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The chaotic systems are among the most important areas that have increased in popularity and are actively used in several fields. One of the most essential components in chaotic systems is the chaotic oscillator which generates chaotic signals. IQ-Math and floating point number systems are preferred number standards. In this study, the Modified Chua chaotic oscillator has been designed to work on FPGA chips using fixed point and floating point number representations, and both system version performances are compared. Euler numeric algorithm has been used to design the Modified Chua chaotic oscillator. In the first section of the study, the Modified Chua chaotic system based on fixed point has been composed the model in the Matlab Simulink and converted to VHDL with the help of Matlab HDL Coder Toolbox. In the second section of the study, the Modified Chua chaotic oscillator has been designed with VHDL based on floating point. Modified Chua chaotic oscillators which are composed with two different number standards have been tested using Xilinx ISE Design Tools in VHDL. Modified Chua chaotic oscillators which have two different number standards and designed, are synthesized for Virtex-6 on ML605 FPGA development board using Xilinx ISE Design Tools 14.2 program. The values that are achieved from the process of synthesizing and the process of maximum operating frequency have been presented. As a result, the study has found that fixed-point representation achieved a maximum operating frequency of 50.242 MHz, while the floating-point representation achieved 273.631 MHz. [ABSTRACT FROM AUTHOR]

A new algorithm is described for detecting hidden changes in the topological structure of the dynamics of a nonlinear system due to the perturbations in the driving signals. The proposed method is based on the use of a double sliding window and wavelet decomposition and calculates the Jensen–Shannon divergence between the probability distributions of the normalised wavelet coefficients of the first half of the sliding window and those of the second half of the sliding window. Applying the proposed approach to the Duffing and Hénon–Heiles systems, real-life signals showed their effectiveness at detecting small discontinuities in the dynamic behaviour of the system even when the system is chaotic. [ABSTRACT FROM AUTHOR]

This scientific study reports on the practical as well as modelling of active hybrid inductor based chaotic oscillators using Current Feedback Amplifier (CFOA) and Op-Amp. Furthermore, a workable design strategy for the chaotic oscillator and its application to secure communication are provided. Here, the transient analysis of chaos waveforms is verified by using MATLAB, Multi-SIM, and LT-Spice tools. The numerical outcomes based on LT-Spice and MATLAB are closely matched. Additionally, results from the multi-SIM platform show how a secure communication network was implemented utilising the master-slave technique of synchronisation. Finally, a comparison analysis is conducted to demonstrate the originality of the suggested work.

Over the past years, many chaos-based secure communication system algorithms have been published for the encryption of different types of data. The problem with these algorithms is either the high complexity that makes them difficult to be implemented or the lack of the high security that prevents them from being applicable in real applications. In this paper, we propose a new highly-secured chaos-based communication system that is based on adaptive control synchronization. The implementation process of the proposed system is detailed including the oscillator dynamic equations, the control laws derivation, the numerical solutions, and the Matlab/Simulink representation. To show the efficiency of the proposed system, its security is analyzed using different security measures. This proofs that the carrier signal is unpredictable by the intruder in both time and frequency domains. The experimental results and the show that the proposed system outperforms the different related and recent work in addition to its stability and robustness to the variation of initial conditions. Finally, we introduce an application on voice encryption to demonstrate the effectiveness of our system. Then We compare the security results of our system with different implementations of recent and related work. [ABSTRACT FROM AUTHOR]

The paper is dedicated to the numerical and experimental study of nonlinear oscillations exhibited by the Vilnius chaotic generator. The motivation for the work is defined by the need for a comprehensive analysis of the dynamics of the oscillators being embedded into chaotic communication systems. These generators should provide low-power operation while ensuring the robustness of the chaotic oscillations, insusceptible to parameter variations and noise. The work focuses on the investigation of the dependence of nonlinear dynamics of the Vilnius oscillator on the operating voltage and component parameter changes. The paper shows that the application of the Method of Complete Bifurcation Groups reveals the complex smooth and non-smooth bifurcation structures, forming regions of robust chaotic oscillations. The novel tool—mode transition graph—is presented, allowing the comparison of experimental and numerical results. The paper demonstrates the applicability of the Vilnius oscillator for the generation of robust chaos, and highlights the need for further investigation of the inherent trade-off between energy efficiency and robustness of the obtained oscillations. [ABSTRACT FROM AUTHOR]

