding of the multi-aspect impact of magnetosonic waves on the magnetospheric particle dynamics.

The asymptotic Lyapunov stability with probability one of multi-degree-of freedom quasi-partially integrable and non-resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises is studied. First, the averaged stochastic differential equations for quasi partially integrable and non-resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises are derived by means of the stochastic averaging method and the stochastic jump-diffusion chain rule. Then, the expression of the largest Lyapunov exponent of the averaged system is obtained by using a procedure similar to that due to Khasminskii and the properties of stochastic integro-differential equations. Finally, the stochastic stability of the original quasi-partially integrable and non-resonant Hamiltonian systems is determined approximately by using the largest Lyapunov exponent. An example is worked out in detail to illustrate the application of the proposed method. The good agreement between the analytical results and those from digital simulation show that the proposed method is effective.

A new procedure for designing optimal bounded control of quasi-nonintegrable Hamiltonian systems with actuator saturation is proposed based on the stochastic averaging method for quasi-nonintegrable Hamiltonian systems and the stochastic maximum principle. First, the stochastic averaging method for controlled quasi-nonintegrable Hamiltonian systems is introduced. The original control problem is converted into one for a partially averaged equation of system energy together with a partially averaged performance index. Then, the adjoint equation and the maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The bounded optimal control forces are obtained from the maximum condition and solving the forward–backward stochastic differential equations (FBSDE). For infinite time-interval ergodic control, the adjoint variable is stationary process, and the FBSDE is reduced to an ordinary differential equation. Finally, the stationary probability density of the Hamiltonian and other response statistics of optimally controlled system are obtained by solving the Fokker–Plank–Kolmogorov equation associated with the fully averaged Ito equation of the controlled system. For comparison, the bang–bang control is also presented. An example of two degree-of-freedom quasi-nonintegrable Hamiltonian system is worked out to illustrate the proposed procedure and its effectiveness. Numerical results show that the proposed control strategy has higher control efficiency and less discontinuous control force than the corresponding bang–bang control at the price of slightly less control effectiveness.

Background Acupuncture is not considered a conventional therapy for post-stroke sequelae but it might have some additional positive effects on early rehabilitation. We conducted this trial to determine whether acupuncture has additional effects in early comprehensive rehabilitation for acute ischemic stroke and dysfunctions secondary to stroke. Methods Two hundred fifty patients were randomized into two groups: acupuncture (AG) or no acupuncture (NAG). Eighteen acupuncture treatment sessions were performed over a 3-week period. The primary outcome was blindly measured with National Institutes of Health Stroke Scale (NIHSS) at week 1, week 3, and week 7. Secondary outcomes included: Fugl-Meyer Assessment (FMA) for motor function, bedside swallowing assessment (BSA) and videofluoroscopic swallowing study (VFSS) for swallowing function, the Mini-Mental State Examination (MMSE) and Montreal Cognitive Assessment (MoCA) for cognitive function, and the adverse reaction of acupuncture for safety assessment. Results Significant improvements from acupuncture treatment were observed in NIHSS (p

A new procedure for designing optimal control of quasi non-integrable Hamiltonian systems under stochastic excitations is proposed based on the stochastic averaging method for quasi non-integrable Hamiltonian systems and the stochastic maximum principle. First, the control problem consisting of 2n-dimensional equations governing the controlled quasi non-integrable system and performance index is converted into a partially averaged one consisting of one-dimensional equation of the controlled system and performance index by using the stochastic averaging method. Then, the adjoint equation and the maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The optimal control forces are determined from the maximum condition and solving the forward–backward stochastic differential equations (FBSDE). For infinite time-interval ergodic control, the adjoint variable is a stationary process and the FBSDE is reduced to a partial differential equation. Finally, the response statistics of optimally controlled system is predicted by solving the Fokker–Plank equation (FPE) associated with the fully averaged Ito equation of the controlled system. An example of two degree-of-freedom (DOF) quasi non-integrable Hamiltonian system is worked out to illustrate the proposed procedure and its effectiveness.

This paper deals with the approximate nonstationary probability density of a class of nonlinear vibrating system excited by colored noise. First, the stochastic averaging method is adopted to obtain the averaged Ito equation for the amplitude of the system. The corresponding Fokker-Planck-Kolmogorov equation governing the evolutionary probability density function is deduced. Then, the approximate solution of the Fokker-Planck-Kolmogorov equation is derived by applying the Galerkin method. The solution is expressed as a sum of a series of expansion in terms of a set of proper basis functions with time-depended coefficients. Finally, an example is given to illustrate the proposed procedure. The validity of the proposed method is confirmed by Monte Carlo Simulation.

Background Acupuncture has been widely used as a treatment for stroke in China for more than 3,000 years. However, previous research has not yet shown that acupuncture is effective as a stroke treatment. We report a protocol for a multicenter, randomized, controlled, and outcome assessor-blind trial to evaluate the efficacy and safety of acupuncture on acute ischemic stroke. Methods/Design In a prospective trial involving three hospitals in the Zhejiang Province (China) 250 patients with a recent (less than 1 week previous) episode of ischemic stroke will be included. Patients will be randomized into two groups: an acupuncture group given scalp acupuncture and electroacupuncture, and a control group given no acupuncture. Eighteen treatment sessions will be performed over a three-week period. The primary outcome will be measured by changes in the National Institutes of Health Stroke Scale score at the one, three, and four-week follow-up. Secondary outcome measures will be: 1) the Fugl-Meyer assessment scale for motor function; 2) the mini-mental state examination and Montreal cognitive assessment for cognitive function; 3) the video-fluoroscopic swallowing study for swallowing ability; and 4) the incidence of adverse events. Discussion This trial is expected to clarify whether or not acupuncture is effective for acute stroke. It will also show if acupuncture can improve motor, cognitive, or swallowing function. Trial registration Chinese Clinical Trial Registry ChiCTR-TRC-12001971.