船舶螺旋桨鸣音时间序列混沌动力特性研究
摘要
螺旋桨鸣音现象对舰船隐蔽性、安静性和舒适性,影响巨大。但由于其产生机理复杂,所以迄今为止,无论是传统的线性理论,还是卡门涡释放理论,都难以对其进行客观合理地描述,其真实有效地控制方程尚未建立,还有许多问题有待深入研究。
近二十年来迅猛发展的混沌动力学理论研究表明:对时间序列分析来说,传统意义上的无规的、没有明确物理意义的随机信号在低维或较低维有可能具有确定性成分,甚至是由简单的非线性控制方程产生,即所谓的混沌现象。那么貌似纷繁复杂的螺旋桨鸣音系统是否是混沌现象,能否利用混沌理论对其进行研究?再考虑到湍流本身已被证明是一种混沌现象,而水声工程中的舰船辐射噪声和混响现象也被证明具有混沌动力特性。有鉴于此,本文分别研究了螺旋桨鸣音系统的重构相空间特性,系统的复杂度特性,利用递归图技术检验了该系统的平稳性特性,应用替代数据方法检验了其非线性特性,并进一步研究了其混沌动力特性,本文解决了螺旋桨鸣音系统“是不是”混沌现象的基本问题,对其混沌动力特性分析研究可以为今后该问题的理论分析和数值模拟研究提供指导,奠定了理论基础,其关联维数揭示了系统控制方程变量的个数,其具有正的Lyapunov指数则说明该系统具有短期可预测性与长期不可预测性的矛盾统一;另外,掌握了该系统的混沌不变量特征也为螺旋桨鸣音时间序列的信号处理、目标探测与识别等提供了技术支持。
(1)螺旋桨鸣音系统相空间重构研究
对于复杂的非线性系统或是混沌动力系统,可以通过对所研究系统的某一维或有限几维时间序列向更高维空间进行嵌入而获得系统的重构相空间拓扑,研究其相空间重构特性。本文分别应用互信息法和本文提出的改进的自相关函数法估计了螺旋桨鸣音系统的最佳嵌入延迟,而后应用最大特征值不变法和平均伪邻近法估计了螺旋桨鸣音系统的最小嵌入维数,并进一步利用时延法重构相空间双参数联合估计策略计算分析了螺旋桨鸣音系统的相空间重构参数。通过上述各种分析方法得到了一致的螺旋桨鸣音系统的最佳嵌入延迟t_D=1和最小嵌入维数d_E=8,此研究成果可为螺旋桨鸣音系统混沌动力特性的后续研究提供理论基础。
(2)螺旋桨鸣音系统时间序列符号动力学研究
应用符号序列分析方法研究螺旋桨鸣音系统时间序列,分别利用分割区间法和差值法对时间序列进行符号化,通过计算分析发现本文所研究的测量信号被噪声污染程度较低,时间序列主要确定性特征明显有效,在研究过程中对噪声可以不予考虑。进一步地,计算了螺旋桨鸣音系统符号序列的Shannon熵和不可逆转性指标,结合Shannon熵的定义,可知:螺旋桨鸣音系统具有强烈非线性特征,同时又具有确定性特征,不能用简单模型进行描述,且具有时间不可逆转性。
应用基于Kolmogorov复杂性定义的Lempel-Ziv复杂度算法,建立了螺旋桨鸣音系统复杂性序列,计算结果表明:随着螺旋桨鸣音系统时间序列声压幅值在零点左侧和右侧频繁跳动,其复杂度序列也频繁地出现局部极大值和极小值,但其整体上还是表现为归一化复杂度在0.73左右,表明螺旋桨鸣音系统具有确定性特征,同时,又表现出复杂的非线性结构特性。
(3)螺旋桨鸣音系统平稳性和非线性研究
利用基于图解法的系统时间序列平稳性检验方法——递归图法,分析螺旋桨鸣音系统的平稳性,利用该方法证明了螺旋桨鸣音系统具有平稳性特征。
在此基础之上,基于替代数据思想分析了螺旋桨鸣音系统的非线性特征:
(a)应用零假设1和检验统计量T_1首次证明了看似随机、无规律可循的螺旋桨鸣音系统具有确定性特征;
(b)应用零假设2和检验统计量T_2和T_3首次证明了螺旋桨鸣音系统具有非线性动力特征;
(c)进一步地,应用零假设3和检验统计量T_(rev)证明了螺旋桨鸣音系统的非线性特征不是由静态非线性测量函数引入的,而是其本身固有的。
