Stochastic Melnikov Research Articles

A class of nonlinear ship stability analysis: Stochastic dynamics with time-delayed control in crosswind and wave conditions

The impact of ocean waves and crosswinds on ships has long been a focal point of research for many scholars. This paper proposes a stochastic ship rolling model influenced by crosswinds, ocean waves, and time delays. The safe operating region for ship navigation is presented through the phase space of the system. A quantitative discussion of the system is conducted using the stochastic Melnikov function, and the stochastic P-D bifurcation of the system is discussed using topological data analysis techniques. The research indicates that incorporating time delay feedback can effectively enhance the system’s stability.

Semi-analytical estimates for the chaotic diffusion in the Second Fundamental Model of Resonance. Application to Earth’s navigation satellites

We discuss the applicability of the Melnikov and Landau–Teller theories in obtaining semi-analytical estimates of the speed of chaotic diffusion in systems driven by the separatrix-like stochastic layers of a resonance belonging to the ‘second fundamental model’ (SFM) (Henrard and Lemaitre, 1983). Stemming from the analytic solution for the SFM in terms of Weierstrass elliptic functions, we introduce stochastic Melnikov and Landau–Teller models allowing to locally approximate chaotic diffusion as a sequence of uncorrelated ‘jumps’ observed in the time series yielding the slow evolution of an ensemble of trajectories in the space of the adiabatic actions of the system. Such jumps occur in steps of one per homoclinic loop. We show how a semi-analytical determination of the probability distribution of the size of the jumps can be arrived at by the Melnikov and Landau–Teller approximate theories. Computing also the mean time required per homoclinic loop, we arrive at estimates of the chaotic diffusion coefficient in such systems. As a concrete example, we refer to the long-term diffusion of a small object (e.g. Earth navigation satellite or space debris) within the chaotic layers of the so-called 2g+h lunisolar resonance, which is of the SFM type. After a suitable normal form reduction of the Hamiltonian, we compute estimates of the speed of diffusion of these objects, which compare well with the results of numerical experiments.

Melnikov’s method for controlled stochastic oscillations of a rocking block with fractional derivative

The problem of controlling a rocking block with fractional dissipation so as to prevent it from leaving a prescribed set of bounded oscillations has a number of interesting applications in many scientific and technological fields. This work develops an analytical framework to describe the influence of control on the behavior of a fractionally dissipative rocking block subject to random acceleration of the ground. Control equations are developed and control strategies leading to optimal suppression of instability are asymptotically described under the assumptions of weak dissipation along with a small energy noise. Under these assumptions, the behavior of the block is interpreted as weakly perturbed rocking oscillations close to a generic periodic solution, and the Melnikov approximations are considered as the necessary conditions of instability. The stochastic Melnikov criterion derived in this paper allows for an approximate calculation of an upper bound to the overturning probability as well as to the representation of an optimal control strategy. The Melnikov criterion for stochastic rocking oscillations in the presence of convenient control strategies are derived explicitly for the first time, using an extension of the Melnikov theory to a model with fractional derivative.

Random Melnikov Method and Induced Chaos in Bistable Vibro-Impact Oscillators with Bounded Noise

In this paper, the Melnikov method for an abstract stochastic nonsmooth hybrid system is derived in detail and employed to study the homoclinic bifurcations and induced chaos in a bistable vibro-impact SD oscillator with bounded noise first proposed by Li et al. 2021, in which the geometric nonlinearity is used to produce the irrational restoring force and an impact map is introduced to describe energy loss during collisions. The initial Poincaré section is selected on the right rigid constraint so that rigorous perturbation analysis can be carried out to avoid the extension of trajectories near the switching manifold. The stochastic Melnikov function with geometrical intuition is obtained to detect the threshold of parameters for homocinic tangency of the perturbed stochastic stable and unstable manifolds by calculating the energy difference of their initial points on the Poincaré section. Finally, the analytical Melnikov analysis combined with numerical simulations is carried out to validate the developed Melnikov method for analyzing the global dynamics of the bistable vibro-impact SD oscillator subject to bounded noise.

