JUMP AND BIFURCATION OF DUFFING OSCILLATOR UNDER NARROW-BAND EXCITATION

The jump and bifurcation of Duffing oscillator with hardening spring subject to narrow-band random excitation are systematically and comprehensively examined. It is shown that, in a certain domain of the space of the oscillator and excitation parameters, there are two types of more probable motions in the stationary response of the Duffing oscillator and jumps may occur. The jump is a transition of the response from one more probable motion to another or vise versa. Outside the domain the stationary response is either nearly Gaussian or like a diffused limit cycle. As the parameters change across the boundary of the domain the qualitative behavior of the stationary response changes and it is a special kind of bifurcation. It is also shown that, for a set of specified parameters, the statistics are unique and they are independent of initial condition. It is pointed out that some previous results and interpretations on this problem are incorrect.

Analysis of a kind of Duffing oscillator system used to detect weak signals

The stability of the periodic solution of the Duffing oscillator system in the periodic phase state is proved by using the Yoshizaw theorem, which establishes a theoretical basis for using this kind of chaotic oscillator system to detect weak signals. The restoring force term of the system affects the weak-signal detection ability of the system directly, the quantitative relationship between the coefficients of the linear and nonlinear items of the restoring force of the Duffing oscillator system and the SNR in the detection of weak signals is obtained through a large number of simulation experiments, then a new restoring force function with better detection results is established.

A circular zone counting method of identifying a Duffing oscillator state transition and determining the critical value in weak signal detection

Identifying state transition and determining the critical value of the Duffing oscillator are crucial to indicating external signal existence and have a great influence on detection accuracy in weak signal detection. A circular zone counting （CZC） method is proposed in this paper, by combining the Duffing oscillator＇s phase trajectory feature and numerical calculation for quickly and accurately identifying state transition and determining the critical value, to realize a high- efficiency weak signal detection. Detailed model analysis and method construction of the CZC method are introduced. Numerical experiments into the reliability of the proposed CZC method compared with the maximum Lyapunov exponent （MLE） method are carried out. The CZC method is demonstrated to have better detecting ability than the MLE method, and furthermore it is simpler and clearer in calculation to extend to engineering application.

Weak wide-band signal detection method based on small-scale periodic state of Duffing oscillator

The conventional Duffing oscillator weak signal detection method, which is based on a strong reference signal, has inherent deficiencies. To address these issues, the characteristics of the Duffing oscillator＇s phase trajectory in a small- scale periodic state are analyzed by introducing the theory of stopping oscillation system. Based on this approach, a novel Duffing oscillator weak wide-band signal detection method is proposed. In this novel method, the reference signal is discarded, and the to-be-detected signal is directly used as a driving force. By calculating the cosine function of a phase space angle, a single Duffing oscillator can be used for weak wide-band signal detection instead of an array of uncoupled Duffing oscillators. Simulation results indicate that, compared with the conventional Duffing oscillator detection method, this approach performs better in frequency detection intervals, and reduces the signal-to-noise ratio detection threshold, while improving the real-time performance of the system.

Extraction of periodic signals in chaotic secure communication using Duffing oscillators

This paper presents a novel approach to extract the periodic signals masked by a chaotic carrier. It verifies that the driven Duffing oscillator is immune to the chaotic carrier and sensitive to certain periodic signals. A preliminary detection scenario illustrates that the frequency and amplitude of the hidden sine wave signal can be extracted from the chaotic carrier by numerical simulation. The obtained results indicate that the hidden messages in chaotic secure communication can be eavesdropped utilizing Duffing oscillators.

