Novel Nonlinearity Extracting Method of Diverse Music Signals Based on Chaotic Techniques for Musical Processing System

The complexity and diversity of different music styles created by different composers make music evaluation a very difficult problem. To quantitatively evaluate the quality of music pronunciation, this work proposes a method for extracting nonlinear features of music signals. By selecting different styles of music signals, based on chaos theory, fractal theory, and Lyapunov exponent, the change in correlation dimension of different kinds of music signals is analyzed from a nonlinear perspective, so as to realize vocal music evaluation and training in music teaching. The results showed that the filter used in the study has a good noise-reduction effect. The power spectrum of the music signal was qualitatively analyzed, and the music signal was characterized by chaos. The difference between the maximum and minimum values of the Lyapunov exponent of the For Alice music signal was 0.244, which is larger than the other music signals. After the correlation dimension analysis, the correlation dimension of the Croatian Rhapsody music signal varied strongly. The average correlation dimension of the five music signals fluctuated above and below the corresponding average correlation dimension value at order 0, and the music signals had stable nonlinear features. The research method of the article is conducive to deepening the understanding of music signal patterns and has a certain promotion effect on music teaching.

Music, as a complex form of human expression, is intrinsically dynamic, with nonlinear characteristics in its structure and behavior over time. Music expresses a variety of emotional qualities and can be represented as a nonlinear dynamical system that frequently exhibits persistence. Traditional analysis approaches are frequently insufficient for capturing these complex dynamics. The research is to utilize Ordinary Differential Equations (ODEs) to investigate the dynamic properties of music, with focus on the nonlinear behavior observed in musical signals. The method involves breaking down music into its core components, examining local characteristics, and generalizing the underlying features. Using ODEs, the system's Lyapunov exponents to measure stability and chaotic behavior, the energy spectrum to analyze oscillatory modes, and the correlation dimension to comprehend the fractality inherent in music signalswere investigated. These strategies use numerical simulations to characterize music's nonlinear features. The findings show that ODE-based approaches can accurately simulate important dynamic characteristics of music, with Lyapunov exponents exposing chaotic behavior the energy spectrum demonstrating oscillatory patterns, and the correlation dimension emphasizing the fractal form of music. Finally, the research verifies the use of differential equations (DE) to explain the dynamic properties of music. The method provides important insights into the nonlinear dynamics of musical signals, paving the way for more detailed models and applications in music analysis.

Music signals are usually generated by mixing many music source signals, such as various instrumental sounds and vocal sounds, and they are often represented as 2-channel signals (i.e., stereo channel signals). Underdetermined source separation for separating the music signals into individual music source signals is a potential technique to develop various applications, such as music transcription, singer discrimination, and vocal extraction. One of the most powerful underdetermined source separation methods is Nonnegative Matrix Factorization (NMF) that models a power spectrogram of an observation signal as a product of two nonnegative matrices; basis and activation matrices. To apply NMF to the stereo channel music signal separation, we have proposed Nonnegative Tensor Factorization (NTF) by further implementing a gain matrix to represent mixing (i.e., panning) information. However, the separation performance of this method is insufficient owing to less prior information to model acoustic characteristics of the individual music source signals. To address this issue, we propose a cepstrum regularization method for NTF to make power spectral envelopes of the separated signals close to those of individual music source signals. We conduct experimental evaluations to investigate the effectiveness of the regularization method and show remaining problems to be addressed.

Nonlinear analysis using Lyapunov exponents in breast thermograms to identify abnormal lesions