通过上述分析,应用替代数据法证明了螺旋桨鸣音系统具有非线性动力特征,而系统非线性作为系统具有混沌动力特性的必要条件,上述分析结果为该系统的混沌动力特性分析研究提供了理论基础。此外,螺旋桨鸣音系统非线性动力特征的证明,也为对该系统的目标探测与识别,系统数值模拟方程组的构建等提供了技术支撑。
(4)螺旋桨鸣音系统混沌动力特性研究
采用相空间重构技术,利用G-P算法估计螺旋桨鸣音系统的相关维数D_2=5.1579,其计算结果为非整数,构成了系统具有混沌动力特性的必要条件。此外,根据上述分析结果,可以判定该系统的拓扑维数下界为6,即生成该复杂系统所必须的独立变量的个数不应小于6个,若要方程组封闭,则控制方程个数也不能小于6个,此研究成果可以为螺旋桨鸣音系统的进一步数值模拟研究提供理论基础。
应用q阶Renyi熵具有单调一致性的特性计算螺旋桨鸣音系统时间序列的关联函数,利用最小二乘法技术得到Kolmogorov熵的稳定估计为K_2=0.6478,即螺旋桨鸣音系统的Kolmogorov熵约为0.6478,为正的、有限值,构成该系统具有混沌动力特性的充分条件,可以作为螺旋桨鸣音系统具有混沌动力特性的判据。
应用最大Lyapunov指数的稳健估算方法,估计螺旋桨鸣音系统的最大Lyapunov指数为λ_(t_D)=0.0771,为正的有限值,构成了动力系统具有混沌动力特性的充分条件,据此也可以判定螺旋桨鸣音系统具有混沌动力特性。
通过上述分析研究,首次利用基于相空间重构技术估计螺旋桨系统时间序列的混沌不变量证明了螺旋桨鸣音系统具有混沌动力特性,此研究结果为螺旋桨鸣音现象的进一步研究奠定了理论基础。
(5)螺旋桨鸣音系统时间序列的时频特性研究:
应用局域波分解技术对螺旋桨鸣音系统时间序列进行局域波分解,通过应用相空间重构技术计算了鸣音系统时间序列及其各局域波分解分量的相关维数并估计了其最大Lyapunov指数,分析研究表明:鸣音系统时间序列的各局域波分解分量的混沌动力特征能够被有效识别。然后,利用局域波时频分析技术分析了螺旋桨鸣音系统时间序列的瞬时能量谱及频谱特性,为螺旋桨鸣音系统的目标探测与识别提供了技术支撑。
总之,本文主要回答了螺旋桨鸣音系统“是否”具有混沌动力特性的基本问题,利用实验采集到的鸣音时间序列采用基于相空间重构的思想证明了该系统具有非线性动力特性和混沌动力特性。本文的研究结果对于正确认识螺旋桨鸣音现象的混沌动力行为具有重要参考价值,也可作为对该系统数值模拟研究的理论基础——提供该系统独立变量的个数不应少于6个的重要信息。此外,本文的研究结果对螺旋桨鸣音系统的目标探测与识别具有良好的借鉴意义。
The phenomenon of the propeller singing has great influences on ship concealment, quietness and comfort. However, it has many problems which deserve further research. So far, the exact mechanism of propeller singing, which are so complex that it cannot be interpreted objectively whether by linear theory or by Kaman vortex street theory, has not been well described. Moreover, the effective control equation of propeller singing has not been established.
