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 e...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.展开更多
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 ...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.展开更多
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 oscilla...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.展开更多
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...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.展开更多
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. ...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.展开更多
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 signa...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.展开更多
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,wh...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.展开更多
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-no...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.展开更多
The chaotic system is sensitive to the initial value,and this property 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 ...The chaotic system is sensitive to the initial value,and this property 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 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 locus chart and time domain chart,the weak signal will be detected.展开更多
By using the idea of open-plus-closed-loop(OPCL) control, a controller composed of an external excitation and a linear feedback is designed to entrain chaotic trajectories of Mathieu-Duffing oscillator to its period...By using the idea of open-plus-closed-loop(OPCL) control, a controller composed of an external excitation and a linear feedback is designed to entrain chaotic trajectories of Mathieu-Duffing oscillator to its periodic and higher periodic orbits. The global basin of entrainment of this open-plus-closed-loop control is proved by combining the Lyapunov stability theory with a comparative theorem of initial value problems for second-order ordinary differential equations. Numerical simulations are performed to verify the theoretical results.展开更多
The dynamical behaviour of a parametrically excited Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with t...The dynamical behaviour of a parametrically excited Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow-flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance, it is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciate choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original system. It is demonstrated that nontrivial steady state responses may lose their stability by saddle-node (SN) or Hopf bifurcation (HB) as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.展开更多
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 dis...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.展开更多
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 an...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.展开更多
We develop in this text a probabilistic approach to deterministic nonlinear systems. By studying the spectral properties of the Frobenius-Perron operator of a Duffing oscillator, relevant dynamic properties of the sys...We develop in this text a probabilistic approach to deterministic nonlinear systems. By studying the spectral properties of the Frobenius-Perron operator of a Duffing oscillator, relevant dynamic properties of the system are identified. Using the characteristics of the Dirac operator, the evolution of the probability density function is obtained for the Duffing oscillator, allowing important aspects of this system to be investigated analytically. A comparison with numerical simulation is carried out in order to validate the results obtained by the analytical approach as well as to verify the nonsymmetric features of the oscillator response.展开更多
This paper discusses a novel technique and implementation to perform nonlinear control for two different forced model state oscillators and actuators. The paper starts by discussing the Duffing oscillator which featur...This paper discusses a novel technique and implementation to perform nonlinear control for two different forced model state oscillators and actuators. The paper starts by discussing the Duffing oscillator which features a second order non-linear differential equation describing complex motion whereas the second model is the Van der Pol oscillator with non-linear damping. A first order actuator is added to both models to expand on the chaotic behavior of the oscillators. In order to control the system without comprising linearization, Lyapunov non-linear control was used. A control Lyapunov function was tailored to the system. This led to improved maneuverability of the controller and the performance of the overall system. The controller was found to be highly efficient in system tracking and had swift response time. Simulations were performed on both the uncontrolled and controlled cases. Both simulation results ultimately confirmed the effectiveness of the proposed controller.展开更多
This paper presents a comprehensive stability analysis of the dynamics of the damped cubic-quintic Duffing oscillator. We employ the derivative expansion method to investigate the slightly damped cubic-quintic Duffing...This paper presents a comprehensive stability analysis of the dynamics of the damped cubic-quintic Duffing oscillator. We employ the derivative expansion method to investigate the slightly damped cubic-quintic Duffing oscillator obtaining a uniformly valid solution. We obtain a uniformly valid solution of the un-damped cubic-quintic Duffing oscillator as a special case of our solution. A phase plane analysis of the damped cubic-quintic Duffing oscillator is undertaken showing some chaotic dynamics which sends a signal that the oscillator may be useful as model for prediction of earth- quake occurrence.展开更多
扩大未击中货车 der Pol 振荡器的动态行为详细数字地被调查。在象分叉图和 Poincar é地图那样的一些数字模拟工具的帮助下,到混乱和奇怪的引起注意的人的各种各样的形状的不同线路被观察。为了描绘这个振荡器系统, Lyapunov 代...扩大未击中货车 der Pol 振荡器的动态行为详细数字地被调查。在象分叉图和 Poincar é地图那样的一些数字模拟工具的帮助下,到混乱和奇怪的引起注意的人的各种各样的形状的不同线路被观察。为了描绘这个振荡器系统, Lyapunov 代表的光谱和 Lyapunov 尺寸的混乱行为,也被采用。展开更多
基金Project supported by the National Natural Science Foundation of China(Grant No.61973037)。
文摘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.
