Chaos Threshold Research Articles

Chaos and attraction domain of fractional Φ6‐van der Pol with time delay velocity

This article investigates the chaotic analysis and attractive domain of a fractional‐order ‐van der Pol with time delay velocity under harmonic excitation. Firstly, eight different types of bifurcation states of the system under different parameters are calculated by using the undisturbed system. Secondly, the Melnikov method is used to explore the effect of time delay velocity on the threshold of chaos in the Smale horseshoe sense under the double‐well potential and three‐well potential of the system. Finally, through numerical analysis of the phase diagram, bifurcation diagram, and maximum Lyapunov exponent, the influence of time delay velocity on system chaos is studied. The results indicate that an increase in the delay velocity coefficient will lead to the system transitioning from a chaotic state to a periodic state, while an increase in the delay velocity term will lead to the system transitioning from a periodic state to a chaotic state. In the study of system bifurcation, it is found that the position of the equilibrium points of the system changes during periodic motion. Therefore, cell mapping is used to draw the attractive domain of the system is studying the influence of initial conditions on the equilibrium point of the system and the results show that there is a close relationship between the attraction domain and the process of chaos occurrence.

Chaos and Sub-Harmonic Bifurcations in a Simple SIR Model with Periodically and Parametrically Nonlinear Incidence Rate

Chaos and sub-harmonic bifurcations of a simple SIR model with periodically and parametrically nonlinear incidence rate are investigated analytically with the Melnikov method and sub-harmonic Melnikov method, respectively. The critical curves of chaos for the model with the periodically and parametrically nonlinear incidence rate are studied analytically for the first time. Some new interesting dynamic phenomena are presented and proved rigorously, including the nonchaotic band, controllable frequencies and even controllable frequency interval, which may exist for some system parameters in this model. The sub-harmonic periodic orbit family is derived. It is presented that sub-harmonic bifurcations with even order and integer order may exist in the system. The sub-harmonic bifurcation conditions are obtained rigorously for two different starting points. Numerical simulations including phase portraits, Poincaré sections, basins of attraction and Lyapunov exponents are given to verify the chaos threshold obtained by the analytical results. It is also shown that the system can undergo chaos through even-order sub-harmonic bifurcations.

Multi-scale analysis of harmonic resonance in cylindrical bubbles under acoustic excitation

In this paper, the dimensionless oscillation equation of a cylindrical bubble is analyzed using the multi-scale method, Lyapunov stability theory, and the Routh–Hurwitz stability criterion. The corresponding second-order analytical solution and stability criterion are obtained. By examining the cases of second-order super-harmonic resonance and 1/2-order sub-harmonic resonance, the harmonic resonance characteristics of cylindrical bubbles and the influencing factors are revealed. The conclusions are summarized as follows: (1) Super-harmonic resonance can exhibit up to three solutions, along with unstable phenomena such as jump and hysteresis. Sub-harmonic resonance, however, shows at most two solutions simultaneously, without jump or hysteresis phenomena. (2) As the acoustic excitation amplitude increases, both the response amplitude and the unstable zone significantly enlarge. An increase in nonlinear coefficients can reduce the response amplitude and increase instability. (3) When the acoustic excitation amplitude reaches a certain threshold, the oscillation mode of the bubble shifts from periodic to chaotic. Under the same initial conditions, the chaos threshold for sub-harmonic resonance is higher than that for super-harmonic resonance.