In the past twenty years, the rapidly developing researches on chaotic dynamics theory have shown that many complicated phenomena which appear irregular, however have deterministic in lower dimensional space, such as for time series analysis, the irregular random signals having no definite physical significance might exist some kind of certainty or even be generated by simple nonlinear governing equations, that is so-called chaotic phenomena. Then, whether the propeller singing signal which appeared complicated is some kind of chaotic phenomena and can be studied by chaotic theory? Besides, the turbulence itself has been proved to be a kind of chaotic phenomena, and the ship-radiated noise and reverberant in underwater acoustic engineering have been certified to possess chaotic dynamic characteristics. Therefore, this article not only studied on the phase-space reconstruction of propeller singing system and its complexity but also evaluated its stability by recurrence plot technique. In this article, it is the first time to use surrogate data analysis to evaluate the nonlinear characteristics of this system and to further study its chaotic dynamics. We proved the existence of chaotic dynamics of this system based on phase-space reconstruction and the estimated result of chaotic invariants for the first time and provided theoretical basis for the signal processing, detection and recognition of the ship propeller singing. Then we also studied instantaneous energy spectrum and frequency spectrum of the time-series of propeller singing system using local wave analysis which proved technological support for the detection and recognition of the propeller singing.
(1) Phase space reconstruction of time series in propeller singing system
To study the property of phase-space reconstruction on the complex nonlinear system or chaotic dynamical system, the phase-space reconstruction topology of the system can be achieved by one certain dimension or several limited dimensions of the time series embedded in more dimensions of this system. In this article, we used mutural information and the modified auto-relativity method proposed by us to estimate the optimal embedded latency for propeller singing system and then we estimated the minimal dimensions embedded in propeller singing system by using the maximal eigenvalue invariant method and average false neighborhood and we used phase-space reconstruction dual-parameter joint estimation method for analyzing the reconstruction parameters of propeller singing system.
(2) Symbolic dynamics study on time series of propeller singing system
The symbolic sequence statistical analysis was applied to analyze the time series of propeller singing system. Based on two kinds of symbolic method (the partition method and the interpolation method), we found the propeller singing signal was almost not destroyed by the noise. Furthermore, our study showed that the propeller singing signal had the feature of intense nonlinearity and determination after calculating and analyzing the Shannon entropy and irreversible parameters.
The complex sequences of propeller singing system were established by Lempel-Ziv complexity algorithm based on the Kolmogorov definition. The results showed that its normalization of complexity measures made narrow width fluctuations around 0.73. That meant, the complex sequences frequently appeared local maximum and minimum with the time series of propeller singing system jumping from left to right around zero point, however, it presented normalization of complexity measures around 0.73 (between 0 and 1) overall. All these indicated that the time series of propeller singing system had deterministic as well as complicated nonlinear characteristics.
(3) Detecting Stationarity and Nonlinearity in Propeller Singing Signal
The stationarity of propeller singing signal is tested by recurrence plot technique for the first time which is based on graphic method to test stationarity of systemic time series.
Moreover, we used surrogate data analysis to evaluate the nonlinear characteristics of this system and further studied features of three null hypotheses and four test statistics. According to surrogate data, the singing time series is proved to have nonlinearity character.
(a) The time series of propeller singing system which appeared random and irregular were proven to have deterministic for the first time by null hypotheses 1 and test statistics T_1.
(b) The time series of propeller singing system were proven the existence of nonlinear dynamical features for the first time by null hypotheses 2 and test statistics T_2 and T_3.
(c) And the nonlinearity of time series is not caused by the static nonlinear measurement function but the intrinsic character itself based on further research by null hypotheses 3 and test statistics T_(rev).
