Stochastic responses of Duffing–Van der Pol vibro-impact system under additive colored noise excitation

A response analysis procedure is developed for a vibro-impact system excited by colored noise. The non-smooth transformation is used to convert the vibro-impact system into a new system without impact term. With the help of the modified quasi-conservative averaging, the total energy of the new system can be approximated as a Markov process, and the stationary probability density function （PDF） of the total energy is derived. The response PDFs of the original system are obtained using the analytical solution of the stationary PDF of the total energy. The validity of the theoretical results is tested through comparison with the corresponding simulation results. Moreover, stochastic bifurcations are also explored.     

Response probability density functions of Duffing–Van der Pol vibro-impact system under correlated Gaussian white noise excitations

This paper aims to investigate the stationary probability density functions (PDFs) of a Duffing–Van der Pol vibro-impact system excited by correlated Gaussian white noise. With the help of non-smooth transformation, the stationary PDFs are formulated analytically by the stochastic averaging of energy envelope. The analytical results are verified by numerical simulation results. Stochastic bifurcations for different parameters are considered, and several special PDF forms are observed in this paper. The first form is the shape of the PDF of total energy can be similar to a crater, which has a minimum and a maximum. The second one is the shape of the joint PDF with three peaks, that is to say, the section of joint PDF has three maximum and two minimum. In addition, the influence of the degree of the correlation of Gaussian white noises is explored.     

Stochastic bifurcations of generalized Duffing–van der Pol system with fractional derivative under colored noise

The stochastic bifurcation of a generalized Duffing–van der Pol system with fractional derivative under color noise excitation is studied. Firstly, fractional derivative in a form of generalized integral with time-delay is approximated by a set of periodic functions. Based on this work, the stochastic averaging method is applied to obtain the FPK equation and the stationary probability density of the amplitude. After that, the critical parameter conditions of stochastic P-bifurcation are obtained based on the singularity theory. Different types of stationary probability densities of the amplitude are also obtained. The study finds that the change of noise intensity, fractional order, and correlation time will lead to the stochastic bifurcation.     

Chaotic motion of the dynamical system under both additive and multiplicative noise excitations

With both additive and multiplicative noise excitations, the effect on the chaotic behaviour of the dynamical system is investigated in this paper. The random Melnikov theorem with the mean-square criterion that applies to a type of dynamical systems is analysed in order to obtain the conditions for the possible occurrence of chaos. As an example, for the Duffing system, we deduce its concrete expression for the threshold of multiplicative noise amplitude for the rising of chaos, and by combining figures, we discuss the influences of the amplitude, intensity and frequency of both bounded noises on the dynamical behaviour of the Duffing system separately. Finally, numerical simulations are illustrated to verify the theoretical analysis according to the largest Lyapunov exponent and Poincaré map.     

Response of a Duffing—Rayleigh system with a fractional derivative under Gaussian white noise excitation

In this paper,we consider the response analysis of a Duffing-Rayleigh system with fractional derivative under Gaussian white noise excitation.A stochastic averaging procedure for this system is developed by using the generalized harmonic functions.First,the system state is approximated by a diffusive Markov process.Then,the stationary probability densities are derived from the averaged Ito stochastic differential equation of the system.The accuracy of the analytical results is validated by the results from the Monte Carlo simulation of the original system.Moreover,the effects of different system parameters and noise intensity on the response of the system are also discussed.     

Oscillation death in a coupled van der Pol–Mathieu system

We report an investigation of the oscillation death (OD) of a parametrically excited coupled van der Pol–Mathieu (vdPM) system. The system can be considered as a pair of harmonically forced van der Pol oscillators under a double-well potential. The two oscillators are coupled with a cubic nonlinearity. We have shown that the system arrives at an OD regime when coupling strength crosses a threshold value at which the system undergoes saddle-node bifurcation and two limit cycles coalesce onto a fixed point of the system. We have further shown that this nonautonomous system possesses a centre manifold corresponding to the OD regime.     

Primary resonance of fractional-order Duffing–van der Pol oscillator by harmonic balance method

The dynamical properties of fractional-order Duffing–van der Pol oscillator are studied, and the amplitude–frequency response equation of primary resonance is obtained by the harmonic balance method. The stability condition for steady-state solution is obtained based on Lyapunov theory. The comparison of the approximate analytical results with the numerical results is fulfilled, and the approximations obtained are in good agreement with the numerical solutions. The bifurcations of primary resonance for system parameters are analyzed. The results show that the harmonic balance method is effective and convenient for solving this problem, and it provides a reference for the dynamical analysis of similar nonlinear systems.     