Homoclinic–Heteroclinic Bifurcations and Chaos in a Coupled SD Oscillator Subjected to Gaussian Colored Noise

The so-called coupled smooth and discontinuous (SD) oscillator whose stiffness term leads to a transcendental function is a simple mass-spring system constrained to a straight line by two parameters, which are the dimensionless distances to the fixed point. This paper studies the homoclinic–heteroclinic chaos in a coupled SD oscillator subjected to Gaussian colored noise. In order to investigate the chaos thresholds analytically, the piecewise linearization approximation is used to fit the transcendental function. Stochastic nonsmooth Melnikov method with homoclinic–heteroclinic orbits is developed to study chaos thresholds of oscillators with tri-stable potential. Based on stochastic Melnikov process, the mean square criterion and the rate of phase-space flux function theory are used to study the chaotic motions of a coupled SD oscillator under weak noise and strong noise, respectively. The obtained results show that it is effective to use the piecewise linear approximation to analyze chaos in the coupled SD oscillator subjected to Gaussian colored noise. It also lays the foundation for chaos research of other nonsmooth mechanical vibration systems under random excitation.

The Melnikov criterion of instability for random rocking of a rigid block with a fractional derivative element

This paper discusses the problem of the cumulative influence of fractional dissipation and external noise on escape of a rigid block from a prescribed set of rocking oscillations. For simplicity, the dynamics of a weakly dissipative slender block affected by low energy noise is considered. Under these assumptions, the generic motion is depicted by the linearized equation of an undamped block with the well-known periodic vibro-impact solution. The stochastic Melnikov criterion based on the generic solution allows calculating the upper bound of escape probability of the structure in the presence of fractional dissipation of the Caputo type. The analytical results demonstrate an explicit dependence of escape probability on the fractional dissipation parameters.

Frequency and Amplitude Identification of Weak Signal Based on the Limit System of Smooth and Discontinuous Oscillator

In this paper, a new method is proposed to identify the frequency and amplitude of weak signals by using a non-smooth system. The variable scale limit system of smooth and discontinuous (SD) oscillator without considering the phase is adopted as the identification system. By using the non-smooth stochastic subharmonic-like Melnikov method, an analytical expression of chaotic threshold under Gaussian white noise is given. Based on the phase diagram and Poincaré section diagram, the influence of noise intensity on the recognition system is studied. According to the non-smooth variable scale-convex-peak frequency identification method, the frequency of the signal to be detected can be accurately identified. Using the numerical simulation, the amplitude of the signal to be measured is identified according to the amplitude bifurcation diagram of the reference signal. The optimal value range of the parameters of the identification system is determined. Through an example of early fault of wheelset bearing of high-speed train, the frequency and amplitude of the early weak fault signal can be identified and the fault location can be determined, which verifies the effectiveness of the above method. The results show that the non-smooth system can identify the frequency and amplitude of the weak signal submerged in strong noise, and it has a wider application and higher accuracy than the continuous system.

The Melnikov criterion of instability for a fractionally damped rigid block with noise-induced response enhancement

This paper discusses the problem of the cumulative influence of fractional dissipation and external noise on possible escape of a rigid block from a prescribed set of rocking oscillations. For simplicity, the dynamics of a weakly dissipative slender block affected by low energy noise and undergoing an almost elastic impact upon the rigid foundation is considered. Under these assumptions, the generic motion is depicted by the linearized equation of an undamped block with the well-known periodic vibro-impact solution. The stochastic Melnikov criterion based on the generic solution allows calculating the upper bound of escape probability of the structure in the presence of fractional dissipation of the Caputo type. The analytical results demonstrate an explicit dependence of escape probability both on structural characteristics as well as on fractional dissipation parameters.

Improved variable scale-convex-peak method for weak signal detection

Based on the rectification method of detectable domain and elimination method of blind domain, the variable scale-convex-peak method for the identification frequency of weak signal is improved. For the initial phase of weak signal to be detected, the analytic expression of chaotic threshold is obtained by using the stochastic Melnikov method. The influence of phase on chaotic threshold is clarified, which leads to that the frequency of weak signal can be identified. Specifically, within a period, the initial phase can be divided into the detectable domain and the blind domain. In the detectable domain, when the frequency identification deviates, the rectification method of detectable domain is introduced. Specifically, the Newton interpolation method is used to give the calculation formula to rectify the deviation, which improves the accuracy of frequency identification. In the blind domain, the elimination method of blind domain is given by constructing the detection equations. The feasibility of the improved variable scale-convex-peak method is verified by numerical simulation and circuit experiment. Furthermore, by comparing the identified frequency value with the theoretical fault frequency value, the fault location of the wheelset bearing of the high-speed train is determined. The improved variable scale-convex-peak method is applied when the initial phase is in the frequency detectable domain or the blind domain, respectively.