It is the first time to prove the nonlinear dynamical characteristics of propeller singing system by surrogate data. That can be the necessary condition of chaotic dynamics and provide the theoretical foundation for further study. Moreover, the analysis results could provide the technical support for the target detection and identification of propeller singing system, and equations formation of systemic numerical simulation.
(4) Study on chaotic dynamics of propeller singing system
Applying for phase-space reconstructing technique, the correlation dimension of propeller singing system was estimated as D_2 = 5.1579 by G-P algorithm. The non-integral results constituted the necessary condition of chaotic dynamics. In addition, the boundary of topological dimension was considered as 6, that is, the number of independent variable was not less than 6 that could generate this complicated system.
We calculated the correlation function of propeller singing system time series by q-order Renyi entropy which had monotonic consistency. The estimation of Kolmogorov entropy was achieved as K_2 = 0.6478 by least square method, that is, Kolmogorov entropy of propeller singing system is about 0.6478. This positive finite value provided the sufficient condition to have chaotic dynamics.
By the estimation of maximum Lyapunov exponent, we obtained a positive and finite maximum Lyapunov exponent withλ(l_D) = 0.0771. This result was considered as the evidence to judge the propeller singing system to have chaotic dynamics.
According to the forementioned analysis, it is the first time to prove the chaotic dynamics of propeller singing system by estimating the chaotic invariant of its time series. It might lay a theoretical foundation for further study the propeller singing phenomenon.
(5) Research on time-frequency characteristics of propeller singing system time series
Propeller singing system time series have been analyzed by local wave decomposition. The fractional correlation dimension and maximum Lyapunov exponent of time series and its intrinsic mode functions (IMF) have been estimated. To solve the de-noising problem, the local wave decomposition method was adopted because of its adaptive filter characteristic for distinguishing the chaotic time series and noise one. The intrinsic energy spectrum and frequency spectrum of propeller singing time series has been got, which revealed the time series has chaotic dynamical character by triple section frequency phenomenon. So it is good for target detection and identification of propeller singing system.
So the existence of chaos in propeller singing system is confirmed in experiments. The results in this dissertation are helpful to understand the nonlinear dynamics of propeller singing, and useful in the numerical simulation study. The method and algorithms about chaotic time series analysis and target detection may be referenced to dealing with other chaotic systems.