Coexistence of attractors in autonomous Van der Pol–Duffing jerk oscillator: Analysis,chaos control and synchronisation in its fractional-order form

Dynamics at infinity and a Hopf bifurcation for a Sprott E system with a very small perturbation constant are studied in this paper. By using Poincaré compactification of polynomial vector fields in \(R^3\), the dynamics near infinity of the singularities is obtained. Furthermore, in accordance with the centre manifold theorem, the subcritical Hopf bifurcation is analysed and obtained. Numerical simulations demonstrate the correctness of the dynamical and bifurcation analyses. Moreover, by choosing appropriate parameters, this perturbed system can exhibit chaotic, quasiperiodic and periodic dynamics, as well as some coexisting attractors, such as a chaotic attractor coexisting with a periodic attractor for \(a>0\), and a chaotic attractor coexisting with a quasiperiodic attractor for \(a=0\). Coexisting attractors are not associated with an unstable equilibrium and thus often go undiscovered because they may occur in a small region of parameter space, with a small basin of attraction in the space of initial conditions.     

Chaotic analysis of Atangana–Baleanu derivative fractional order Willis aneurysm system

A new Willis aneurysm system is proposed, which contains the Atangana–Baleanu(AB) fractional derivative.we obtain the numerical solution of the Atangana–Baleanu fractional Willis aneurysm system(ABWAS) with the AB fractional integral and the predictor–corrector scheme.Moreover, we research the chaotic properties of ABWAS with phase diagrams and Poincare sections.The different values of pulse pressure and system order are used to evaluate and compare their effects on ABWAS.The simulations verify that the changes of pulse pressure and system order are the significant reason for ABWAS'states varying from chaotic to steady.In addition, compared with Caputo fractional WAS(FWAS),ABWAS shows less state that is chaotic.Furthermore, the results of bifurcation diagrams of blood flow damping coefficient and reciprocal heart rate show that the blood flow velocity tends to stabilize with the increase of blood flow damping coefficient or reciprocal heart rate, which is consistent with embolization therapy and drug therapy for clinical treatment of cerebral aneurysms.Finally, in view of the fact that ABWAS in chaotic state increases the possibility of rupture of cerebral aneurysms, a reasonable controller is designed to control ABWAS based on the stability theory.Compared with the control results of FWAS by the same method, the results show that the blood flow velocity in the ABWAS system varies in a smaller range.Therefore, the control effect of ABWAS is better and more stable.The new Willis aneurysm system with Atangana–Baleanu fractional derivative provides new information for the further study on treatment and control of brain aneurysms.     

Chaotic behaviour of nonlinear coupled reaction–diffusion system in four-dimensional space

In recent years, nonlinear coupled reaction–diffusion (CRD) system has been widely investigated by coupled map lattice method. Previously, nonlinear behaviour was observed dynamically when one or two of the three variables in the discrete system change. In this paper, we consider the chaotic behaviour when three variables change, which is called as four-dimensional chaos. When two parameters in the discrete system are unknown, we first give the existing condition of the chaos in four-dimensional space by the generalized definitions of spatial periodic orbits and spatial chaos. In addition, the chaotic behaviour will vary with the parameters. Then we propose a generalized Lyapunov exponent in four-dimensional space to characterize the different effects of parameters on the chaotic behaviour, which has not been studied in detail. In order to verify the chaotic behaviour of the system and the different effects clearly, we simulate the dynamical behaviour in two- and three-dimensional spaces.     

Enhancement of the shot noise of a quantum dot–Luttinger lead system

We investigate the joint effects of the intralead electron interaction and Coulombic dot–lead interaction on the shot noise of a quantum dot coupled to Luttinger liquid leads. A formula of the shot noise is derived by applying the nonequilibrium Green function technique. The shot noise is enhanced by the dot–lead interaction. For a weak or moderately strong interaction the differential shot noise demonstrates resonant-like behavior as a function of bias and gate voltages. In the limit of strong interaction resonant behavior disappears and the differential shot noise and Fano factor scale as a power law in bias voltage. Under some parameters, the differential shot noise may become negative around resonant peaks, and the physical reason is analyzed.     