The measurement of bio-impedance spectra at ultra low frequencies (sub-Hz) is known to require a considerably long time with the classical frequency-sweep method or other narrow-band periodic excitation signals. In this work, an impedance measurement technique based on using wide-band chaotic signals is proposed and experimentally validated over the frequency range 10 m H z - 1 H z . The technique was tested in both potentiostatic and galvanostatic modes, first using commercial components and then using an enhanced Howland current pump designed and fabricated in a 65nm CMOS technology. The accuracy of the proposed technique was assessed on fruit samples compared to measurements conducted using a research-grade Biologic VSP-300 electro-chemical station. [ABSTRACT FROM AUTHOR]

This paper strives to achieve a comprehensive review of chaos in analog circuits and lumped electronic networks. Readers will be guided from the beginning of the investigations of simple electronic circuits to the current trends in the research into chaos. The author tries to provide the key references related to this issue, including papers describing modern numerical algorithms capable of localizing chaotic and hyperchaotic motion in complex mathematical models, interesting full on-chip implementations of chaotic systems, possible practical applications of entropic signals, fractional-order chaotic systems and chaotic oscillators with mem-elements. [ABSTRACT FROM AUTHOR]

This paper proposes novel forecasting models for fractional-order chaotic oscillators, such as Duffing’s, Van der Pol’s, Tamaševičius’s and Chua’s, using feedforward neural networks. The models predict a change in the state values which bears a weighted relationship with the oscillator states. Such an arrangement is a suitable candidate model for out-of-sample forecasting of system states. The proposed neural network-assisted weighted model is applied to the above oscillators. The improved out-of-sample forecasting results of the proposed modeling strategy compared with the literature are comprehensively analyzed. The proposed models corresponding to the optimal weights result in the least mean square error (MSE) for all the system states. Further, the MSE for the proposed model is less in most of the oscillators compared with the one reported in the literature. The proposed prediction model’s out-of-sample forecasting plots show the best tracking ability to approximate future state values.

Memristor is a non-linear circuit element in which voltage-current relationship is determined by the previous values of the voltage and current, generally the history of the circuit. The nonlinearity in this component can be considered as a fractional-order form, which yields a fractional memristor (fracmemristor). In this paper, a fractional-order memristor in a chaotic oscillator is applied, while the other electronic elements are of integer order. The fractional-order range is determined in a way that the circuit has chaotic solutions. Also, the statistical and dynamical features of this circuit are analyzed. Tools like Lyapunov exponents and bifurcation diagram show the existence of multistability and antimonotonicity, two less common properties in chaotic circuits.

In recent years designing new multistable chaotic oscillators has been of noticeable interest. A multistable system is a double-edged sword which can have many benefits in some applications while in some other situations they can be even dangerous. In this paper, we introduce a new multistable two-dimensional oscillator. The forced version of this new oscillator can exhibit chaotic solutions which makes it much more exciting. Also, another scarce feature of this system is the complex basins of attraction for the infinite coexisting attractors. Some initial conditions can escape the whirlpools of nearby attractors and settle down in faraway destinations. The dynamical properties of this new system are investigated by the help of equilibria analysis, bifurcation diagram, Lyapunov exponents’ spectrum, and the plot of basins of attraction. The feasibility of the proposed system is also verified through circuit implementation.

This paper strives to achieve a comprehensive review of chaos in analog circuits and lumped electronic networks. Readers will be guided from the beginning of the investigations of simple electronic circuits to the current trends in the research into chaos. The author tries to provide the key references related to this issue, including papers describing modern numerical algorithms capable of localizing chaotic and hyperchaotic motion in complex mathematical models, interesting full on-chip implementations of chaotic systems, possible practical applications of entropic signals, fractional-order chaotic systems and chaotic oscillators with mem-elements.