The authors computed the correlation dimensions of the synchronous 12-lead ECG of 30 healthy persons and 30 coronary heart disease (CHD) patients. They found that the D/sub 2/ of the same person is different for different lead ECG signals. Therefore, it is not appropriate to describe the human heart, a nonlinear dynamic system with complex structures, by only a single fractal dimension. Synchronous 12-lead analysis of correlation dimension can present a more detailed description of the heart's dynamic characteristics than can a simple one-lead analysis. It is important to specify the particular lead position when one refers to the D/sub 2/'s of the ECG signal. The authors also find that the differences among D/sub 2/'s of 12-lead ECG signals are relatively small for healthy people, compared with the same lead for CHD patients. For CHD patients, the situation is complicated. The differences among D/sub 2/'s for 12-lead ECG signals of some patients are large. Correlation dimensions of chest-lead ECG signals are obviously smaller than those of limb-lead ECG signals. This suggests that the complexity of the ECG attractor is reduced with cardiac muscle ischemia, It may be that D/sub 2/'s of synchronous 12-lead ECG signals may be able to provide valuable diagnostic information about the location of ischemia or infarction. The authors' research shows that the correlation dimension is a valuable parameter in analyzing nonlinear dynamic characteristics of the ECG signal and may provide new insights and methods in diagnosis of cardiac disease.

Advancing knowledge about the phase space topologies of nonlinear hydrodynamic or dynamical systems has raised the question of whether the structure of the attractors in which the solutions are eventually confined can be characterized rigorously and economically. It is shown by applying the Lyapunov exponents, Lyapunov dimension, and correlation dimension to several low-order truncated spectral models that these quantities give useful information about the phase space structure and predictability characteristics of such attractors. The Lyapunov exponents measure the average exponential rate of convergence or divergence of nearby solution trajectories in an appropriate phase space. The Lyapunov dimension dL incorporates the dynamical information of the Lyapunov exponents to give an estimate of the dimension of the system attractor, while the correlation dimension v is a more geometrically motivated measure that is simple to compute and related to more classical dimensions. The Lyapunov exponents detect bifurcations between solution regimes and also subtle predictability differences between attractors. As measures of chaotic attractor dimension, v>dL in all cases, and the ratio v/dL is smallest at values of the forcing just above the transition to chaos. Changes in the Lyapunov dimension are concentrated in a small range of forcing values, while the correlation dimension varies more uniformly. The value of dL is tied closely to the number of positive Lyapunov exponents, while v is more sensitive to the magnitude of the chaotic component of the system. Variations in these measures for a hierarchy of convection models support the idea that the appearance of strong chaos in two-dimensional models is truncation-related, and can be delayed to arbitrarily large forcing if enough modes are included.

Introduction. Problems and methods of finding Н∞ – control are the basis of modern control theory. They are actively used to develop robust controllers, especially in aircraft control systems under limited external actions. These methods allow for adapting control systems to changing environmental conditions, which is critically important for providing the reliability and safety of aircraft operation. Current research is aimed at improving approaches to the synthesis of controllers covering both linear and nonlinear dynamic systems. In this context, special attention is paid to the integration of new mathematical methods, such as linear matrix inequalities and frequency analysis, which allows for optimizing the system response to various external actions and providing protection against unexpected conditions. It is important to note that, despite the progress made in this area, significant problems remain unsolved regarding the analysis and synthesis of controllers for nonlinear systems. This necessitates further research and development in this promising area. In this paper, in order to fill the existing gap, sufficient conditions for the existence of control for one of the frequently encountered classes of nonlinear systems are formulated and proven, which will then be used as a theoretical basis for developing approximate algorithms for finding it.Materials and Methods. The basic research tool was the Н∞ – control synthesis methods based on the minimax approach, which consisted in finding the control law under the worst external action. In this context, it was proposed to prove sufficient conditions for the existence of control using the extension principle. However, due to the computational difficulties that might arise when applying those conditions, it was decided to simplify the initial formulation of the problem. The simplification process was performed by approximate replacing the nonlinear system with another nonlinear system, which was similar in structure to the linear one, using the factorization procedure. This approach made it possible to use the solution of the Riccati equation, whose coefficients depended on the state vector, for the synthesis of controllers. To solve model examples and applied problems, a software package was developed using the MATLAB mathematical package.Results. The article solved the problem of synthesis of Н∞ – control of the state of nonlinear continuous dynamic systems, linear in control and disturbance. Sufficient conditions for the existence of Н∞ – control were formulated and proved on the basis of the extension principle. An approximate method was proposed that provided solving the problem of finding control laws for dynamic systems that were nonlinear in state, similar to the methods used for linear systems. Analytical solutions were found for two model examples, which were illustrated by graphs of transient processes to demonstrate the results of numerical modeling of the considered nonlinear dynamic systems in the presence of external actions.Discussion and Conclusion. The proposed approximate algorithm for synthesizing state and output controllers guarantees the required quality of transient processes and asymptotic stability of closed nonlinear control systems. This significantly expands the class of dynamic systems for which it is possible to synthesize controllers capable of resisting various external actions. The methods presented in this paper can be effectively applied to solve a variety of control problems, including the design of autopilots and automatic navigation systems for aircraft, even under conditions of limited external actions.

