Search results for: fractional chaotic systems

Authors: Meng Hu, Lili Wang

Abstract:

In this paper, applying He-s energy balance method to determine frequency formulation relations of nonlinear oscillators with discontinuous term or fractional potential. By calculation and computer simulations, compared with the exact solutions show that the results obtained are of high accuracy.

Keywords: He's energy balance method, periodic solution, nonlinear oscillator, discontinuous, fractional potential.

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Authors: Nisa Özge Önal, Kamil Karaçuha, Göksu Hazar Erdinç, Banu Bahar Karaçuha, Ertuğrul Karaçuha

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The study aims to use a mathematical approach with the fractional calculus which is developed to have the ability to continuously analyze the factors related to the children’s height development. Until now, tracking the development of the child is getting more important and meaningful. Knowing and determining the factors related to the physical development of the child any desired time would provide better, reliable and accurate results for childcare. In this frame, 7 groups for height percentile curve (3th, 10th, 25th, 50th, 75th, 90th, and 97th) of Turkey are used. By using discrete height data of 0-18 years old children and the least squares method, a continuous curve is developed valid for any time interval. By doing so, in any desired instant, it is possible to find the percentage and location of the child in Percentage Chart. Here, with the help of the fractional calculus theory, a mathematical model is developed. The outcomes of the proposed approach are quite promising compared to the linear and the polynomial method. The approach also yields to predict the expected values of children in the sense of height.

Keywords: Children growth percentile, children physical development, fractional calculus, linear and polynomial model.

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Authors: R.M. López-Gutiérrez, C. Cruz-Hernández, C. Posadas-Castillo, E.E.García-Guerrero

Abstract:

Encoded information based on synchronization of coupled chaotic Nd:YAG lasers in master-slave configuration is numerically studied. Encoding, transmission, and decoding of information in optical chaotic communication with a single channel is presented. We analyze the robustness of the encrypted audio transmission in a channel noise. In order to illustrate this synchronization robustness, we present two cases of study: synchronization and transmission with a single channel without and with noise in the channel.

Keywords: Encryption, Secure coomunication, Chaos, Synchronization, Complex networks, Nd:YAG laser.

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Authors: Samira Mabtoul, Elhassan Ibn Elhaj, Driss Aboutajdine

Abstract:

In this paper a new robust digital image watermarking algorithm based on the Complex Wavelet Transform is proposed. This technique embeds different parts of a watermark into different blocks of an image under the complex wavelet domain. To increase security of the method, two chaotic maps are employed, one map is used to determine the blocks of the host image for watermark embedding, and another map is used to encrypt the watermark image. Simulation results are presented to demonstrate the effectiveness of the proposed algorithm.

Keywords: Image watermarking, Chaotic map, DT-CWT.

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Authors: Pei. P. Tan, Don. U.A. Galagedera, Elizabeth A.Maharaj

Abstract:

This paper examines predictability in stock return in developed and emergingmarkets by testing long memory in stock returns using wavelet approach. Wavelet-based maximum likelihood estimator of the fractional integration estimator is superior to the conventional Hurst exponent and Geweke and Porter-Hudak estimator in terms of asymptotic properties and mean squared error. We use 4-year moving windows to estimate the fractional integration parameter. Evidence suggests that stock return may not be predictable indeveloped countries of the Asia-Pacificregion. However, predictability of stock return insome developing countries in this region such as Indonesia, Malaysia and Philippines may not be ruled out. Stock return in the Thailand stock market appears to be not predictable after the political crisis in 2008.

Keywords: Asia-Pacific stock market, long-memory, return predictability, wavelet

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Authors: Ruming Yin, Jian Yuan, Qiuhua Yang, Xiuming Shan, Xiqin Wang

Abstract:

Linear cryptanalysis methods are rarely used to improve the security of chaotic stream ciphers. In this paper, we apply linear cryptanalysis to a chaotic stream cipher which was designed by strictly using the basic design criterion of cryptosystem – confusion and diffusion. We show that this well-designed chaos-based stream cipher is still insecure against distinguishing attack. This distinguishing attack promotes the further improvement of the cipher.

Keywords: Stream cipher, chaos, linear cryptanalysis, distinguishing attack.