Solving the Optimal Control Problems of Nonlinear Duffing Oscillators By Using an Iterative Shape Functions Method

In the optimal control problem of nonlinear dynamical system,the Hamiltonian formulation is useful and powerful to solve an optimal control force.However,the resulting Euler-Lagrange equations are not easy to solve,when the performance index is complicated,because one may encounter a two-point boundary value problem of nonlinear differential algebraic equations.To be a numerical method,it is hard to exactly preserve all the specified conditions,which might deteriorate the accuracy of numerical solution.With this in mind,we develop a novel algorithm to find the solution of the optimal control problem of nonlinear Duffing oscillator,which can exactly satisfy all the required conditions for the minimality of the performance index.A new idea of shape functions method(SFM)is introduced,from which we can transform the optimal control problems to the initial value problems for the new variables,whose initial values are given arbitrarily,and meanwhile the terminal values are determined iteratively.Numerical examples confirm the high-performance of the iterative algorithms based on the SFM,which are convergence fast,and also provide very accurate solutions.The new algorithm is robust,even large noise is imposed on the input data.

Blind parameter estimation of pseudo-random binary code-linear frequency modulation signal based on Duffing oscillator at low SNR

Conventional parameter estimation methods for pseudo-random binary code-linear frequency modulation(PRBC-LFM)signals require prior knowledge,are computationally complex,and exhibit poor performance at low signal-to-noise ratios(SNRs).To overcome these problems,a blind parameter estimation method based on a Duffing oscillator array is proposed.A new relationship formula among the state of the Duffing oscillator,the pseudo-random sequence of the PRBC-LFM signal,and the frequency difference between the PRBC-LFM signal and the periodic driving force signal of the Duffing oscillator is derived,providing the theoretical basis for blind parameter estimation.Methods based on amplitude method,short-time Fourier transform method,and power spectrum entropy method are used to binarize the output of the Duffing oscillator array,and their performance is compared.The pseudo-random sequence is estimated using Duffing oscillator array synchronization,and the carrier frequency parameters are obtained by the relational expressions and characteristics of the difference frequency.Simulation results show that this blind estimation method overcomes limitations in prior knowledge and maintains good parameter estimation performance up to an SNR of-35 dB.

Research and Simulation on Weak Signal Detection Based on Duffing Oscillator and Damping Ratio Perturbation

The chaotic system is sensitive to the initial value, and this can be applied in the weak signal detection. There are periodic, critical and chaotic states in a chaotic system. When the system is in the critical state, a small perturbation of system,n parameter may lead to a qualitative change of the system＇s state. This paper introduces a new method to detect weak signals by the way of disturbing the damping ratio. The authors choose the duffing equation, using MATLAB to carry on the simulation, to study the changes of the system when the signal to be measured is added to the damping ratio. By means of observing the phase loots chart and time damin chart, the weak signal will be detected.

Application of the Duffing Chaotic Oscillator Model for Early Fault Diagnosis-Ⅰ. Basic Theory

In this paper, the well-known Duffing equation and the nonlinear equation describing vibration of the human eardrum are introduced from elastic nonlinear system theory. According to the fact that the human ear can distinguish weak sound with small difference, the idea that the Duffing oscillator can be used to detect a weak signal and diagnose early fault of machinery is proposed. In order to obtain a model for weak signal detection via the Duffing oscillator, the first step is to seek all forms of solutions of the Duffing equation. The second step is to study global bifurcations of the Duffing equation using qualitative analysis theory of a dynamic system. That is to say, a series of bifurcations thresholds of the Duffing equation can be analyzed by the Melnikov function and a subharmonics Melnikov function. Then the three types of bifurcations thresholds varying with damping and external exciting amplitude are discussed. The analysis concludes that the bifurcation threshold corresponding to the maximum orbit of solutions outside the homo-clinic orbit of the Duffing equation can be used to detect a weak signal. Finally, the implementing model of the Duffing oscillator for weak signal detection is given.