Fluorescence anisotropy of indole molecules under two-photon excitation in the spectral range of 485–510 nm

Decay of polarized fluorescence in indole dissolved in propylene glycol under two-photon excitation by femtosecond laser pulses in the wavelength range of 485–510 nm has been studied. It is shown that under the experimental conditions used the fluorescence decay signal can be well described by a single excited state lifetime τ_f and a single rotation diffusion time τ_rot. By processing the data obtained, the times τ_f and τ_rot as well as anisotropy parameter r ₀ characterizing the symmetry of two-photon excitation of indole molecules have been determined. Decreasing of the anisotropy parameter r₀ down to zero under two-photon excitation energy higher than 5.1 eV has been observed. Interpretation of the obtained results have been done on the basis of ab initio quantum-mechanical computations. A model of energy relaxation under the condition of twophoton excitation of indole in a polar solvent has been discussed.     

Spontaneous excitation of a detector in circular motion coupled to massless Rarita–Schwinger fields in vacuum

We study the spontaneous excitation of a detector (modeled by a two-level atom) in circular motion coupled nonlinearly to vacuum massless Rarita–Schwinger fields in the ultrarelativistic limit and demonstrate that the spontaneous excitation occurs for ground-state atoms in circular motion in vacuum but the excitation rate is not of a pure thermal form as that of the atoms in linear uniform acceleration. An interesting feature is that terms of odd powers in acceleration appear in the excitation rate whereas in the linear acceleration case there are only terms of even powers present. On the other hand, what makes the present case unique in comparison to the atom’s coupling to other fields that are previously studied is the appearance of the terms proportional to the seventh and ninth powers of acceleration in the mean rate of change of atomic energy which are absent in the scalar, electromagnetic and Dirac field cases.     

Dynamical analysis of bounded and unbounded orbits in a generalized Hénon–Heiles system

The Hénon–Heiles potential was first proposed as a simplified version of the gravitational potential experimented by a star in the presence of a galactic center. Currently, this system is considered a paradigm in dynamical systems because despite its simplicity exhibits a very complex dynamical behavior. In the present paper, we perform a series expansion up to the fifth-order of a potential with axial and reflection symmetries, which after some transformations, leads to a generalized Hénon–Heiles potential. Such new system is analyzed qualitatively in both regimes of bounded and unbounded motion via the Poincaré sections method and plotting the exit basins. On the other hand, the quantitative analysis is performed through the Lyapunov exponents and the basin entropy, respectively. We find that in both regimes the chaoticity of the system decreases as long as the test particle energy gets far from the critical energy. Additionally, we may conclude that despite the inclusion of higher order terms in the series expansion, the new system shows wider zones of regularity (islands) than the ones present in the Hénon–Heiles system.     

The analysis of stochastic resonance and bearing fault detection based on linear coupled bistable system under lévy noise

In this paper, the stochastic resonance (SR) phenomenon of the linear coupled bistable system induced by Lévy noise is analyzed. Meanwhile, the characteristics of Lévy noise is also analyzed according to its probability density functions (PDFs) of different stability index α, symmetry parameter β, scale parameter σ and location index μ. The mean of signal-noise ratio increase (MSNRI) is regarded as an index to measure the SR phenomenon. Then, the rules for MSNRI affected by noise intensity D are explored under different charastic indexes of Lévy noise, system parameters a, b, c and coupling coefficient r. The results are beneficial to the numerical simulation of single-frequency and multi-frequency weak signals detection based on single bistable system and linear coupled system respectively. It is found that the performance of the proposed system is better than single bistable system and results of bearing fault detection could also verify the conclusion.     