Weak signal detection is an important topic, which has been widely studied in various fields. Different from other signal processing methods, stochastic resonance (SR) can utilize noise to enhance the characteristic frequency. Inspired by the unique advantage of SR, the strongly coupled Duffing and Van der pol SR system (SCD-VSR) is investigated. The simulation results show that the relationship between the output average signal–noise ratio increase (MSNRI) and different jump values of trichotomous noise presents different odd symmetrical distribution. It is also found that a double SR phenomenon could be observed when the damping coefficient of Van der pol system is small. Moreover, as the damping coefficient of the Duffing system increases, the output response would become gradually smooth. In addition,a smaller damping force coupling coefficient combined with a large restoring force coupling coefficient would achieve better system response. In the case of detecting an analog signal, MSNRI of SCD-VSR is larger than that of both classical bistable SR system (CBSR) and coupled Duffing SR system (CDSR). In addition, the experiments suggest that SCD-VSR could obtain a higher MSNRI and better detection effect, which implies the performance is superior to CBSR and CDSR. [ABSTRACT FROM AUTHOR]

Memristor is a non-linear circuit element in which voltage-current relationship is determined by the previous values of the voltage and current, generally the history of the circuit. The nonlinearity in this component can be considered as a fractional-order form, which yields a fractional memristor (fracmemristor). In this paper, a fractional-order memristor in a chaotic oscillator is applied, while the other electronic elements are of integer order. The fractional-order range is determined in a way that the circuit has chaotic solutions. Also, the statistical and dynamical features of this circuit are analyzed. Tools like Lyapunov exponents and bifurcation diagram show the existence of multistability and antimonotonicity, two less common properties in chaotic circuits. [ABSTRACT FROM AUTHOR]

The efficiency of the hardware implementations of fractional-order systems heavily relies on the efficiency of realizing the fractional-order derivative operator. In this work, a generic hardware implementation of the fractional-order derivative based on the Grünwald–Letnikov's approximation is proposed and verified on a field-programmable gate array. The main advantage of this particular realization is its flexibility in applications which enable easy real-time configuration of the values of the fractional orders, step sizes, and/or other system parameters without changing the hardware architecture. Different approximation techniques are used to improve the hardware performance including piece-wise linear/quadratic methods. As an application, a variable-order chaotic oscillator is implemented and verified using fractional orders that vary in time. [ABSTRACT FROM AUTHOR]

Since the dynamical behavior of chaotic and stochastic systems is very similar, it is sometimes difficult to determine the nature of the movement. One of the best-studied stochastic processes is Brownian motion, a random walk that accurately describes many phenomena that occur in nature, including quantum mechanics. In this paper, we propose an approach that allows us to analyze chaotic dynamics using the Langevin equation describing dynamics of the phase difference between identical coupled chaotic oscillators. The time evolution of this phase difference can be explained by the biased Brownian motion, which is accepted in quantum mechanics for modeling thermal phenomena. Using a deterministic model based on chaotic Rössler oscillators, we are able to reproduce a similar time evolution for the phase difference. We show how the phenomenon of intermittent phase synchronization can be explained in terms of both stochastic and deterministic models. In addition, the existence of phase multistability in the phase synchronization regime is demonstrated.

In this study, we present a new method for assessing the functional connectivity between the drive and the noisy response chaotic oscillators with significant frequency mismatch. This method is based on the footprint of the drive oscillator on the response oscillator. Specifically, we calculate the magnitude squared coherence between the intrinsic frequency of the drive oscillator and the oscillation frequency of the dynamics of the spectral energy of the response oscillator. The spectral energy of the response oscillator is estimated using a modified S-transform algorithm with a short sliding window. We apply the proposed approach to master–slave Rössler systems and coupled Van der Pol oscillators with a frequency ratio of ∼ 1:4 and show that when the slave signal is contaminated with white Gaussian noise at different signal-to-noise ratios (SNR), the new algorithm is well suited to assess the presence of coupling with a priori known direction in noisy, unidirectionally coupled chaotic oscillators, especially in the case of weak and moderate coupling. [ABSTRACT FROM AUTHOR]

In recent years designing new multistable chaotic oscillators has been of noticeable interest. A multistable system is a double-edged sword which can have many benefits in some applications while in some other situations they can be even dangerous. In this paper, we introduce a new multistable two-dimensional oscillator. The forced version of this new oscillator can exhibit chaotic solutions which makes it much more exciting. Also, another scarce feature of this system is the complex basins of attraction for the infinite coexisting attractors. Some initial conditions can escape the whirlpools of nearby attractors and settle down in faraway destinations. The dynamical properties of this new system are investigated by the help of equilibria analysis, bifurcation diagram, Lyapunov exponents' spectrum, and the plot of basins of attraction. The feasibility of the proposed system is also verified through circuit implementation. [ABSTRACT FROM AUTHOR]