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Authors: Raja Muhammad Asif Zahoor, Junaid Ali Khan, I. M. Qureshi

Abstract:

In this article an evolutionary technique has been used for the solution of nonlinear Riccati differential equations of fractional order. In this method, genetic algorithm is used as a tool for the competent global search method hybridized with active-set algorithm for efficient local search. The proposed method has been successfully applied to solve the different forms of Riccati differential equations. The strength of proposed method has in its equal applicability for the integer order case, as well as, fractional order case. Comparison of the method has been made with standard numerical techniques as well as the analytic solutions. It is found that the designed method can provide the solution to the equation with better accuracy than its counterpart deterministic approaches. Another advantage of the given approach is to provide results on entire finite continuous domain unlike other numerical methods which provide solutions only on discrete grid of points.

Keywords: Riccati Equation, Non linear ODE, Fractional differential equation, Genetic algorithm.

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Authors: C. C. Wang, N. S. Pai, H. T. Yau, T. T. Liao, M. J. Jang, C. W. Lee, W. M. Hong

Abstract:

Horizontal platform system (HPS) is popularly applied in offshore and earthquake technology, but it is difficult and time-consuming for regulation. In order to understand the nonlinear dynamic behavior of HPS and reduce the cost when using it, this paper employs differential transformation method to study the bifurcation behavior of HPS. The numerical results reveal a complex dynamic behavior comprising periodic, sub-harmonic, and chaotic responses. Furthermore, the results reveal the changes which take place in the dynamic behavior of the HPS as the external torque is increased. Therefore, the proposed method provides an effective means of gaining insights into the nonlinear dynamics of horizontal platform system.

Keywords: horizontal platform system, differentialtransformation method, chaotic.

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Authors: Elif Derya UBEYLI, Inan GULER

Abstract:

A new approach based on the consideration that electroencephalogram (EEG) signals are chaotic signals was presented for automated diagnosis of electroencephalographic changes. This consideration was tested successfully using the nonlinear dynamics tools, like the computation of Lyapunov exponents. This paper presented the usage of statistics over the set of the Lyapunov exponents in order to reduce the dimensionality of the extracted feature vectors. Since classification is more accurate when the pattern is simplified through representation by important features, feature extraction and selection play an important role in classifying systems such as neural networks. Multilayer perceptron neural network (MLPNN) architectures were formulated and used as basis for detection of electroencephalographic changes. Three types of EEG signals (EEG signals recorded from healthy volunteers with eyes open, epilepsy patients in the epileptogenic zone during a seizure-free interval, and epilepsy patients during epileptic seizures) were classified. The selected Lyapunov exponents of the EEG signals were used as inputs of the MLPNN trained with Levenberg- Marquardt algorithm. The classification results confirmed that the proposed MLPNN has potential in detecting the electroencephalographic changes.

Keywords: Chaotic signal, Electroencephalogram (EEG) signals, Feature extraction/selection, Lyapunov exponents

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Authors: H. B. Kekre, Tanuja K. Sarode

Abstract:

This paper presents a very simple and efficient algorithm for codebook search, which reduces a great deal of computation as compared to the full codebook search. The algorithm is based on sorting and centroid technique for search. The results table shows the effectiveness of the proposed algorithm in terms of computational complexity. In this paper we also introduce a new performance parameter named as Average fractional change in pixel value as we feel that it gives better understanding of the closeness of the image since it is related to the perception. This new performance parameter takes into consideration the average fractional change in each pixel value.

Keywords: Vector Quantization, Data Compression, Encoding, Searching.

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Authors: Koushlendra Kumar Singh, Manish Kumar Bajpai, Rajesh K. Pandey

Abstract:

The edges of low contrast images are not clearly distinguishable to human eye. It is difficult to find the edges and boundaries in it. The present work encompasses a new approach for low contrast images. The Chebyshev polynomial based fractional order filter has been used for filtering operation on an image. The preprocessing has been performed by this filter on the input image. Laplacian of Gaussian method has been applied on preprocessed image for edge detection. The algorithm has been tested on two test images.

Keywords: Chebyshev polynomials, Fractional order differentiator, Laplacian of Gaussian (LoG) method, Low contrast image.

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Authors: Anwar H. Jarndal, Ahmed S. Elwakil

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In this paper, a fraction-order model for pad parasitic effect of GaN HEMT on Si substrate is developed and validated. Open de-embedding structure is used to characterize and de-embed substrate loading parasitic effects. Unbiased device measurements are implemented to extract parasitic inductances and resistances. The model shows very good simulation for S-parameter measurements under different bias conditions. It has been found that this approach can improve the simulation of intrinsic part of the transistor, which is very important for small- and large-signal modeling process.