Since chaotic phenomena of heart rate have been observed, investigations by time series analysis have been focused on the non-linear dynamics of the cardiac control system. Without the necessity to understand the functional structures of the complex regulatory network that controls heart rate, non-linear methods are a useful approach to characterize the processes inside this black box. We applied established linear and non-linear methods to obtain comprehensive information regarding heart rate control and its relation to the respiratory system. To study normal regulation under various conditions, 42 healthy children were investigated during different vigilance stages. The parameters of heart rate power spectra were estimated, the linear intensity of cardiorespiratory coupling was concluded from the coherence spectra. As to non-linear properties of heart rate, the largest Lyapunov exponents as well as the correlation dimension were determined. Similarly, the correlation dimension of the respiratory signals was evaluated. The total power of the heart rate spectrum was found to be greatest during REM, it decreased during wakefulness and was low in nonREM-sleep. These variations are mainly accounted for by low frequency power. The “complexity” of heart rate, as indicated by the correlation dimension, is diminished during sleep phases, whereas the Lyapunov exponents are less affected. The cardiorespiratory coherence is strongly modulated by vigilance with an increase during nonREM and lowest values during REM. The complexity of respiration was also affected by vigilance. A different behavior of heart rate complexity was found during REM-phases. Concluded from spectral analysis, we suggest a specific setting of autonomic heart rate regulation for each vigilance stage. A low dimensional deterministic chaos is present in heart rate time series. More independent control loops were found to be active during wakefulness. Revealed by parameters of the non-linear dynamics, different stages of vigilance determine different operating points in the cardiorespiratory coordination.

In terms of the chaos theory, the phase-space-reconstruction method has been employed to describe the multi-dimensional phase space for the time series of air pollution index (API) during the past 10 years in Lanzhou, northwest China. The mutual information and Cao method were used to determine the reconstruction parameters, and the characteristic quantities including the Lyapunov exponent and the correlation dimension were calculated respectively. As a result, the correlation dimensions were fractioned, and the maximum Lyapunov exponent (λ 1) > 0. It shows that these presented the obvious chaotic characteristics that resulted from the evolution of non-linear chaotic dynamic system in the time series of air pollution index over the past 10 years. In the meanwhile, three or even four main dynamic variables were discussed here that could effectively interpret the changes of air pollution index time series and their causes. Some reasonable preventive countermeasures were thus put forward. These findings might provide a scientific basis for probing further into the regional complexity and evolution of the time series of air pollution index.

Musical instruments are usually distinguished by their sound produced through human perception which may lead to misinterpretation due to auditory perception bias and other disturbances. Therefore, recognition using music signals is carried out to help characterize signals from different musical instruments. Fractal analysis is a mathematical tool used to study complex and irregular patterns in various systems. In this study, fractal analysis was used to study and analyze musical notes signal data from different instruments. The fractal analysis method used is the box counting method. The traditional musical instruments involved in this study are the seruling, cak lempong, kompang, and gambus. Matlab software was used to analyze the musical data signal. First, the fractal dimension of the music data signal in the form of time domain is calculated. Then, the mean and standard deviation of the fractal dimension values were determined to recognize different musical instruments. Additionally, different image resolutions and box sizes are also used to calculate the fractal dimension of musical instrument from the time domain data. The error of best fit line, E will also be calculated to ensure the reliability of box counting dimensions using leastsquares regression method. The results show that the value of the fractal dimension for all 18 data is between 1.2636 to 1.7543. In terms of musical instrument recognition, the recognition of seruling and kompang using mean fractal dimension is successful. However, this method is less suitable for identification for cak lempong and gambus due to the high standard deviation. By using different image resolutions and box sizes, the accuracy of fractal dimension values will be affected. Increasing these parameters increases the accuracy of the results. The results of the study show that an image with a resolution of 1024 x 1024 pixels and a scaling factor of 9 is suitable to analyze the fractal characteristics of musical instrument data signals by using the box counting method.