Parameter estimation method for a linear frequency modulation signal with a Duffing oscillator based on frequency periodicity

In view of the complexity of existing linear frequency modulation(LFM)signal parameter estimation methods and the poor antinoise performance and estimation accuracy under a low signal-to-noise ratio(SNR),a parameter estimation method for LFM signals with a Duffing oscillator based on frequency periodicity is proposed in this paper.This method utilizes the characteristic that the output signal of the Duffing oscillator excited by the LFM signal changes periodically with frequency,and the modulation period of the LFM signal is estimated by autocorrelation processing of the output signal of the Duffing oscillator.On this basis,the corresponding relationship between the reference frequency of the frequencyaligned Duffing oscillator and the frequency range of the LFM signal is analyzed by the periodic power spectrum method,and the frequency information of the LFM signal is determined.Simulation results show that this method can achieve high-accuracy parameter estimation for LFM signals at an SNR of-25 dB.

ON DOUBLE PEAK PROBABILITY DENSITY FUNCTIONS OF DUFFING OSCILLATOR TO COMBINED DETERMINISTIC AND RANDOM EXCITATIONS

The principal resonance of Duffing random external excitation was investigated. oscillator to combined deterministic and The random excitation was taken to be white noise or harmonic with separable random amplitude and phase. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The one peak probability density function of each of the two stable stationary solutions was calculated by the linearization method. These two one-peak-density functions were combined using the probability of realization of the two stable stationary solutions to obtain the double peak probability density function. The theoretical analysis are verified by numerical results.

Time Dependent Gaussian Equivalent Linearization of Duffing Oscillator Using Continuous Wavelet Transform

Evolutionary response analysis of Duffing oscillator using Gaussian equivalent linearization in wavelet based time-frequency frame work is presented here. Cubic （i.e., odd type） non-linearity associated with stiffness and damping is modeled. The goal of this research is to develop the mathematical model of an equivalent linear system which is applicable for different non-stationary input processes （i.e., either summation of amplitude modulated stationary orthogonal processes or digitally simulated non-stationary processes）. The instantaneous parameters of the ELTVS （equivalent linear time varying system） are evaluated by minimizing the error between the displacements of non-linear and equivalent linear systems in wavelet domain. For this purpose, three different basis functions （i.e., Mexican Hat, Morlet and a modified form of Littlewood-Paley） are used. The unknown parameters （i.e., natural frequency and damping） of the ELTVS are optimized in stochastic least square sense. Numerical results are presented for different types of input to show the applicability and accuracy of the proposed wavelet based linearization technique.

NONSTATIONARY EFFECTS ON SAFE BASINS OF A FORCED SOFTENING DUFFING OSCILLATOR

The safe basin of a forced softening Duffing oscillator is studiednumerically. The changes of safe basins are observed under bothstationary and nonstationary variations of the external excitationfrequency. The kind of nonstationary variations of the excitationfrequency can greatly change the erosion rate and the shape of thesafe basin. The other effects of nonstatinary variations on the safebasin are also discussed.

非线性振动、非线性波与Jacobi椭圆函数(续)

Succinct and efficient method to obtain analytic solutions of nonlinear vibrations and nonlinear waves by Jacobian elliptic functions are introduced. Important typical examples are given and explained, including simple pendulum, Duffing oscillator, cnoidal wave and solitary wave solutions of KdV equation, sine-Gordon equation, nonlinear Schrdinger equation, sech^2 profile solitons, kink and anti-kink solitons, breather, interaction of a kink and an anti-kink, and envelop solitons.

