几类不同系统的随机稳定性及多重振动共振现象
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摘要
随机稳定性是随机动力学领域的重要研究内容之一,它刻画随机动力系统在平稳解邻域内的稳定性,其研究指标为最大Lyapunov指数和矩Lyapunov指数,分别确定系统响应的几乎肯定渐近稳定性和p阶矩渐近稳定性。由于噪声在自然界的普遍存在性,随机稳定性问题在自然科学及工程科学等领域都具有重要的研究价值。
一些受高频信号和微弱低频信号同时激励的非线性系统,其响应在低频信号频率处的幅值随加入高频信号幅值的变化呈现非线性关系,出现类似于“共振”的现象,这种现象定义为振动共振。利用振动共振机理,可以实现系统中微弱低频信号的放大。双频信号在通讯技术、激光物理、离子物理、声学、神经科学等领域具有广泛的应用,因此研究振动共振具有重要的意义,实现对振动共振的控制在诸多领域具有潜在的价值。
本文重点研究不同噪声参激系统的随机稳定性与非线性系统的多重振动共振及其控制两方面的内容,主要进行了以下工作:
(1)研究了高斯白噪声参激余维2分岔系统的随机稳定性问题。基于一维扩散过程的奇异边界理论,讨论了一维相扩散过程存在奇异边界的所有情形。针对不同情形,推导出了参激矩阵的取值条件,得到了系统不变测度的解析解,并研究了其P-分岔问题。最后根据最大Lyapunov指数的结果,给出了各种情形下系统几乎肯定渐近稳定的边界。
(2)研究了实噪声(n维Ornstein-Uhlenbeck向量过程的可积函数)参激余维2分岔系统的随机稳定性问题。通过进行n维线性滤波系统的谱分析,得到了实噪声的谱密度函数,用摄动法得到了系统不变测度的标准FPK方程及最大Lyapunov指数的求解式。通过讨论一维相扩散过程存在奇异边界的所有情形,给出了对应参激矩阵的取值条件,得到了不变测度的解析式及系统响应的最大Lyapunov指数。研究了不变测度的P-分岔,给出了系统响应几乎肯定渐近稳定的边界。
(3)研究了一类非高斯色噪声参激的van der Pol-Duffing振子的随机稳定性。将噪声近似简化之后,通过尺度变换和线性随机变换求得了矩Lyapunov指数的二阶近似解,给出了系统响应p阶矩稳定性和几乎肯定渐近稳定的条件。
(4)研究了受双频信号激励的一维多稳态系统,发现了一种新颖且规律性较强的多重振动共振现象。在欠阻尼和过阻尼情形下,通过解析分析和数值模拟揭示了序列性多重振动共振现象发生的机理,重点研究了阻尼系数对共振效应的影响,给出了多重振动共振现象潜在的应用价值。
(5)研究了高频信号和微弱低频信号同时激励下,线性时滞反馈对不同非线性系统中振动共振现象的影响,结果表明通过调节时滞参数可以引起系统的输出发生多重振动共振现象,从而实现对振动共振的有效控制。时滞反馈不仅可以控制振动共振的发生或消失,而且还可以增强振动共振的程度,从而进一步放大微弱低频信号。研究还发现系统对低频信号响应的幅值增益随时滞参数的变化同时呈现两种不同的周期性关系,它们分别等于两激励信号的周期。在一些不存在经典振动共振现象的非线性系统中,引入线性时滞反馈之后,通过调节时滞参数也可以引发多重振动共振现象。
Stochastic stability is one of the most important issues in the field of stochastic dynamics. It characterizes the stability of a random system in the vicinity of the steady state. The measure indicators are maximal Lyapunov exponent and moment Lyapunov exponent, and they determine the almost assure asymptotically stability and pth moment asymptotically stability respectively. Due to the universal existence of noise in the natural world, the problem of stochastic stability has significant research value in natural science and engineering fields.
In some nonlinear systems that excited simultaneously by high- and low-frequency signals, the response amplitude at the low-frequency is a nonlinear function of the amplitude of the high-frequency signal, and it presents“resonance”by adjusting the high-frequency signal. This phenomenon is defined as vibrational resonance (VR). Based on the mechanism of VR, the weak low-frequency signal can be amplified in the system. Biharmonic signals are widely applied in communication technology, laser physics, ion physics, acoustics, neuroscience, etc. Hence, the investigation of VR and its control have important and potential values in a wide range of fields. In the present paper, besides the stochastic stability of some systems that are parametrically excited by different kinds of noise, the vibrational multiresonance and its control in nonlinear systems are also investigated. The main works and results are as following:
(1) The stochastic stability of a co-dimensional two-bifurcation system driven by a white noise is studied. Based on the theory of singular boundary, all possible singularities that exist in the one-dimensional phase diffusion process are detailed discussed. For all the cases, the different conditions of the matrix that included in the noise excitation term are deduced, and the analytical expressions of the invariant measure are obtained. In addition, the P-bifurcation for the invariant measure is researched completely. Finally, according to the results of the maximal Lyapunov exponent, the regions of the almost assure asymptotically stability are given for all the cases.