Periodic‐phase‐diagram similarity method for weak signal detection

AbstractThe periodic‐phase‐diagram similarity method is proposed to identify the frequency of weak harmonic signals. The key technology is to find a set of optimal coefficients for Duffing system, which leads to the periodic motion under the influence of weak signal and strong noise. Introducing the phase diagram similarity, the influences of strong noise on the similarity of periodic phase diagram are discussed. The principle of highest similarity of periodic phase diagram with the same frequency is detected by discussing the persistence of similarity of periodic motion phase diagram under the strong noise and the periodic‐phase‐diagram similarity method is constructed. The weak signals of early fault and strong noise are input into Duffing system to obtain the identified system. The stochastic subharmonic Melnikov method is extended to obtain the conditions of the optimal coefficients for the identified system. Based on the results, the constructed frequency conversion harmonic weak signals are considered to form a datum periodic system. With the change of frequency in the datum periodic system, the phase diagram similarity of the two constructed systems can be calculated. Based on the periodic‐phase‐diagram similarity method, the frequency of weak harmonic signals can be identified by the principle of highest similarity of periodic phase diagram with the same frequency. The results of numerical simulation and the early fault diagnosis results of actual bearings verify the feasibility of the periodic‐phase‐diagram similarity method. The accuracy of the detection effect is 97%, and the minimum signal‐to‐noise ratio is −80.71 dB.

The TVICMs method for weak signal detection based on a nonlinear stochastic delay differential system

A nonlinear stochastic delay differential system with symmetry SD oscillator is investigated to detect weak signal. Firstly, the stochastic Melnikov function and the chaos threshold are obtained by using the stochastic mean square criterion. Secondly, the stochastic center manifold and the stochastic averaging method are used to obtain the one-dimensional Itoˆ equation and the corresponding FPK equation. The stationary probability density of the FPK equation is studied and the corresponding nonlinear stochastic dynamical behaviors are analyzed. Finally, the highlight of this paper is to detect the weak signal with ‘transient vacancy in chaotic motions’ (TVICMs) method successfully and extract the features information such as frequency and amplitude on the premise of unknown features information. Astonishingly, numerical simulation shows that the time delay can enhance the detection capability of the system and the effectiveness of the new method.

Variable scale-convex-peak method for weak signal detection

A variable scale-convex-peak method is constructed to identify the frequency of weak harmonic signal. The key of this method is to find a set of optimal identification coefficients to make the transition of dynamic behavior topologically persistent. By the stochastic Melnikov method, the lower bound of the chaotic threshold continuous function is obtained in the mean-square sense. The intermediate value theorem is applied to detect the optimal identification coefficients. For the designated identification system, there is a valuable co-frequency-convex-peak in bifurcation diagram, which indicates the state transition of chaos-period-chaos. With the change of the weak signal amplitude and external noise intensity in a certain range, the convex peak phenomenon is still maintained, which leads to the identification of frequency. Furthermore, the proposition of the existence of reversible scaling transformation is introduced to detect the frequency of the harmonic signal in engineering. The feasibility of constructing the hardware and software platforms of the variable scale-convex-peak method is verified by the experimental results of circuit design and the results of early fault diagnosis of actual bearings, respectively.

Chaos analysis of ship rolling motion in stochastic beam seas

In this paper, the chaotic motion of ship roll in stochastic beam seas, which is regarded as a bounded noise is investigated in detail. The stochastic Melnikov approach is applied to the model and the criterion for the chaos in the mean-square sense is derived. The chaotic thresholds of noise parameters obtained by means of the stochastic Melnikov process are verified by the numerical safe basin. Besides of the factors of noise disturbances, the effects of the other parameters in the system on safe basin are also discussed systematically. The varieties of coefficients of the restoring moment can also induce the erosion of safe basin and lead to the occurrence of the chaos. On the other hand, the increase of damping coefficients can enhance the safe domain of ship sailing.

Coherence resonance of a magnet-induced buckled piezoelectric energy harvester under stochastic parametric excitation

This study takes the magnet-induced buckled piezoelectric energy harvester as the object of research and employs the stochastic Melnikov method to explore the necessary conditions for occurrence of...

Onset of colored-noise-induced chaos in the generalized Duffing system

The effects of colored noise, red noise and green noise, on the onset of chaos are investigated theoretically and confirmed numerically in the generalized Duffing system with a fractional-order deflection. Analytical predictions concerning the chaotic thresholds in the parameter space are derived by using the stochastic Melnikov method combined with the mean-square criterion. To qualitatively confirm the analytical results, numerical simulations obtained from the mean largest Lyapunov exponent are used as test beds. We show that colored noise can induce chaos, and the effects for the case of red noise on the onset of chaos differ from those for the case of green noise. The most noteworthy result of this work is the formula, which relates the chaotic thresholds among red, green and white noise, holds for noise-induced chaos in the Duffing system. We also show that Gaussian white noise can induce chaos more easily than colored noise.