Keywords: Fractional-order modeling, GaN HEMT, Si-substrate, open de-embedding structure.

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Authors: Navid Javidtash, Abdolmohamad Davodi, Mojtaba Hakimzadeh, Abdolreza Roozbeh

Abstract:

Economic dispatch (ED) is considered to be one of the key functions in electric power system operation. This paper presents a new hybrid approach based genetic algorithm (GA) to economic dispatch problems. GA is most commonly used optimizing algorithm predicated on principal of natural evolution. Utilization of chaotic queue with GA generates several neighborhoods of near optimal solutions to keep solution variation. It could avoid the search process from becoming pre-mature. For the objective of chaotic queue generation, utilization of tent equation as opposed to logistic equation results in improvement of iterative speed. The results of the proposed approach were compared in terms of fuel cost, with existing differential evolution and other methods in literature.

Keywords: Economic Dispatch(ED), Optimization, Fuel Cost, Genetic Algorithm (GA).

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Authors: P. Chaipanya, P. Uthansakul, M. Uthansakul

Abstract:

This paper proposes a vertical beamforming concept to a cellular network employing Fractional Frequency Reuse technique including with cell sectorization. Two different beams are utilized in cell-center and cell-edge, separately. The proposed concept is validated through computer simulation in term of SINR and channel capacity. Also, comparison when utilizing horizontal and vertical beam formation is in focus. The obtained results indicate that the proposed concept can improve the performance of the cellular networks comparing with the one using horizontal beamforming.

Keywords: Beamforming, Fractional Frequency Reuse, Inter- Cell Interference, cell sectorization.

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Authors: Alhasan Hakami, Arun Kumar, Sung J Shim, Yousef Abu Nahleh

Abstract:

In recent years, many high intensity earthquakes have occurred around the world, such as the 2011 earthquake in Tohoku, Japan. These large-scale disasters caused huge casualties and losses. In addition, inefficient disaster response operations also caused the second wave of casualties and losses, and expanded the damage. Effective disaster management can be used to respond to the chaotic situation, and reduce the damage; however, some inefficient disaster response operations are still used. Therefore, this case study chose the 921 earthquake for analyzing disaster emergency logistics problems and proposed the Soft Systems Methodology (SSM) to solve disaster emergency logistics problems. Moreover, it analyses the effect of human factors on system operation, and suggests a solution to improve the system.

Keywords: Soft systems methodology, emergency logistics.

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Authors: A.Goel, M. Agrawal, P. Gupta Poddar

Abstract:

Selected Mapping (SLM) is a PAPR reduction technique, which converts the OFDM signal into several independent signals by multiplication with the phase sequence set and transmits one of the signals with lowest PAPR. But it requires the index of the selected signal i.e. side information (SI) to be transmitted with each OFDM symbol. The PAPR reduction capability of the SLM scheme depends on the selection of phase sequence set. In this paper, we have proposed a new phase sequence set generation scheme based on M-ary chaotic sequence and a mapping scheme to map quaternary data to concentric circle constellation (CCC) is used. It is shown that this method does not require SI and provides better SER performance with good PAPR reduction capability as compared to existing SLMOFDM methods.

Keywords: Orthogonal frequency division multiplexing (OFDM), Peak-to-average power ratio (PAPR), Selected mapping (SLM), Side information (SI)

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Authors: Puntani Pongsumpun, I-Ming Tang

Abstract:

The incidences of dengue hemorrhagic disease (DHF) over the long term exhibit a seasonal behavior. It has been hypothesized that these behaviors are due to the seasonal climate changes which in turn induce a seasonal variation in the incubation period of the virus while it is developing the mosquito. The standard dynamic analysis is applied for analysis the Susceptible-Exposed- Infectious-Recovered (SEIR) model which includes an annual variation in the length of the extrinsic incubation period (EIP). The presence of both asymptomatic and symptomatic infections is allowed in the present model. We found that dynamic behavior of the endemic state changes as the influence of the seasonal variation of the EIP becomes stronger. As the influence is further increased, the trajectory exhibits sustained oscillations when it leaves the chaotic region.