(2) The stochastic stability of a co-dimensional two-bifurcation system subjected to parametric excitation by a real noise that is assumed to be an integrable function of an n-dimensional Ornstein-Uhlenbeck vector process is investigated. Via the spectral analysis for an n-dimensional linear filter system, the spectral density functions for the real noise are given. Then, the standard FPK equation of the invariant measure and the calculation expression of the maximal Lyapunov exponent are obtained. Through discussing all the possible singular boundaries that exist in the one-dimensional phase diffusion process, the corresponding conditions of the matrix that included in the noise term, the expressions of the invariant measure and the maximal Lyapunov exponents are given for all the cases. In addition, the P-bifurcation of the invariant measure is investigated. And finally, the regions for the almost assure asymptotically stability of the system are obtained.
(3) The stochastic stability of a van der Pol-Duffing oscillator that under the parametric excitation by a non-Gaussian colored noise is studied. After the simplification of the noise, through scale transformation and linear stochastic transformation, the second-order solution of the moment Lyapunov is obtained, and the conditions for the pth moment asymptotically stability and almost assure asymptotically are given finally.
(4) A novel and much more regular vibrational multiresonance in a multistable system that driven by biharmonic signals is reported. For the underdamped and overdamped cases, the mechanism of the sequential vibrational multiresonance is discovered by both theoretical and numerical methods. The effect of the damping coefficient on the multiresonance is a focus in the investigation. The potential application values of vibrational multiresonance are discussed.
(5) Under the excitations of both high- and weak low-frequency signals, the effects of linear time delay feedback on the VR in different kinds of systems are investigated. The results show that the vibrational multiresonance appears through adjusting the delay parameter, and then the VR phenomenon can be effectively controlled. Via modulating the delay parameter, the VR phenomenon not only can occur or vanish but also can be enhanced and then improve the weak low-frequency signal further. With the variation of the delay parameter, the response amplitude of the system to the weak low-frequency signal presents vibrational multiresonance, and it is periodic in the delay with two different periods, i.e., the periods of the two-frequency exciting signals. The investigation also shows that the linear time delay feedback can induce VR even in the nonlinear system in which there is no classical VR existing.
一些受高频信号和微弱低频信号同时激励的非线性系统,其响应在低频信号频率处的幅值随加入高频信号幅值的变化呈现非线性关系,出现类似于“共振”的现象,这种现象定义为振动共振。利用振动共振机理,可以实现系统中微弱低频信号的放大。双频信号在通讯技术、激光物理、离子物理、声学、神经科学等领域具有广泛的应用,因此研究振动共振具有重要的意义,实现对振动共振的控制在诸多领域具有潜在的价值。
本文重点研究不同噪声参激系统的随机稳定性与非线性系统的多重振动共振及其控制两方面的内容,主要进行了以下工作:
(1)研究了高斯白噪声参激余维2分岔系统的随机稳定性问题。基于一维扩散过程的奇异边界理论,讨论了一维相扩散过程存在奇异边界的所有情形。针对不同情形,推导出了参激矩阵的取值条件,得到了系统不变测度的解析解,并研究了其P-分岔问题。最后根据最大Lyapunov指数的结果,给出了各种情形下系统几乎肯定渐近稳定的边界。
(2)研究了实噪声(n维Ornstein-Uhlenbeck向量过程的可积函数)参激余维2分岔系统的随机稳定性问题。通过进行n维线性滤波系统的谱分析,得到了实噪声的谱密度函数,用摄动法得到了系统不变测度的标准FPK方程及最大Lyapunov指数的求解式。通过讨论一维相扩散过程存在奇异边界的所有情形,给出了对应参激矩阵的取值条件,得到了不变测度的解析式及系统响应的最大Lyapunov指数。研究了不变测度的P-分岔,给出了系统响应几乎肯定渐近稳定的边界。
(3)研究了一类非高斯色噪声参激的van der Pol-Duffing振子的随机稳定性。将噪声近似简化之后,通过尺度变换和线性随机变换求得了矩Lyapunov指数的二阶近似解,给出了系统响应p阶矩稳定性和几乎肯定渐近稳定的条件。
(4)研究了受双频信号激励的一维多稳态系统,发现了一种新颖且规律性较强的多重振动共振现象。在欠阻尼和过阻尼情形下,通过解析分析和数值模拟揭示了序列性多重振动共振现象发生的机理,重点研究了阻尼系数对共振效应的影响,给出了多重振动共振现象潜在的应用价值。
(5)研究了高频信号和微弱低频信号同时激励下,线性时滞反馈对不同非线性系统中振动共振现象的影响,结果表明通过调节时滞参数可以引起系统的输出发生多重振动共振现象,从而实现对振动共振的有效控制。时滞反馈不仅可以控制振动共振的发生或消失,而且还可以增强振动共振的程度,从而进一步放大微弱低频信号。研究还发现系统对低频信号响应的幅值增益随时滞参数的变化同时呈现两种不同的周期性关系,它们分别等于两激励信号的周期。在一些不存在经典振动共振现象的非线性系统中,引入线性时滞反馈之后,通过调节时滞参数也可以引发多重振动共振现象。
Stochastic stability is one of the most important issues in the field of stochastic dynamics. It characterizes the stability of a random system in the vicinity of the steady state. The measure indicators are maximal Lyapunov exponent and moment Lyapunov exponent, and they determine the almost assure asymptotically stability and pth moment asymptotically stability respectively. Due to the universal existence of noise in the natural world, the problem of stochastic stability has significant research value in natural science and engineering fields.