This paper presents the experimental results obtained from the synchronization of two memristor-based Wien-bridge oscillators. The experimental results show that synchronization is feasible between two chaotic oscillators. This is very important for implementing chaotic-based secure communication systems. The experiment can be easily introduced in an undergraduate laboratory course to demonstrate the feasibility of synchronizing chaotic memristor-based oscillators. [ABSTRACT FROM AUTHOR]

Recently, chaotic systems with hidden attractors and multistability have been of great interest in the field of chaos and nonlinear dynamics. Two special categories of systems with multistability are systems with extreme multistability and systems with megastability. In this paper, the simplest (yet) megastable chaotic oscillator is designed and introduced. Dynamical properties of this new system are completely investigated through tools like bifurcation diagram, Lyapunov exponents, and basin of attraction. It is shown that between its countable infinite coexisting attractors, only one is self-excited and the rest are hidden. [ABSTRACT FROM AUTHOR]

In this paper, a new two-dimensional nonlinear oscillator with an unusual sequence of rational and irrational parameters is introduced. This oscillator has endless coexisting limit cycles, which make it a megastable dynamical system. By periodically forcing this system, a new system is designed which is capable of exhibiting an infinite number of coexisting asymmetric torus and strange attractors. This system is implemented by an analog circuit, and its Hamiltonian energy is calculated. [ABSTRACT FROM AUTHOR]

In this work, we demonstrate the possibility of performing two-dimensional rotation on a chaotic system. This enables the rotation of its attractor in space without changing its chaotic dynamics. In particular, the rotated system preserves the same eigenvalues at all equilibrium points and its largest Lyapunov exponent remains unchanged. Two chaotic systems, one of which is the classical Lorenz system, are used to illustrate and validate the rotation operation using numerical simulations and further experimentally using a digital FPGA platform. [ABSTRACT FROM AUTHOR]

In the present work, we present a new algorithm for assessing causality in uni-directionally coupled chaotic oscillators with small frequency mismatch embedded in heavy white Gaussian noise. This method is based on the correlation between changes in the phase dynamics of the slave oscillator and the dynamics of the phase difference between the oscillators. To recover the phase at low signal-to-noise ratio, a nonlinear adaptive denoising algorithm based on finding sinusoidal fits to the local neighbourhood of the reconstructed phase space is used. Application of the proposed approach to master-slave Rössler systems showed that the new algorithm is well-suited for assessing the presence and direction of coupling in highly noisy uni-directionally coupled chaotic oscillators, especially in the case of weak and moderate coupling. [ABSTRACT FROM AUTHOR]

In this paper, by modifying a known two-dimensional oscillator, we obtain an interesting new oscillator with coexisting limit cycles and point attractors. Then by changing this new system to its forced version and choosing a proper set of parameters, we introduce a chaotic system with some very interesting features. In this system, not only can we see the coexistence of different types of attractors, but also a fascinating phenomenon: some initial conditions can escape from the gravity of nearby attractors and travel far away before being trapped in an attractor beyond the usual access. [ABSTRACT FROM AUTHOR]

In this paper, inspired by a newly proposed two-dimensional nonlinear oscillator with an infinite number of coexisting attractors, a modified nonlinear oscillator is proposed. The original system has an exciting feature of having layer−layer coexisting attractors. One of these attractors is self-excited while the rest are hidden. By forcing this system with its twin, a new four-dimensional nonlinear system is obtained which has an infinite number of coexisting torus attractors, strange attractors, and limit cycle attractors. The entropy, energy, and homogeneity of attractors’ images and their basin of attractions are calculated and reported, which showed an increase in the complexity of attractors when changing the bifurcation parameters.