基金The project supported by National Natural Science Foundation of China
文摘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.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 40374045 and 40574051), and by the Jilin Technology Development Plan (Grant No 20050526),
文摘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.
基金Project supported by the National Natural Science Foundation of China(Grant No.61673066)
文摘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.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61172047 and 61071025)
文摘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.
基金supported by the National Natural Science Foundation of China (Grant Nos 60577019 and 60777041) the International Cooperation Project of Shanxi Province,China
文摘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.
文摘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.
基金the National Natural Science Foundation of China(Grant Nos.61973037 and 61673066).
文摘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.
文摘The chaotic system is sensitive to the initial value,and this property 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 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 locus chart and time domain chart,the weak signal will be detected.
基金Project supported by the National Natural Science Foundation of China (No. 10672193)
文摘By using the idea of open-plus-closed-loop(OPCL) control, a controller composed of an external excitation and a linear feedback is designed to entrain chaotic trajectories of Mathieu-Duffing oscillator to its periodic and higher periodic orbits. The global basin of entrainment of this open-plus-closed-loop control is proved by combining the Lyapunov stability theory with a comparative theorem of initial value problems for second-order ordinary differential equations. Numerical simulations are performed to verify the theoretical results.
基金Project supported by the Scientific Research Foundation for Returned Overseas Chinese Scholar of Ministry of Eduction, China (No.2006-331)
文摘The dynamical behaviour of a parametrically excited Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow-flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance, it is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciate choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original system. It is demonstrated that nontrivial steady state responses may lose their stability by saddle-node (SN) or Hopf bifurcation (HB) as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.
文摘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.
基金Project supported by the National Natural Science Foundation of China (Key Program) (No.10332030)the Natural Science Foundation of Guangdong Province of China (No.04011640)
文摘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.
文摘We develop in this text a probabilistic approach to deterministic nonlinear systems. By studying the spectral properties of the Frobenius-Perron operator of a Duffing oscillator, relevant dynamic properties of the system are identified. Using the characteristics of the Dirac operator, the evolution of the probability density function is obtained for the Duffing oscillator, allowing important aspects of this system to be investigated analytically. A comparison with numerical simulation is carried out in order to validate the results obtained by the analytical approach as well as to verify the nonsymmetric features of the oscillator response.
文摘This paper discusses a novel technique and implementation to perform nonlinear control for two different forced model state oscillators and actuators. The paper starts by discussing the Duffing oscillator which features a second order non-linear differential equation describing complex motion whereas the second model is the Van der Pol oscillator with non-linear damping. A first order actuator is added to both models to expand on the chaotic behavior of the oscillators. In order to control the system without comprising linearization, Lyapunov non-linear control was used. A control Lyapunov function was tailored to the system. This led to improved maneuverability of the controller and the performance of the overall system. The controller was found to be highly efficient in system tracking and had swift response time. Simulations were performed on both the uncontrolled and controlled cases. Both simulation results ultimately confirmed the effectiveness of the proposed controller.
文摘This paper presents a comprehensive stability analysis of the dynamics of the damped cubic-quintic Duffing oscillator. We employ the derivative expansion method to investigate the slightly damped cubic-quintic Duffing oscillator obtaining a uniformly valid solution. We obtain a uniformly valid solution of the un-damped cubic-quintic Duffing oscillator as a special case of our solution. A phase plane analysis of the damped cubic-quintic Duffing oscillator is undertaken showing some chaotic dynamics which sends a signal that the oscillator may be useful as model for prediction of earth- quake occurrence.
基金supported by the National Natural Science Foundation of China under Grant No.10875078the Natural Science Foundation of Zhejiang Province of China under Grant No.Y7080455
文摘扩大未击中货车 der Pol 振荡器的动态行为详细数字地被调查。在象分叉图和 Poincar é地图那样的一些数字模拟工具的帮助下,到混乱和奇怪的引起注意的人的各种各样的形状的不同线路被观察。为了描绘这个振荡器系统, Lyapunov 代表的光谱和 Lyapunov 尺寸的混乱行为,也被采用。