Dynamical response of a neuron–astrocyte coupling system under electromagnetic induction and external stimulation

Previous studies have observed that electromagnetic induction can seriously affect the electrophysiological activity of the nervous system. Considering the role of astrocytes in regulating neural firing, we studied a simple neuron–astrocyte coupled system under electromagnetic induction in response to different types of external stimulation. Both the duration and intensity of the external stimulus can induce different modes of electrical activity in this system, and thus the neuronal firing patterns can be subtly controlled. When the external stimulation ceases, the neuron will continue to fire for a long time and then reset to its resting state. In this study, "delay" is defined as the delayed time from the firing state to the resting state, and it is highly sensitive to changes in the duration or intensity of the external stimulus. Meanwhile, the self-similarity embodied in the aforementioned sensitivity can be quantified by fractal dimension. Moreover, a hysteresis loop of calcium activity in the astrocyte is observed in the specific interval of the external stimulus when the stimulus duration is extended to infinity, since astrocytic calcium or neuron electrical activity in the resting state or during periodic oscillation depends on the initial state. Finally, the regulating effect of electromagnetic induction in this system is considered. It is clarified that the occurrence of "delay" depends purely on the existence of electromagnetic induction. This model can reveal the dynamic characteristics of the neuron–astrocyte coupling system with magnetic induction under external stimulation. These results can provide some insights into the effects of electromagnetic induction and stimulation on neuronal activity.     

Degradation of current–voltage and low frequency noise characteristics under negative bias illumination stress in InZnO thin film transistors

The instabilities of indium–zinc oxide thin film transistors under bias and/or illumination stress are studied in this paper. Firstly, illumination experiments are performed, which indicates the variations of current–voltage characteristics and electrical parameters(such as threshold voltage and sub-threshold swing) are dominated by the stress-induced ionized oxygen vacancies and acceptor-like states. The dependence of degradation on light wavelength is also investigated. More negative shift of threshold voltage and greater sub-threshold swing are observed with the decrease of light wavelength.Subsequently, a negative bias illumination stress experiment is carried out. The degradation of the device is aggravated due to the decrease of recombination effects between ionized oxygen vacancies and free carriers. Moreover, the contributions of ionized oxygen vacancies and acceptor-like states are separated by using the mid-gap method. In addition, ionized oxygen vacancies are partially recombined at room temperature and fully recombined at high temperature. Finally, low-frequency noise is measured before and after negative bias illumination stress. Experimental results show few variations of the oxide trapped charges are generated during stress, which is consistent with the proposed mechanism.     

Control synthesis of magnetic Fe3O4–chitosan nanoparticles under UV irradiation in aqueous system

Novel magnetic Fe₃O₄–chitosan nanoparticles were synthesized via photochemical method in an emulsifier-free aqueous system at room temperature for the first time. The scanning electron microscopy (SEM) and transmission electron microscopy (TEM) results showed that the Fe₃O₄–chitosan nanoparticles were in regular shape with a mean diameter of 41 nm, whereas the average size in aqueous solution measured by photocorrelation spectroscopy (PCS) was 64 nm, which indicated that the nanoparticles had water-swelling properties. X-ray diffraction (XRD) patterns indicated that the Fe₃O₄ nanoparticles were pure Fe₃O₄ with a spinel structure, and the irradiation under UV light did not result in a phase change. The Fe₃O₄–chitosan nanoparticles were also characterized by Fourier transform infrared (FTIR) spectra, thermogravimetric analysis (TGA) and vibrating sample magnetometer (VSM). Magnetic measurement revealed that the saturated magnetization (Ms) of the Fe₃O₄–chitosan nanoparticles reached 48.6 emu/g and the nanoparticles showed the characteristics of superparamagnetism. The stability test showed these novel nanoparticles had high magnetic stability. The PCS and TGA results indicated that the size and chitosan content of Fe₃O₄–chitosan nanoparticles formed was pH- and chitosan/Fe₃O₄ ratio-dependent, which could be used to synthesize magnetic Fe₃O₄–chitosan nanoparticles with different size to meet the requirements of different applications.     

Comparison of backstepping and modified active control in projective synchronization of chaos in an extended Bonhöffer–van der Pol oscillator

In this article, projective synchronization of double–scroll attractor of an extended Bonöffer–van der Pol oscillator (BVPO) is considered via the backstepping and active control techniques. In each synchronization scheme, a single control function is designed to achieve projective synchronization between two Bonhöffer–van der Pol oscillator evolving from different initial conditions. To obtain a single control function via the active control, the coefficient of the error dynamics is chosen such that the number of control functions is reduced from three to one, thereby, reducing control function complexity in design. The results show that the transient error dynamics convergence and synchronization time are achieved faster via the backstepping than that of the active control technique. However, the control function obtained via the active control is simpler with a more stable synchronization time and hence, it is more suitable for practical implementation. Numerical simulations are presented to confirm the effectiveness of the analytical results.