Keywords: Chaotic behavior, dengue hemorrhagic fever, extrinsic incubation period, SEIR model.

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Authors: Yusuke Uchiyama, Hidetoshi Konno

Abstract:

We have investigated statistical properties of the defect turbulence in 1D CGLE wherein many body interaction is involved between local depressing wave (LDW) and local standing wave (LSW). It is shown that the counting number fluctuation of LDW is subject to the sub-Poisson statistics (SUBP). The physical origin of the SUBP can be ascribed to pair extinction of LDWs based on the master equation approach. It is also shown that the probability density function (pdf) of inter-LDW distance can be identified by the hyper gamma distribution. Assuming a superstatistics of the exponential distribution (Poisson configuration), a plausible explanation is given. It is shown further that the pdf of amplitude of LDW has a fattail. The underlying mechanism of its fluctuation is examined by introducing a generalized fractional Poisson configuration.

Keywords: sub-Poisson statistics, hyper gamma distribution, fractional Poisson configuration.

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Authors: Farhad Asadi, S. Hossein Sadati

Abstract:

This paper presents a prediction performance of feedforward Multilayer Perceptron (MLP) and Echo State Networks (ESN) trained with extended Kalman filter. Feedforward neural networks and ESN are powerful neural networks which can track and predict nonlinear signals. However, their tracking performance depends on the specific signals or data sets, having the risk of instability accompanied by large error. In this study we explore this process by applying different network size and leaking rate for prediction of nonlinear or chaotic signals in MLP neural networks. Major problems of ESN training such as the problem of initialization of the network and improvement in the prediction performance are tackled. The influence of coefficient of activation function in the hidden layer and other key parameters are investigated by simulation results. Extended Kalman filter is employed in order to improve the sequential and regulation learning rate of the feedforward neural networks. This training approach has vital features in the training of the network when signals have chaotic or non-stationary sequential pattern. Minimization of the variance in each step of the computation and hence smoothing of tracking were obtained by examining the results, indicating satisfactory tracking characteristics for certain conditions. In addition, simulation results confirmed satisfactory performance of both of the two neural networks with modified parameterization in tracking of the nonlinear signals.

Keywords: Feedforward neural networks, nonlinear signal prediction, echo state neural networks approach, leaking rates, capacity of neural networks.

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Authors: T. Yamaguchi, Y. Kurosawa, S. Maruyama, K. Tobita, Y. Hirano, K. Yokouchi, K. Kihara, T. Sunaga

Abstract:

This paper describes dynamic analysis using proposed fast finite element method for a shock absorbing structure including a sponge. The structure is supported by nonlinear concentrated springs. The restoring force of the spring has cubic nonlinearity and linear hysteresis damping. To calculate damping properties for the structures including elastic body and porous body, displacement vectors as common unknown variable are solved under coupled condition. Under small amplitude, we apply asymptotic method to complex eigenvalue problem of this system to obtain modal parameters. And then expressions of modal loss factor are derived approximately. This approach was proposed by one of the authors previously. We call this method as Modal Strain and Kinetic Energy Method (MSKE method). Further, using the modal loss factors, the discretized equations in physical coordinate are transformed into the nonlinear ordinary coupled equations using normal coordinate corresponding to linear natural modes. This transformation yields computation efficiency. As a numerical example of a shock absorbing structures, we adopt double skins with a sponge. The double skins are supported by nonlinear concentrated springs. We clarify influences of amplitude of the input force on nonlinear and chaotic responses.

Keywords: Dynamic response, Nonlinear and chaotic motions, Finite Element analysis, Numerical analysis.

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Authors: D. M. S. Bandara, Yunqi Lei, Ye Luo

Abstract:

Fingerprints are suitable as long-term markers of human identity since they provide detailed and unique individual features which are difficult to alter and durable over life time. In this paper, we propose an algorithm to encrypt and decrypt fingerprint images by using a specially designed Elliptic Curve Cryptography (ECC) procedure based on block ciphers. In addition, to increase the confusing effect of fingerprint encryption, we also utilize a chaotic-behaved method called Arnold Cat Map (ACM) for a 2D scrambling of pixel locations in our method. Experimental results are carried out with various types of efficiency and security analyses. As a result, we demonstrate that the proposed fingerprint encryption/decryption algorithm is advantageous in several different aspects including efficiency, security and flexibility. In particular, using this algorithm, we achieve a margin of about 0.1% in the test of Number of Pixel Changing Rate (NPCR) values comparing to the-state-of-the-art performances.