近二十年来迅猛发展的混沌动力学理论研究表明:对时间序列分析来说,传统意义上的无规的、没有明确物理意义的随机信号在低维或较低维有可能具有确定性成分,甚至是由简单的非线性控制方程产生,即所谓的混沌现象。那么貌似纷繁复杂的螺旋桨鸣音系统是否是混沌现象,能否利用混沌理论对其进行研究?再考虑到湍流本身已被证明是一种混沌现象,而水声工程中的舰船辐射噪声和混响现象也被证明具有混沌动力特性。有鉴于此,本文分别研究了螺旋桨鸣音系统的重构相空间特性,系统的复杂度特性,利用递归图技术检验了该系统的平稳性特性,应用替代数据方法检验了其非线性特性,并进一步研究了其混沌动力特性,本文解决了螺旋桨鸣音系统“是不是”混沌现象的基本问题,对其混沌动力特性分析研究可以为今后该问题的理论分析和数值模拟研究提供指导,奠定了理论基础,其关联维数揭示了系统控制方程变量的个数,其具有正的Lyapunov指数则说明该系统具有短期可预测性与长期不可预测性的矛盾统一;另外,掌握了该系统的混沌不变量特征也为螺旋桨鸣音时间序列的信号处理、目标探测与识别等提供了技术支持。
(1)螺旋桨鸣音系统相空间重构研究
对于复杂的非线性系统或是混沌动力系统,可以通过对所研究系统的某一维或有限几维时间序列向更高维空间进行嵌入而获得系统的重构相空间拓扑,研究其相空间重构特性。本文分别应用互信息法和本文提出的改进的自相关函数法估计了螺旋桨鸣音系统的最佳嵌入延迟,而后应用最大特征值不变法和平均伪邻近法估计了螺旋桨鸣音系统的最小嵌入维数,并进一步利用时延法重构相空间双参数联合估计策略计算分析了螺旋桨鸣音系统的相空间重构参数。通过上述各种分析方法得到了一致的螺旋桨鸣音系统的最佳嵌入延迟t_D=1和最小嵌入维数d_E=8,此研究成果可为螺旋桨鸣音系统混沌动力特性的后续研究提供理论基础。
(2)螺旋桨鸣音系统时间序列符号动力学研究
应用符号序列分析方法研究螺旋桨鸣音系统时间序列,分别利用分割区间法和差值法对时间序列进行符号化,通过计算分析发现本文所研究的测量信号被噪声污染程度较低,时间序列主要确定性特征明显有效,在研究过程中对噪声可以不予考虑。进一步地,计算了螺旋桨鸣音系统符号序列的Shannon熵和不可逆转性指标,结合Shannon熵的定义,可知:螺旋桨鸣音系统具有强烈非线性特征,同时又具有确定性特征,不能用简单模型进行描述,且具有时间不可逆转性。
应用基于Kolmogorov复杂性定义的Lempel-Ziv复杂度算法,建立了螺旋桨鸣音系统复杂性序列,计算结果表明:随着螺旋桨鸣音系统时间序列声压幅值在零点左侧和右侧频繁跳动,其复杂度序列也频繁地出现局部极大值和极小值,但其整体上还是表现为归一化复杂度在0.73左右,表明螺旋桨鸣音系统具有确定性特征,同时,又表现出复杂的非线性结构特性。
(3)螺旋桨鸣音系统平稳性和非线性研究
利用基于图解法的系统时间序列平稳性检验方法——递归图法,分析螺旋桨鸣音系统的平稳性,利用该方法证明了螺旋桨鸣音系统具有平稳性特征。
在此基础之上,基于替代数据思想分析了螺旋桨鸣音系统的非线性特征:
(a)应用零假设1和检验统计量T_1首次证明了看似随机、无规律可循的螺旋桨鸣音系统具有确定性特征;
(b)应用零假设2和检验统计量T_2和T_3首次证明了螺旋桨鸣音系统具有非线性动力特征;
(c)进一步地,应用零假设3和检验统计量T_(rev)证明了螺旋桨鸣音系统的非线性特征不是由静态非线性测量函数引入的,而是其本身固有的。
通过上述分析,应用替代数据法证明了螺旋桨鸣音系统具有非线性动力特征,而系统非线性作为系统具有混沌动力特性的必要条件,上述分析结果为该系统的混沌动力特性分析研究提供了理论基础。此外,螺旋桨鸣音系统非线性动力特征的证明,也为对该系统的目标探测与识别,系统数值模拟方程组的构建等提供了技术支撑。
(4)螺旋桨鸣音系统混沌动力特性研究
采用相空间重构技术,利用G-P算法估计螺旋桨鸣音系统的相关维数D_2=5.1579,其计算结果为非整数,构成了系统具有混沌动力特性的必要条件。此外,根据上述分析结果,可以判定该系统的拓扑维数下界为6,即生成该复杂系统所必须的独立变量的个数不应小于6个,若要方程组封闭,则控制方程个数也不能小于6个,此研究成果可以为螺旋桨鸣音系统的进一步数值模拟研究提供理论基础。
应用q阶Renyi熵具有单调一致性的特性计算螺旋桨鸣音系统时间序列的关联函数,利用最小二乘法技术得到Kolmogorov熵的稳定估计为K_2=0.6478,即螺旋桨鸣音系统的Kolmogorov熵约为0.6478,为正的、有限值,构成该系统具有混沌动力特性的充分条件,可以作为螺旋桨鸣音系统具有混沌动力特性的判据。
应用最大Lyapunov指数的稳健估算方法,估计螺旋桨鸣音系统的最大Lyapunov指数为λ_(t_D)=0.0771,为正的有限值,构成了动力系统具有混沌动力特性的充分条件,据此也可以判定螺旋桨鸣音系统具有混沌动力特性。
通过上述分析研究,首次利用基于相空间重构技术估计螺旋桨系统时间序列的混沌不变量证明了螺旋桨鸣音系统具有混沌动力特性,此研究结果为螺旋桨鸣音现象的进一步研究奠定了理论基础。
(5)螺旋桨鸣音系统时间序列的时频特性研究:
应用局域波分解技术对螺旋桨鸣音系统时间序列进行局域波分解,通过应用相空间重构技术计算了鸣音系统时间序列及其各局域波分解分量的相关维数并估计了其最大Lyapunov指数,分析研究表明:鸣音系统时间序列的各局域波分解分量的混沌动力特征能够被有效识别。然后,利用局域波时频分析技术分析了螺旋桨鸣音系统时间序列的瞬时能量谱及频谱特性,为螺旋桨鸣音系统的目标探测与识别提供了技术支撑。
总之,本文主要回答了螺旋桨鸣音系统“是否”具有混沌动力特性的基本问题,利用实验采集到的鸣音时间序列采用基于相空间重构的思想证明了该系统具有非线性动力特性和混沌动力特性。本文的研究结果对于正确认识螺旋桨鸣音现象的混沌动力行为具有重要参考价值,也可作为对该系统数值模拟研究的理论基础——提供该系统独立变量的个数不应少于6个的重要信息。此外,本文的研究结果对螺旋桨鸣音系统的目标探测与识别具有良好的借鉴意义。