This study characterized drought propagation in the Han River basin based on chaos theory and nonlinear dynamic systems. The nonlinear complexity of drought propagation, specifically between meteorological and hydrological droughts, was identified using convergent cross-mapping and phase-space reconstruction techniques. The analysis results on the causality of drought propagation between meteorological and hydrological droughts revealed that among the 24 middle-sized watersheds of the Han River basin, 11 watersheds had bidirectional causality, and the other 13 watersheds had one-directional causality. This study also confirmed that there were large correlations between the two drought types when 12 of the 13 watersheds with one-directional causality had a lag time of 4 months and the remaining watershed had a delay time of 7 months. The complexity of the interaction between different droughts was confirmed through adjustments of key parameters, such as the embedding dimension, time delay, correlation dimension, and Lyapunov exponent. The approach proposed in this study made it possible to analyze the nonlinear dynamic characteristics of drought propagation, which have not been analyzed in previous studies.

Background: It is better to consider the formation of coal and rock nonlinear complex system dynamics role in the cutter, using nonlinear dynamics theory and methods, in order to study rocks and coal cutter picks nonlinear dynamical systems effectively. Some patents have been discussed in this article. Objective: The purpose of the article is to study the characteristics of cutting load, especially the load characteristics of different cutting material parameters for the coal seam with coal and rock interface cutting. Method: The phase space was reconstructed by cutting load test data, through which the strange attractor phase diagram of cutting load was obtained. Moreover, this method can describe the load characteristics of cutting conditions above well. Based on determining the embedding dimension, the cutting load correlation dimensions and Lyapunov exponents of different working conditions were obtained. Results: Analyses show that the value ranges of correlation dimension and Lyapunov exponents are in accord with the chaotic characteristics, which proves that the single pick cutting system has chaotic characteristics; the corresponding value has some variation. In the scope of this study, different cutting conditions can be distinguished by these values. Finally, the cutting system characteristics can be studied by taking the complexity as measure standard, and it can be obtained that the complexity of cutting system will increase with the cutting object variation, which provides guidance significance for the system chaotic characteristics study. Conclusion: The similarities and differences of coal seam with coal and rock interface, homogeneous coal seam and homogeneous rock seam can be seen by the above contrast. Keywords: Coal seam with coal and rock interface, complexity, correlation dimension, cutting load, Lyapunov exponents, phase space reconstructed.