Keywords: Arnold cat map, biometric encryption, block cipher, elliptic curve cryptography, fingerprint encryption, Koblitz’s Encoding.

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Authors: N. Ghatwary, A. Ahmed, H. Jalab

Abstract:

In this paper, an approach for the liver tumor detection in computed tomography (CT) images is represented. The detection process is based on classifying the features of target liver cell to either tumor or non-tumor. Fractional differential (FD) is applied for enhancement of Liver CT images, with the aim of enhancing texture and edge features. Later on, a fusion method is applied to merge between the various enhanced images and produce a variety of feature improvement, which will increase the accuracy of classification. Each image is divided into NxN non-overlapping blocks, to extract the desired features. Support vector machines (SVM) classifier is trained later on a supplied dataset different from the tested one. Finally, the block cells are identified whether they are classified as tumor or not. Our approach is validated on a group of patients’ CT liver tumor datasets. The experiment results demonstrated the efficiency of detection in the proposed technique.

Keywords: Fractional differential (FD), Computed Tomography (CT), fusion.

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Authors: Guoyuan Qi , Yskandar Hamam, Barend Jacobus van Wyk, Shengzhi Du

Abstract:

The majority of existing predictors for time series are model-dependent and therefore require some prior knowledge for the identification of complex systems, usually involving system identification, extensive training, or online adaptation in the case of time-varying systems. Additionally, since a time series is usually generated by complex processes such as the stock market or other chaotic systems, identification, modeling or the online updating of parameters can be problematic. In this paper a model-free predictor (MFP) for a time series produced by an unknown nonlinear system or process is derived using tracking theory. An identical derivation of the MFP using the property of the Newton form of the interpolating polynomial is also presented. The MFP is able to accurately predict future values of a time series, is stable, has few tuning parameters and is desirable for engineering applications due to its simplicity, fast prediction speed and extremely low computational load. The performance of the proposed MFP is demonstrated using the prediction of the Dow Jones Industrial Average stock index.

Keywords: Forecast, model-free predictor, prediction, time series

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Authors: Abolhassan Razminia, Mohammad-Ali Sadrnia

Abstract:

Chua’s circuit is one of the most important electronic devices that are used for Chaos and Bifurcation studies. A central role of secure communication is devoted to it. Since the adaptive control is used vastly in the linear systems control, here we introduce a new trend of application of adaptive method in the chaos controlling field. In this paper, we try to derive a new adaptive control scheme for Chua’s circuit controlling because control of chaos is often very important in practical operations. The novelty of this approach is for sake of its robustness against the external perturbations which is simulated as an additive noise in all measured states and can be generalized to other chaotic systems. Our approach is based on Lyapunov analysis and the adaptation law is considered for the feedback gain. Because of this, we have named it NAFT (Nonlinear Adaptive Feedback Technique). At last, simulations show the capability of the presented technique for Chua’s circuit.

Keywords: Chaos, adaptive control, nonlinear control, Chua's circuit.

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Authors: T. Ganesan, M. S. Aris, I. Elamvazuthi, Momen Kamal Tageldeen

Abstract:

Terms set in power purchase agreements (PPA) challenge power utility companies in balancing between the returns (from maximizing power production) and securing long term supply contracts at capped production. The production limitation set in the PPA has driven efforts to maximize profits through efficient and economic power production. In this paper, a combined industrial-scale gas turbine (GT) - absorption chiller (AC) system is considered to cool the GT air intake for reducing the plant’s heat rate (HR). This GT-AC system is optimized while considering power output limitations imposed by the PPA. In addition, the proposed formulation accounts for uncertainties in the ambient temperature using Type-2 fuzzy programming. Using the enhanced chaotic differential evolution (CEDE), the Pareto frontier was constructed and the optimization results are analyzed in detail.

Keywords: Absorption chillers, turbine inlet air cooling, power purchase agreement, multiobjective optimization, type-2 fuzzy programming, chaotic differential evolution.