The phenomenon of the propeller singing has great influences on ship concealment, quietness and comfort. However, it has many problems which deserve further research. So far, the exact mechanism of propeller singing, which are so complex that it cannot be interpreted objectively whether by linear theory or by Kaman vortex street theory, has not been well described. Moreover, the effective control equation of propeller singing has not been established.
In the past twenty years, the rapidly developing researches on chaotic dynamics theory have shown that many complicated phenomena which appear irregular, however have deterministic in lower dimensional space, such as for time series analysis, the irregular random signals having no definite physical significance might exist some kind of certainty or even be generated by simple nonlinear governing equations, that is so-called chaotic phenomena. Then, whether the propeller singing signal which appeared complicated is some kind of chaotic phenomena and can be studied by chaotic theory? Besides, the turbulence itself has been proved to be a kind of chaotic phenomena, and the ship-radiated noise and reverberant in underwater acoustic engineering have been certified to possess chaotic dynamic characteristics. Therefore, this article not only studied on the phase-space reconstruction of propeller singing system and its complexity but also evaluated its stability by recurrence plot technique. In this article, it is the first time to use surrogate data analysis to evaluate the nonlinear characteristics of this system and to further study its chaotic dynamics. We proved the existence of chaotic dynamics of this system based on phase-space reconstruction and the estimated result of chaotic invariants for the first time and provided theoretical basis for the signal processing, detection and recognition of the ship propeller singing. Then we also studied instantaneous energy spectrum and frequency spectrum of the time-series of propeller singing system using local wave analysis which proved technological support for the detection and recognition of the propeller singing.
(1) Phase space reconstruction of time series in propeller singing system
To study the property of phase-space reconstruction on the complex nonlinear system or chaotic dynamical system, the phase-space reconstruction topology of the system can be achieved by one certain dimension or several limited dimensions of the time series embedded in more dimensions of this system. In this article, we used mutural information and the modified auto-relativity method proposed by us to estimate the optimal embedded latency for propeller singing system and then we estimated the minimal dimensions embedded in propeller singing system by using the maximal eigenvalue invariant method and average false neighborhood and we used phase-space reconstruction dual-parameter joint estimation method for analyzing the reconstruction parameters of propeller singing system.