Chaotic dynamics of granules-beam coupled vibration: Route and threshold

Although granular materials are the second most processed in industry after water, the theoretical study of granules-structure interactions is not as advanced as that of fluid-structure interactions due to the lack of a unified view of the constitutive relation of granular materials. In the previous work [1], the theoretical model of granules-beam coupled vibration was developed and verified by experiments. However, it was also found that the system exhibits significant stiffness-softening Duffing characteristics even under micro-vibration, which implies that chaotic responses may be reached under certain conditions. To reveal the route and critical conditions for the system to enter chaos, in the present work, the chaotic dynamics of the system are studied. In qualitative analysis, Melnikov method is applied to analyze the instability behavior of the perturbed heteroclinic orbit of the system, thus the critical condition for the system to enter Smale horseshoe chaos, i.e., the Melnikov criterion, is obtained. The validity of the criterion is verified numerically. In experimental studies, the existence of chaotic responses and the route to chaos for granules-beam coupled vibrations is revealed. The experimental results suggest that the system response first experiences symmetry-breaking then undergoes a complete period-doubling cascade, and finally enters single-scroll chaos. In addition, although the Additional Dynamic Load (ADL) generated by granular media is highly complicated, a general and simple evolution pattern of the chaos threshold is found by parametric experiments, which is also supported by the Melnikov criterion. In short, the chaotic dynamics of granules-beam coupled vibration is revealed, which is a contribution to the engineering vibration aspect. On the theoretical side, an impressive result lies in that for the first time, the Melnikov criterion of such fractional-order systems is obtained in a global and closed form, which provides, respectively, improvement suggestions and reference results for the existing and future research on chaotic dynamics of fractional-order systems, especially of the Duffing-type systems.

Stochastic bifurcation and chaos study for nonlinear ship rolling motion with random excitation and delayed feedback controls

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 Itoˆ-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.

Analysis of torsional vibration characteristics for wind turbine drivetrain under external excitation

This work studies the torsional vibration characteristics analysis of the wind turbine drivetrain with the variation of internal parameter and external excitation. The electromechanically coupled torsional vibration model for wind turbine drivetrain is first established by taking into account the torsional vibration angle of the PMSG. Then, frequency response analysis of main parametric resonance is illustrated by multiple scale method, and the influences of the system parameters involving power factor angle, number of pole pairs, torsional stiffness, and wind speed excitation on the torsional vibration characteristics of wind turbine drivetrain are investigated. Moreover, the critical condition for homoclinic chaos in terms of the system parameters is derived by means of Melnikov’s method. The function relationship and the boundary curve of chaos threshold are obtained. Meanwhile, the effects of excitation amplitude and damping factor on the system transition to chaos are studied in detail. The analytical predictions including phase portrait, bifurcation diagram and Lyapunov exponent are investigated. The research results can offer a theoretical basis for further the parameter design and control of wind turbine drivetrain.

Chaotic Dynamics of a Duffing Oscillator Subjected to External and Nonlinear Parametric Excitations With Delayed Feedbacks

Abstract Duffing oscillator with delayed feedback is widely used in engineering. Chaos in such system plays an important role in the dynamic response of the system, which may lead to the collapse of the system. Therefore, it is necessary and significant to study the chaotic dynamical behaviors of such systems. Chaotic dynamics of the Duffing oscillator subjected to periodic external and nonlinear parameter excitations with delayed feedback are investigated both analytically and numerically in this paper. With the Melnikov method, the critical value of chaos arising from heteroclinic intersection is derived analytically. The feature of the critical curves separating chaotic and nonchaotic regions on the excitation frequency and the time delay is investigated analytically in detail. Under the corresponding system parameters, the monotonicity of the critical value to the excitation frequency, displacement time delay, and velocity time delay is obtained rigorously. The chaos threshold obtained by the analytical method is verified by numerical simulations.

Bifurcation and Chaotic Behavior of Duffing System with Fractional-Order Derivative and Time Delay

In this paper, the abundant nonlinear dynamical behaviors of a fractional-order time-delayed Duffing system under harmonic excitation are studied. By constructing Melnikov function, the necessary conditions of chaotic motion in horseshoe shape are detected, and the chaos threshold curve is obtained by comparing the results obtained through the Melnikov theory and numerical iterative algorithm. The results show that the trend of change is the same, which confirms the accuracy of the chaos threshold curve. It could be found that when the excitation frequency ω is larger than a certain value, the Melnikov theory is not valid for these values. Furthermore, by numerical simulation, some numerical results are obtained, including phase portraits, the largest Lyapunov exponents, and the bifurcation diagrams, Poincare maps, time histories, and frequency spectrograms at some typical points. These numerical simulation results show that the system exhibits some new complex dynamical behaviors, including entry into the state of chaotic motion from single period to period-doubling bifurcation and chaotic motion and periodic motion alternating under the necessary condition of chaotic occurrence. In addition, the effects of time delay, fractional-order coefficient, fractional order, linear viscous damping coefficient, and linear stiffness coefficient on the chaotic threshold curve are discussed, respectively. Those results reveal that there exist abundant nonlinear dynamic behaviors in this fractional-order system, and by adjusting these parameters reasonably, the system could be transformed from chaotic motion to non-chaotic motion.