Homoclinic and heteroclinic chaos in nonlinear systems driven by trichotomous noise**Project supported by the National Natural Science Foundation of China (Grant No. 11672231), the Natural Science Foundation of Shaanxi Province, China (Grant No. 2016JM1010), the Fundamental Research Funds for the Central Universities, China (Grant No. 3102015ZY073), and the Seed Foundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical

The homoclinic and heteroclinic chaos in nonlinear systems subjected to trichotomous noise excitation are studied. The Duffing system and the Josephson-junction system are taken for example to calculate the corresponding amplitude thresholds for the onset of chaos on the basis of the stochastic Melnikov process with the mean-square criterion. It is shown that the amplitude threshold for the onset of chaos can be adjusted by changing the internal parameters of trichotomous noise, thereby inducing or suppressing chaotic behaviors in the two systems driven by trichotomous noise. The effects of trichotomous noise on the systems are verified by vanishing the mean largest Lyapunov exponent and demonstrated by phase diagrams and time histories.

Study on the Safety Degree of Ship Capsizing in Stochastic Waves

To quantify ship capsizing, from the energy perspective, the safety degree of a ship in waves is estimated based on stochastic Melnikov function and phase space transport theory. Considering the influence of nonlinear damping moment, nonlinear restoring moment, as well as the random waves, a nonlinear single degree of freedom differential equation for ship rolling is established. Transform the random wave moment from time domain to frequency domain by fast Fourier transformation, the random Melnikov function and rate of phase flux are extended to include the effects of navigation speed and heading angle and the safety degree of ship capsizing is quantified according to its statistical characteristics. Through an example, the accuracy of Melnikov function and phase space transport theory are verified and the effects of ship speed and heading angle on phase space transport rate are also quantified. This method is demonstrated properly to quantify the safety degree of ship capsizing and some valuable reference can be provided for the further research on ship stability criteria.

Stochastic chaos induced by diffusion processes with identical spectral density but different probability density functions.

Stochastic chaos induced by diffusion processes, with identical spectral density but different probability density functions (PDFs), is investigated in selected lightly damped Hamiltonian systems. The threshold amplitude of diffusion processes for the onset of chaos is derived by using the stochastic Melnikov method together with a mean-square criterion. Two quasi-Hamiltonian systems, namely, a damped single pendulum and damped Duffing oscillator perturbed by stochastic excitations, are used as illustrative examples. Four different cases of stochastic processes are taking as the driving excitations. It is shown that in such two systems the spectral density of diffusion processes completely determines the threshold amplitude for chaos, regardless of the shape of their PDFs, Gaussian or otherwise. Furthermore, the mean top Lyapunov exponent is employed to verify analytical results. The results obtained by numerical simulations are in accordance with the analytical results. This demonstrates that the stochastic Melnikov method is effective in predicting the onset of chaos in the quasi-Hamiltonian systems.

Identification of horseshoes chaos in a cable-stayed bridge subjected to randomly moving loads

In this paper, the dynamic response of cable-stayed bridge loaded by a train of moving forces with stochastic velocity is investigated. The cable-stayed bridge is modelled by Rayleigh beam with linear elastic supports. The stochastic Melnikov method is derived and the mean-square criterion is used to determine the effects of stochastic velocity and cables number on the threshold condition for the inhibition of smale horseshoes chaos in the system. The results indicate that the intensity of the random component of the loads velocity can be contributed to the enlargement of the possible chaotic domain of the system, and/or increases the chances to have a regular behavior of the system. On the other hand, the presence of cables in cable-stayed bridges system increases it degree of safety and paradoxically can be contributed to its destabilization. Numerical simulations of the governing equations are carried out to confirm the analytical prediction. The effect of loads number on the system response is also investigated.

Effect of bounded noise on chaotic motions of stochastically perturbed slowly varying oscillator

The effect of bounded noise on the chaotic behavior of a class of slowly varying oscillators is investigated. The stochastic Melnikov method is employed and then the criteria in both mean and mean-square sense are derived. The threshold amplitude of bounded noise given by stochastic Melnikov process is in good comparison with one determined by the numerical simulation of top Lyapunov exponents. The presence of noise scatters the chaotic domain in parameter space and the larger noise intensity results in a sparser and more irregular region. Both the simple cell mapping method and the generalized cell mapping method are applied to demonstrate the effects of noises on the attractors. Results show that the attractors are diffused and smeared by bounded noise and if the noise intensity increases, the diffusion is exacerbated.