(2) Symbolic dynamics study on time series of propeller singing system
The symbolic sequence statistical analysis was applied to analyze the time series of propeller singing system. Based on two kinds of symbolic method (the partition method and the interpolation method), we found the propeller singing signal was almost not destroyed by the noise. Furthermore, our study showed that the propeller singing signal had the feature of intense nonlinearity and determination after calculating and analyzing the Shannon entropy and irreversible parameters.
The complex sequences of propeller singing system were established by Lempel-Ziv complexity algorithm based on the Kolmogorov definition. The results showed that its normalization of complexity measures made narrow width fluctuations around 0.73. That meant, the complex sequences frequently appeared local maximum and minimum with the time series of propeller singing system jumping from left to right around zero point, however, it presented normalization of complexity measures around 0.73 (between 0 and 1) overall. All these indicated that the time series of propeller singing system had deterministic as well as complicated nonlinear characteristics.
(3) Detecting Stationarity and Nonlinearity in Propeller Singing Signal
The stationarity of propeller singing signal is tested by recurrence plot technique for the first time which is based on graphic method to test stationarity of systemic time series.
Moreover, we used surrogate data analysis to evaluate the nonlinear characteristics of this system and further studied features of three null hypotheses and four test statistics. According to surrogate data, the singing time series is proved to have nonlinearity character.
(a) The time series of propeller singing system which appeared random and irregular were proven to have deterministic for the first time by null hypotheses 1 and test statistics T_1.
(b) The time series of propeller singing system were proven the existence of nonlinear dynamical features for the first time by null hypotheses 2 and test statistics T_2 and T_3.
(c) And the nonlinearity of time series is not caused by the static nonlinear measurement function but the intrinsic character itself based on further research by null hypotheses 3 and test statistics T_(rev).
It is the first time to prove the nonlinear dynamical characteristics of propeller singing system by surrogate data. That can be the necessary condition of chaotic dynamics and provide the theoretical foundation for further study. Moreover, the analysis results could provide the technical support for the target detection and identification of propeller singing system, and equations formation of systemic numerical simulation.
(4) Study on chaotic dynamics of propeller singing system
Applying for phase-space reconstructing technique, the correlation dimension of propeller singing system was estimated as D_2 = 5.1579 by G-P algorithm. The non-integral results constituted the necessary condition of chaotic dynamics. In addition, the boundary of topological dimension was considered as 6, that is, the number of independent variable was not less than 6 that could generate this complicated system.
We calculated the correlation function of propeller singing system time series by q-order Renyi entropy which had monotonic consistency. The estimation of Kolmogorov entropy was achieved as K_2 = 0.6478 by least square method, that is, Kolmogorov entropy of propeller singing system is about 0.6478. This positive finite value provided the sufficient condition to have chaotic dynamics.
By the estimation of maximum Lyapunov exponent, we obtained a positive and finite maximum Lyapunov exponent withλ(l_D) = 0.0771. This result was considered as the evidence to judge the propeller singing system to have chaotic dynamics.
According to the forementioned analysis, it is the first time to prove the chaotic dynamics of propeller singing system by estimating the chaotic invariant of its time series. It might lay a theoretical foundation for further study the propeller singing phenomenon.
(5) Research on time-frequency characteristics of propeller singing system time series
Propeller singing system time series have been analyzed by local wave decomposition. The fractional correlation dimension and maximum Lyapunov exponent of time series and its intrinsic mode functions (IMF) have been estimated. To solve the de-noising problem, the local wave decomposition method was adopted because of its adaptive filter characteristic for distinguishing the chaotic time series and noise one. The intrinsic energy spectrum and frequency spectrum of propeller singing time series has been got, which revealed the time series has chaotic dynamical character by triple section frequency phenomenon. So it is good for target detection and identification of propeller singing system.
So the existence of chaos in propeller singing system is confirmed in experiments. The results in this dissertation are helpful to understand the nonlinear dynamics of propeller singing, and useful in the numerical simulation study. The method and algorithms about chaotic time series analysis and target detection may be referenced to dealing with other chaotic systems.
引文
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