In some nonlinear systems that excited simultaneously by high- and low-frequency signals, the response amplitude at the low-frequency is a nonlinear function of the amplitude of the high-frequency signal, and it presents“resonance”by adjusting the high-frequency signal. This phenomenon is defined as vibrational resonance (VR). Based on the mechanism of VR, the weak low-frequency signal can be amplified in the system. Biharmonic signals are widely applied in communication technology, laser physics, ion physics, acoustics, neuroscience, etc. Hence, the investigation of VR and its control have important and potential values in a wide range of fields. In the present paper, besides the stochastic stability of some systems that are parametrically excited by different kinds of noise, the vibrational multiresonance and its control in nonlinear systems are also investigated. The main works and results are as following:
(1) The stochastic stability of a co-dimensional two-bifurcation system driven by a white noise is studied. Based on the theory of singular boundary, all possible singularities that exist in the one-dimensional phase diffusion process are detailed discussed. For all the cases, the different conditions of the matrix that included in the noise excitation term are deduced, and the analytical expressions of the invariant measure are obtained. In addition, the P-bifurcation for the invariant measure is researched completely. Finally, according to the results of the maximal Lyapunov exponent, the regions of the almost assure asymptotically stability are given for all the cases.
(2) The stochastic stability of a co-dimensional two-bifurcation system subjected to parametric excitation by a real noise that is assumed to be an integrable function of an n-dimensional Ornstein-Uhlenbeck vector process is investigated. Via the spectral analysis for an n-dimensional linear filter system, the spectral density functions for the real noise are given. Then, the standard FPK equation of the invariant measure and the calculation expression of the maximal Lyapunov exponent are obtained. Through discussing all the possible singular boundaries that exist in the one-dimensional phase diffusion process, the corresponding conditions of the matrix that included in the noise term, the expressions of the invariant measure and the maximal Lyapunov exponents are given for all the cases. In addition, the P-bifurcation of the invariant measure is investigated. And finally, the regions for the almost assure asymptotically stability of the system are obtained.
(3) The stochastic stability of a van der Pol-Duffing oscillator that under the parametric excitation by a non-Gaussian colored noise is studied. After the simplification of the noise, through scale transformation and linear stochastic transformation, the second-order solution of the moment Lyapunov is obtained, and the conditions for the pth moment asymptotically stability and almost assure asymptotically are given finally.
(4) A novel and much more regular vibrational multiresonance in a multistable system that driven by biharmonic signals is reported. For the underdamped and overdamped cases, the mechanism of the sequential vibrational multiresonance is discovered by both theoretical and numerical methods. The effect of the damping coefficient on the multiresonance is a focus in the investigation. The potential application values of vibrational multiresonance are discussed.
(5) Under the excitations of both high- and weak low-frequency signals, the effects of linear time delay feedback on the VR in different kinds of systems are investigated. The results show that the vibrational multiresonance appears through adjusting the delay parameter, and then the VR phenomenon can be effectively controlled. Via modulating the delay parameter, the VR phenomenon not only can occur or vanish but also can be enhanced and then improve the weak low-frequency signal further. With the variation of the delay parameter, the response amplitude of the system to the weak low-frequency signal presents vibrational multiresonance, and it is periodic in the delay with two different periods, i.e., the periods of the two-frequency exciting signals. The investigation also shows that the linear time delay feedback can induce VR even in the nonlinear system in which there is no classical VR existing.
引文
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