Bifurcation Analysis of a Wind Turbine Generator Drive System with Stochastic Excitation Under Both Displacement and Velocity Delayed Feedback

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.

Chaos analysis for a class of impulse Duffing-van der Pol system

Abstract Chaotic dynamics of an impulse Duffing-van der Pol system is studied in this paper. With the Melnikov method, the existence condition of transversal homoclinic point is obtained, and chaos threshold is presented. In addition, numerical simulations including phase portraits and time histories are carried out to verify the analytical results. Bifurcation diagrams are also given, from which it can be seen that the system may undergo chaotic motions through period doubling bifurcations.

Stochastic heating of free electrons in multiple electromagnetic waves: A simple physical analysis.

It is established that charged particles crossing the interference field of two colliding electromagnetic (EM) waves can behave chaotically, leading to a stochastic heating of the particle distribution. A fine understanding of the stochastic heating process is crucial to the optimization of many physical applications requiring a high EM energy deposition to these charged particles. Predicting key stochastic heating features (particle distribution, chaos threshold) is usually achieved using a heavy Hamiltonian formalism required to model particle dynamics in chaotic regimes. Here, we explore an alternative and more intuitive path, which makes it possible to reduce the equationsof motion of particles to rather simple and well-known physical systems such as Kapitza and gravity pendulums. Starting from these simple systems, we first show how to estimate chaos thresholds by deriving a model of the stretching and folding dynamics of the pendulum bob in phase space. Based on this first model, we then derive a random walk model for particle dynamics above the chaos threshold, which can predict major features of stochastic heating for any EM polarization and angle θ_{i}.

Weak signal detection based on variable-situation-potential with time-delay feedback and colored noise

For a long time, weak signal detection has been a research direction of great interest to scholars. This paper proposes a variable stochastic differential equation with double potential wells with time-delay feedback control, and its stochastic dynamics are studied and analyzed. Under the background of colored noise, the theoretical chaos threshold is calculated by the numerical integration method based on the Melnikov function. The influences of colored noise and time-delay feedback on the threshold are also discussed. Appropriate time-delay feedback can reduce the threshold, which is more conducive to detecting weak signals in the system. On this basis, we established a set of schemes for detecting weak signals: First, the frequency of the signal to be detected is determined by the “transient vacancy” of the system’s chaotic motion. Secondly, on the basis of the known frequency and phase of the weak signal, the amplitude of the weak signal is obtained. Finally, from the aspect of numerical simulation, this method has obvious advantages in detecting weak signals.

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.

Nonlinear dynamic responses of an inclined beam to harmonic excitation in temperature field

Abstract Using both analytical and numerical methods, nonlinear dynamic behaviours including chaotic motions and subharmonic bifurcations of an inclined beam subjected to harmonic excitation in temperature field are investigated in this paper. Based on the Galerkin method, the mathematical model of motion is derived. Melnikov method is adopted to give an analytical expression of conditions for chaotic motions of the inclined beam. The chaotic feature on the inclined angle is studied in detail. It is presented that there exists a unique excitation frequency $\omega ^*$, such that the critical value of chaos is the monotone decreasing function of the inclination angle when the excitation frequency $\omega <\omega ^*$; whereas $\omega>\omega ^*$, it is the monotone increasing function of the inclination angle. The subharmonic bifurcations are also studied. It is obtained that subharmonic bifurcations of even orders or odd orders may occur for this system. With the techniques of elliptic functions, it is proved rigorously that this system may undergo chaos through finite subharmonic bifurcations. Numerical simulations are given to verify the chaos threshold obtained by the analytical method.