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Authors: C. Kwan, X. Li, D. Lao, Y. Deng, Z. Ren, B. Raj, R. Singh, R. Stern

Abstract:

Automated operations based on voice commands will become more and more important in many applications, including robotics, maintenance operations, etc. However, voice command recognition rates drop quite a lot under non-stationary and chaotic noise environments. In this paper, we tried to significantly improve the speech recognition rates under non-stationary noise environments. First, 298 Navy acronyms have been selected for automatic speech recognition. Data sets were collected under 4 types of noisy environments: factory, buccaneer jet, babble noise in a canteen, and destroyer. Within each noisy environment, 4 levels (5 dB, 15 dB, 25 dB, and clean) of Signal-to-Noise Ratio (SNR) were introduced to corrupt the speech. Second, a new algorithm to estimate speech or no speech regions has been developed, implemented, and evaluated. Third, extensive simulations were carried out. It was found that the combination of the new algorithm, the proper selection of language model and a customized training of the speech recognizer based on clean speech yielded very high recognition rates, which are between 80% and 90% for the four different noisy conditions. Fourth, extensive comparative studies have also been carried out.

Keywords: Non-stationary, speech recognition, voice commands.

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Authors: Tae H. Lee, J.H. Park, S.M. Lee, H.Y. Jung

Abstract:

This paper addresses functional projective lag synchronization of Lorenz system with four unknown parameters, where the output of the master system lags behind the output of the slave system proportionally. For this purpose, an adaptive control law is proposed to make the states of two identical Lorenz systems asymptotically synchronize up. Based on Lyapunov stability theory, a novel criterion is given for asymptotical stability of the null solution of an error dynamics. Finally, some numerical examples are provided to show the effectiveness of our results.

Keywords: Adaptive function projective synchronization, Chaotic system, Lag synchronization, Lyapunov method

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Authors: Eiichi Sasaki, Shin-ichi Takehiro, Michio Yamada

Abstract:

We study bifurcation structure of the zonal jet flow the streamfunction of which is expressed by a single spherical harmonics on a rotating sphere. In the non-rotating case, we find that a steady traveling wave solution arises from the zonal jet flow through Hopf bifurcation. As the Reynolds number increases, several traveling solutions arise only through the pitchfork bifurcations and at high Reynolds number the bifurcating solutions become Hopf unstable. In the rotating case, on the other hand, under the stabilizing effect of rotation, as the absolute value of rotation rate increases, the number of the bifurcating solutions arising from the zonal jet flow decreases monotonically. We also carry out time integration to study unsteady solutions at high Reynolds number and find that in the non-rotating case the unsteady solutions are chaotic, while not in the rotating cases calculated. This result reflects the general tendency that the rotation stabilizes nonlinear solutions of Navier-Stokes equations.

Keywords: rotating sphere, two-dimensional flow, bifurcationstructure

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Authors: Olga I. Moskalenko, Maksim O. Zhuravlev, Alexey A. Koronovskii, Alexander E. Hramov

Abstract:

Intermittent behavior near the boundary of phase synchronization in the presence of noise is studied. In certain range of the coupling parameter and noise intensity the intermittency of eyelet and ring intermittencies is shown to take place. Main results are illustrated using the example of two unidirectional coupled Rössler systems. Similar behavior is shown to take place in two hydrodynamical models of Pierce diode coupled unidirectional.

Keywords: Chaotic oscillators, phase synchronization, noise, intermittency of intermittencies, control.

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Authors: M. Sasajima, T. Yamaguchi, Y. Koike, A. Hara

Abstract:

This paper describes vibration analysis using the finite element method for a small earphone, especially for the diaphragm shape with a low-rigidity. The viscoelastic diaphragm is supported by multiple nonlinear concentrated springs with linear hysteresis damping. The restoring forces of the nonlinear springs have cubic nonlinearity. The finite elements for the nonlinear springs with hysteresis are expressed and are connected to the diaphragm that is modeled by linear solid finite elements in consideration of a complex modulus of elasticity. Further, the discretized equations in physical coordinates are transformed into the nonlinear ordinary coupled equations using normal coordinates corresponding to the linear natural modes. We computed the nonlinear stationary and non-stationary responses due to the internal resonance between modes with large amplitude in the nonlinear springs and elastic modes in the diaphragm. The non-stationary motions are confirmed as the chaos due to the maximum Lyapunov exponents with a positive number. From the time histories of the deformation distribution in the chaotic vibration, we identified nonlinear modal couplings.

Keywords: Nonlinear Vibration, Finite Element Method, Chaos , Small Earphone.

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