A Speech Cryptosystem Using the New Chaotic System with a Capsule-Shaped Equilibrium Curve

Chaotic neural networks consisting of a great number of chaotic neurons are able to reproduce the rich dynamics observed in biological nervous systems. In recent years, the memristor has attracted much interest in the efficient implementation of artificial synapses and neurons. This work addresses adaptive synchronization of a class of memristor-based neural chaotic systems using a novel adaptive backstepping approach. A systematic design procedure is presented. Simulation results have demonstrated the effectiveness of the proposed adaptive synchronization method and its potential in practical application of memristive chaotic oscillators in secure communication.

Audio encryption has gained popularity in a variety of fields including education, banking over the phone, military, and private audio conferences. Data encryption algorithms are necessary for processing and sending sensitive information in the context of secure speech conversations. In recent years, the importance of security in any communications system has increased. To transfer data securely, a variety of methods have been used. Chaotic system-based encryption is one of the most significant encryption methods used in the field of security. Chaos-based communication is a promising application of chaos theory and nonlinear dynamics. In this research, a chaotic algorithm for the new chaotic chameleon system was proposed, studied, and implemented. The chameleon chaotic system has been preferred to be employed because it has the property of changing from self-excited (SA) to hidden-attractor (HA) which increases the complexity of the system dynamics and gives strength to the encryption algorithm. A chaotic chameleon system is one in which, depending on the parameter values, the chaotic attractor alternates between being a hidden attractor and a self-excited attractor. This is an important feature, so it is preferable to use it in cryptography compared to other types of chaotic systems. This model was first implemented using a Field Programmable Gate Array (FPGA), which is the first time it has been implemented in practical applications. The chameleon system model was implemented using MATLAB Simulink and the Xilinx System Generator model. Self-excited, hidden, and coexisting attractors are shown in the proposed system. Vivado software was used to validate the designs, and Xilinx ZedBoard Zynq-7000 FPGA was used to implement them. The dynamic behavior of the proposed chaotic system was also studied and analysis methods, including phase portrait, bifurcation diagrams, and Lyapunov exponents. Assessing the quality of the suggested method by doing analyses of many quality measures, including correlation, differential signal-to-noise ratio (SNR), entropy, histogram analysis, and spectral density plot. The numerical analyses and simulation results demonstrate how well the suggested method performs in terms of security against different types of cryptographic assaults.

Recently, Leonov and Kuznetsov introduced a new class of nonlinear dynamical systems, which is called systems with hidden attractors, in contrary to the well-known class of systems with self-excited attractors. In this class, dynamical systems with infinite number of equilibrium points, with stable equilibria, or without equilibrium are classified. Since then, the study of chaotic systems with hidden attractors has become an attractive research topic because this new class of dynamical systems could play an important role not only in theoretical problems but also in engineering applications. In this direction, the proposed chapter presents the bidirectional and unidirectional coupling schemes between two identical dynamical chaotic systems with no-equilibrium points. As it is observed, when the value of the coupling coefficient is increased in both coupling schemes, the coupled systems undergo a transition from desynchronization mode to complete synchronization. Also, the simulation results reveal the richness of the coupled system’s dynamical behavior, especially in the bidirectional case, showing interesting nonlinear dynamics, with a transition between periodic, quasiperiodic and chaotic behavior as the coupling coefficient increases, as well as synchronization phenomena, such as complete and anti-phase synchronization. Various tools of nonlinear theory for the study of the proposed coupling method, such as bifurcation diagrams, phase portraits and Lyapunov exponents have been used.