Chaotic dynamics and subharmonic bifurcations of current‐carrying conductors subjected to harmonic excitation

AbstractWith both analytical and numerical methods, global dynamics including chaotic motions and subharmonic bifurcations of current‐carrying conductors subjected to harmonic excitation are investigated in this paper. The system parameter conditions for chaos are obtained with the Melnikov method. The monotonicity of the critical value for chaos on the damping, alternating current, and excitation amplitude is studied in detail for three cases. Some interesting dynamic phenomena such as “uncontrollable frequency interval” and “controllable frequency” are presented analytically. Subharmonic bifurcations of odd order or even order are also investigated with the subharmonic Melnikov method. The evolution of subharmonic bifurcation to chaos is studied. It is proved rigorously that the system may undergo chaotic motions through finite or infinite subharmonic bifurcations. Numerical simulations are given to verify the chaos threshold obtained with the analytical method.

Analytical Threshold for Chaos of Piecewise Duffing Oscillator with Time-Delayed Displacement Feedback

In this paper, the bifurcation and chaotic motion of a piecewise Duffing oscillator with delayed displacement feedback under harmonic excitation are studied. Based on the Melnikov method, the necessary critical conditions for the chaotic motion in the system are obtained, and the chaos threshold curve is obtained by calculation and numerical simulation. The accuracy of the analytical result is proved by some typical numerical simulation results, including the local bifurcation diagrams, phase portraits, Poincaré maps, and the largest Lyapunov exponents. The effects of excitation frequency and time delay of the displacement feedback are analytically discussed. It could be found that the critical excitation amplitude will increase obviously with the increase of the excitation frequency, and under the selection of certain parameters, the critical excitation amplitude takes the time delay of 0.58 as the inflection point, which decreases at first and then increases.

Chaos threshold analysis of Duffing oscillator with fractional-order delayed feedback control

Chaos threshold analysis of Duffing oscillator with fractional-order delayed feedback control

Homoclinic bifurcation for a bi-stable piezoelectric energy harvester subjected to galloping and base excitations

In this paper, we studied the homoclinic bifurcation and nonlinear characteristics of a bistable piezoelectric energy harvester while it is concurrently excited by galloping and base excitation. Firstly, the electromechanical model of the energy harvester is established analytically by the energy approach, the Kirchhoff's law and quasi-steady hypothesis. Then, by the Melnikov method, the threshold for underlying snap-through in the system is derived, and the necessary conditions for homoclinic bifurcation and chaos are presented. The threshold is a determinant for the occurrence of high-energy oscillation. The analysis results reveal that the wind speed and the distance between magnets could affect the threshold for inter-well chaos and high energy oscillation. Finally, numerical simulation and experiments are carried out. Both results from numerical simulation and experiment support the theoretical prediction from Melnikov theory. The study could provide a guideline for the optimum design of the bi-stable piezoelectric energy harvester for wind and vibration in practice.

Chaos and subharmonic bifurcations of a soft Duffing oscillator with a non-smooth periodic perturbation and harmonic excitation.

Chaotic dynamics and subharmonic bifurcations of a soft Duffing oscillator with a non-smooth periodic perturbation and a harmonic excitation are investigated analytically in this paper. With the Fourier series, the system is expanded to the equivalent smooth system, and chaos arising from heteroclinic intersections is studied with the Melnikov method. The chaotic feature on the system parameters is investigated in detail. Some new interesting dynamic phenomena, such as chaotic bands for some excitation frequencies, are presented. The relationship between the frequency range of chaotic bands and the amplitude of the excitation as well as the damping is obtained analytically. Particularly, for some system parameters satisfying a particular relationship, chaos cannot occur for any excitation amplitudes or frequencies. Subharmonic bifurcations are investigated with a subharmonic Melnikov method. It is analytically proved that the system may undergo chaotic motions through infinite or finite odd order subharmonic bifurcations. Numerical simulations are given to verify the chaos threshold and revolution from subharmonic bifurcations to chaos obtained by analytical methods.

Chaos of the Rayleigh–Duffing oscillator with a non-smooth periodic perturbation and harmonic excitation

With both analytical and numerical methods, chaotic motions of the Rayleigh–Duffing oscillator with a non-smooth periodic perturbation and harmonic excitation are investigated in this paper. Chaos arising from homoclinic or heteroclinic intersections is studied with the Melnikov method. Chaos threshold is obtained and chaotic feature on the system parameters is investigated in detail. Some new interesting dynamic phenomena including “controllable frequency interval”, “chaotic band” and “uncontrollable parameters” are presented and proved rigorously. Numerical simulations are given to verify chaos threshold obtained by the analytical results.