Cumulative solutions of nonlinear longitudinal vibration in isotropic solid bars

Based on the strain invariant relationship and taking the high-order elastic energy into account, a nonlinear wave equation is derived, in which the excitation, linear damping, and the other nonlinear terms are regarded as the first-order correction to the linear wave equation. To solve the equation, the biggest challenge is that the secular terms exist not only in the fundamental wave equation but also in the harmonic wave equation （unlike the Duffing oscillator, where they exist only in the fundamental wave equation）. In order to overcome this difficulty and to obtain a steady periodic solution by the perturbation technique, the following procedures are taken： （i） for the fundamental wave equation, the secular term is eliminated and therefore a frequency response equation is obtained; （ii） for the harmonics, the cumulative solutions are sought by the Lagrange variation parameter method. It is shown by the results obtained that the second- and higher-order harmonic waves exist in a vibrating bar, of which the amplitude increases linearly with the distance from the source when its length is much more than the wavelength; the shift of the resonant peak and the amplitudes of the harmonic waves depend closely on nonlinear coefficients; there are similarities to a certain extent among the amplitudes of the odd- （or even-） order harmonics, based on which the nonlinear coefficients can be determined by varying the strain and measuring the amplitudes of the harmonic waves in different locations.

Regular nonlinear response of the driven Duffng oscillator to chaotic time series

Nonlinear response of the driven Duffing oscillator to periodic or quasi-periodic signals has been well studied. In this paper, we investigate the nonlinear response of the driven Duffing oscillator to non-periodic, more specifically, chaotic time series. Through numerical simulations, we find that the driven Duffing oscillator can also show regular nonlinear response to the chaotic time series with different degree of chaos as generated by the same chaotic series generating model, and there exists a relationship between the state of the driven Duffing oscillator and the chaoticity of the input signal of the driven Duffing oscillator. One real-world and two artificial chaotic time series are used to verify the new feature of Duffing oscillator. A potential application of the new feature of Duffing oscillator is also indicated.

Application of Lyapunov exponent algorithm in balise signal chaotic oscillator detection

To improve the detection accuracy of the balise uplink signal transmitted in a strong noise environment,we use chaotic oscillator to detect the balise uplink signal based on the characteristics of the chaotic system that is sensitive to initial conditions and immune to noise.Combining with the principle of Duffing oscillator system used in weak signal detection and uplink signal feature,the methods and steps of using Duffing oscillator to detect the balise signal are presented.Furthermore,the Lyapunov exponent algorithm is used to calculate the critical threshold of the Duffing oscillator detection system.Thus,the output states of the system can be quantitatively judged to achieve demodulation of the balise signal.The simulation results show that the chaotic oscillator detection method for balise signal based on Lyapunov exponent algorithm not only improves the accuracy and efficiency of threshold setting,but also ensures the reliability of balise signal detection.

Adaptive stochastic resonance method for weak signal detection based on particle swarm optimization

In order to solve the parameter adjustment problems of adaptive stochastic resonance system in the areas of weak signal detection,this article presents a new method to enhance the detection efficiency and availability in the system of two-dimensional Duffing based on particle swarm optimization.First,the influence of different parameters on the detection performance is analyzed respectively.The correlation between parameter adjustment and stochastic resonance effect is also discussed and converted to the problem of multi-parameter optimization.Second,the experiments including typical system and sea clutter data are conducted to verify the effect of the proposed method.Results show that the proposed method is highly effective to detect weak signal from chaotic background,and enhance the output SNR greatly.

Bistability in Coupled Oscillators Exhibiting Synchronized Dynamics

We report some new results associated with the synchronization behavior of two coupled double-well Duffing oscillators （DDOs）. Some sufficient algebraic criteria for global chaos synchronization of the drive and response DDOs via linear state error feedback control are obtained by means of Lyapunov stability theory. The synchronization is achieved through a bistable state in which a periodic attractor co-exists with a chaotic attractor. Using the linear perturbation analysis, the prevalence of attractors in parameter space and the associated bifurcations are examined. Subcritical and supercritical Hopf bifurcations and abundance of Arnold tongues -- a signature of mode locking phenomenon are found.

Chaotic synchronization via linear controller

A technical framework of constructing a linear controller for chaotic synchronization by utilizing the stability theory of cascade-connected system is presented. Based on the method developed in the paper, two simple and linear feedback controllers, as examples, are derived for the synchronization of Liu chaotic system and Duffing oscillator, respectively. This method is quite flexible in constructing a control law. Its effectiveness is also illustrated by the simulation results.