Some quite recent results from the area of Deterministic Chaos have considerable significance for practitioners of Information Theory. In 1959 Kolmogorov observed that Shannon’s probabilistic theory of information could be applied to symbolic encodings of the phase-space descriptions of physical non-linear dynamical systems so that one might characterise a process in terms of its Kolmogorov-Sinai entropy. Pesin’s theorem in 1977, proved that for certain deterministic non-linear dynamical systems exhibiting chaotic behaviour, the Kolmogorov-Sinai entropy h KS is given by the sum of the positive Lyapunov exponents for the process, i.e. h KS = ∑ i λ + i [3]. For a number of simple non-linear processes the Lyapunov exponents may be computed very precisely. Thus a non-linear dynamical systems may be viewed as an information source whose corresponding source entropy is accurately known. The existence of simple ‘calibrated’ sources such as the logistic map (ẋ = f(x) = rx(1−x)), [3] provides a means for precisely evaluating the performance of compression schemes, but also information measures such as the grammar based measures described in [1][4]. In respect of [1][2], the authors have computed the average T-entropy from sample encodings of the symbolic dynamics for the logistic map, and compared these values directly with the corresponding known Lyapunov exponents. As the Figure below shows, the average T-entropy for strings of 10 bits long closely agree with the positive Lyapunov exponents for the one dimensional dynamical system. The values are plotted here as a function of the system parameter r at increments of 0.0001 . The difference between the T-entropy and Lyapunov exponents averages about 1% RMS of full scale over the whole range, r ∈ [0, ln(2)]. Such agreement may be interpreted as strong evidence of the link between this grammar based information measure for finite strings and the probabilistic entropy measure of Shannon for information sources. That the process imbues individual finite sample strings with its corresponding information characteristics, echoes the our quotation from Kolmogorov. Clearly the T-entropy reflects the fine structure of chaotic attractors. Thus Deterministic Information Theory [2] appears to offer a new approach for evaluating the limits of compression for individual finite strings, while Deterministic Chaos Theory results further allow one to select precisely calibrated information sources to be used to assess the performance of specific compression algorithms.

It is well known that the structural phase transitions in ferroelectric materials are connected with strong nonlinear properties. So we should expect all features of nonlinear dynamical systems such as period-doubling-cascades and chaotic behaviour in a dynamical system that contains ferroelectric materials. The experimentally investigated nonlinear dynamical system is a series resonance circuit, consisting of a linear inductance and a nonlinear capacitance. Different ferroelectric materials have been used as a nonlinear capacitance. There are two directions for our investigations: (i) The response-function of the nonlinear dynamical system and analysis methods of the nonlinear dynamics for dissipative chaotic systems (e. g. phase portrait, Fourier spectra, correlation dimension and Lyapunov exponents) are used for the study of structural phase-transitions and for the characterization of the large signal behaviour of ferroelectric materials. (ii) Controlling the chaotic behaviour of the system by applying a small perturbation of an external control parameter to it. This way the chaotic motion of the system can be switched over to a regular one by stabilizing one of the many regular instable states of the system.

The most important numerical tools needed in the analysis of chaotic systems performing chaos synchronization and chaotic communications are discussed in this chapter. Basic concepts, theoretical framework, and computer algorithms are reviewed. The subjects covered include the concepts and numerical simulations of stochastic nonlinear systems, the complexity of a chaotic attractor measured by Lyapunov exponents and correlation dimension, the robustness of synchronization measured by the transverse Lyapunov exponents in parameter-matched systems and parameter-mismatched systems, the quality of synchronization measured by the correlation coefficient and the synchronization error, and the treatment of channel noise for quantifying the performance of a chaotic communication system. For a dynamical system described by stochastic differential equations, the integral of a stochastic term is very different from that of a deterministic term. The difference and connection between two different stochastic integrals in the Ito and Stratonovich senses, respectively, are discussed. Numerical algorithms for the simulation of stochastic differential equations are developed. Two quantitative measures, namely, the Lyapunov exponents and the correlation dimension, for a chaotic attractor are discussed. Numerical methods for calculating these parameters are outlined. The robustness of synchronization is measured by the transverse Lyapunov exponents. Because perfect parameter matching between a transmitter and a receiver is generally not possible in a real system, a new concept of measuring the robustness of synchronization by comparing the unperturbed and perturbed receiver attractors is introduced for a system with parameter mismatch. For the examination of the quality of synchronization, the correlation coefficient and the synchronization error obtained by comparing the transmitter and the receiver outputs are used. The performance of a communication system is commonly measured by the bit-error rate as a function of signal-to-noise ratio. In addition to the noise in the transmitter and the receiver, the noise of the communication channel has to be considered in evaluating the bit-error rate and signal-to-noise ratio of the system. An approach to integrating the linear and nonlinear effects of the channel noise into the system consistently is addressed. Optically injected single-mode semiconductor lasers are used as examples to demonstrate the use of these numerical tools.