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Chaotic systems and chaos-based applications have been commonly used in the fields of engineering recently. The most essential part of them is the chaotic oscillator that has very critical role in some applications such as chaotic communications and cryptography. In this study, Sundarapandian–Pehlivan chaotic system has been modeled and simulated in three distinct platforms to show the advantages of FPGA-based chaotic oscillator with respect to alternative solutions. In the first stage, the chaotic system has been modeled numerically by the help of fourth order of Runge–Kutta (RK4) method. Additionally, phase portraits of the system have been obtained and Lyapunov exponents have been examined. Secondly, the system has been modeled by using PSpice for the implementation of the chaotic system with analog circuit elements. Then, Pspice simulation results have been compared with the numerical outcome to justify the designed model. Furthermore, the chaotic system has been physically confirmed with real analog circuit elements. Signals obtained from the physical system have been verified with both numerical and PSpice results. It has been also modeled by the help of method of RK4 in a hardware description language (VHDL) and the model further has been synthesized and tested for Xilinx Virtex-6 FPGA chip. Finally, the chaotic oscillator designed has been tested for True Random Number Generators (TRNG) and the maximum operating frequency has been achieved as 293 MHz with a speed of 58.76 Mbit/s. Besides, the random bit sets produced by TRNG have been further verified by FIPS-140-1 and NIST-800-22 statistical standards and it has been proved that the proposed design can be used in embedded cryptologic applications. [ABSTRACT FROM AUTHOR]

This paper describes the design, realization and use of an analogical, fully reconfigurable experimental setup to analyze the complex dynamics of networks of chaotic circuits. It reports details of the implementation and characterization of the setup, together with representative results, showing its flexibility and potential. The setup allows to choose arbitrarily the coupling strength and interconnection structure among the circuits, the type of link and to select the parameters of the node dynamics. It has a modular structure, and it can accommodate up to 32 nodes interconnected by at most 32 links. The collective dynamics of a relatively large set of different network structures and configurations has been investigated using the setup. Synchronization, pattern formation and other interesting collective phenomena were observed experimentally, their evidence being reported here as an illustration of the potential of the proposed setup. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Compared to the traditional single scroll and double scroll chaotic systems, multi-scroll chaotic systems present more complex structure and dynamic behavior, possess good application prospect in information security and secure communications. Therefore, theoretical research and circuit implementation of multi-scroll chaotic attractor generation has become a hot spot in the research field of chaos at present domain. In this paper, we briefly overview the recent progress that has been reported in the study of multi-scroll chaotic oscillators based on current-mode devices. Multi-scroll chaotic oscillators are listed according to their electronic implementations. Finally, we list multi-scroll chaotic oscillators based on current-mode devices, and prospect development trends of multi-scroll chaotic oscillators based on current-mode devices in the future. [ABSTRACT FROM AUTHOR]

The Derivative-free nonlinear Kalman Filter is used for developing a communication system that is based on a chaotic modulator such as the Duffing system. In the transmitter’s side, the source of information undergoes modulation (encryption) in which a chaotic signal generated by the Duffing system is the carrier. The modulated signal is transmitted through a communication channel and at the receiver’s side demodulation takes place, after exploiting the estimation provided about the state vector of the chaotic oscillator by the Derivative-free nonlinear Kalman Filter. Evaluation tests confirm that the proposed filtering method has improved performance over the Extended Kalman Filter and reduces significantly the rate of transmission errors. Moreover, it is shown that the proposed Derivative-free nonlinear Kalman Filter can work within a dual Kalman Filtering scheme, for performing simultaneously transmitter–receiver synchronisation and estimation of unknown coefficients of the communication channel. [ABSTRACT FROM AUTHOR]

There has recently been an increase in the number of new chaotic system designs and chaos-based engineering applications. In this study, since homoclinic and heteroclinic orbits did not exist and analyses like Shilnikov method could not be used, a 3D chaotic system without equilibrium points was included and thus different engineering applications especially for encryption studies were realized. The 3D chaotic system without equilibrium points represents a new different phenomenon and an almost unexplored field of research. First of all, chaotic system without equilibrium points was examined as the basis and electronic circuit application of the chaotic system was realized and oscilloscope outputs of phase portraits were obtained. Later, chaotic system without equilibrium points was modelled on Labview Field Programmable Gate Array (FPGA) and then FPGA chip statistics, phase portraits and oscilloscope outputs were derived. With another study, VHDL and RK-4 algorithm were used and a new FPGA-based chaotic oscillators design was achieved. Results of Labview-based design on FPGA- and VHDL-based design were compared. Results of chaotic oscillator units designed here were gained via Xilinx ISE Simulator. Finally, a new chaos-based RNG design was achieved and internationally accepted FIPS-140-1 and NIST-800-22 randomness tests were run. Furthermore, video encryption application and security analyses were carried out with the RNG designed here. [ABSTRACT FROM AUTHOR]