Back to table of contents Previous article Next article LETTERFull AccessGlobal Versus Local Perspectives on SchizophreniaSareh ZendehrouhFatemeh BakouieShahriar Gharibzadeh,Sareh ZendehrouhSearch for more papers by this authorFatemeh BakouieSearch for more papers by this authorShahriar GharibzadehSearch for more papers by this author,Published Online:1 Apr 2009AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InEmail To the Editor: Recent neurobiological studies indicate that schizophrenia may be a neurodevelopmental and progressive disorder with multiple biochemical abnormalities involving dopamine, serotonin, glutamate, and gamma-aminobutyric acid secretion. In postmortem tissue studies, structural abnormalities and alterations in synaptic connectivity have been observed in the intracortical circuitry of the prefrontal dorsal cortex of schizophrenia patients. These morphological changes could be the sequelae of earlier environmental insults and genetic processes. There are probably multiple susceptibility genes, each of small effect, which act in conjunction with environmental factors as obstetric abnormalities, intrauterine infection, and abnormal nutrition. Candidate identified genes could influence neurodevelopment, synaptic plasticity, and neurotransmission. 1 The recent findings are consistent with notions that epigenetic factors play a major role in the disease process and epigenetic factors may continue to influence the expression of the affected genes in adulthood. 2 It is also demonstrated that schizophrenia is a dynamical disease—i.e., that important aspects of schizophrenia can be understood on the basis of concepts of the theory of nonlinear dynamical systems. 3 Chaotic dynamical systems are characterized by a lawful but delicate sensitivity to initial conditions. This leads to the observation that initially similar behaviors evolve into a striking divergence of behavioral patterns over time. Therefore, small differences in the input can result in an entirely different sequence of outputs. 4 We believe that genes act as initial conditions and environmental factors are the control parameters of a chaotic human brain system, which can force the system to special states with particular characteristics (e.g., schizophrenia). Since in chaotic dynamical systems, control parameters can direct the system to special states, environmental factors play a key role in aggravation or decline of schizophrenia symptoms. This complies with some reports that showed high relapse rate of schizophrenia in families that expressed high emotion, 5 which can be considered as a positive control factor. Based on our hypothesis, we propose that schizophrenia could be considered a chaotic model and the global features of the disease could be extracted instead of paying attention to local detailed features. In such a perspective, the manner of managing the disease will be changed. For example, we think that it is not necessary to include the effects of each part of the brain circuits and to correct any change in the amount of neurotransmitters; instead it is better to model the disease state as a chaotic system and recognize the interaction of different environmental factors, such as maternal health, birth complications, emotional states, familial relationships, etc., on it and try to minimize the pathological effects of them. We stress that it is not the compartments but the global interactions of different elements that play major roles in the disease and deserve attention. Modeling the behavior of schizophrenia on the basis of chaos theory to control the disease seems to be a good place to start.Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran

Hidden attractors are associated with multistability phenomena, which have considerable application prospects in engineering. By modifying a simple three-dimensional continuous quadratic dynamical system, this paper reports a new autonomous chaotic system with two stable node-foci that can generate double-wing hidden chaotic attractors. We discuss the rich dynamics of the proposed system, which have some interesting characteristics for different parameters and initial conditions, through the use of dynamic analysis tools such as the phase portrait, Lyapunov exponent spectrum, and bifurcation diagram. The topological classification of the periodic orbits of the system is investigated by a recently devised variational method. Symbolic dynamics of four and six letters are successfully established under two sets of system parameters, including hidden and self-excited chaotic attractors. The system is implemented by a corresponding analog electronic circuit to verify its realizability.

Chapter 16 - Dynamics, Synchronization and Fractional Order Form of a Chaotic System With Infinite Equilibria

Memristor-based chaotic and hyperchaotic systems are of great interest in the recent years, and addition of meminductor and memcapacitors to the family has widened the applications. In this paper, we propose a new chaotic system with fractional-order memristor and memcapacitor components. Nonlinear chaotic properties of the proposed system are investigated with equilibrium points, eigenvalues, Lyapunov exponents, bifurcation and bicoherence plots. We show that a small model disturbance can make the system to show self-excited and hidden attractors. We use the Adomian Decomposition method for implementing the proposed system in Field Programmable Gate Arrays.

Chaotic sequences and chaotic dynamical systems are attractive candidates for use in signal synthesis and analysis as well as in communications applications. In previous works, various methods for the estimation of chaotic sequences under noise were developed. However, although the methods were different, their qualitative performance was the same: for high signal-to-noise ratio (SNR) the performance was good, but below some threshold SNR, a sharp degradation in performance occurred. We prove and quantify the existence of this threshold effect and derive lower bounds on the value of the threshold SNR. Using information-theoretic tools, we prove that for any ergodic chaotic system, there is a certain threshold SNR level below which the ratio between the mean-square error obtained by any estimator of the system's initial state at the output of additive white Gaussian noise (AWGN) channel and the Cramer-Rao bound increases exponentially fast as the number of observations N grows without bound. We explain the connection between the existence of a threshold effect in the estimation of chaotic sequences and the converse to the joint source-channel coding theorem. We derive lower bounds on SNR/sub th/, the value of the threshold SNR, as a function of the system's Lyapunov exponent. Our bounds have two versions, one for a finite number of observations, and one for the asymptotic regime as N/spl rarr//spl infin/. For the asymptotic regime, the bound we develop on the threshold SNR incorporates the key parameters of chaotic systems, the Lyapunov exponent, and the chaotic sequence power spectrum into a simple formula. We demonstrate our results on the chaotic system governed by the r-diadic map.

A gallery of chaotic systems with an infinite number of equilibrium points

Recently, Leonov and Kuznetsov have introduced a new definition “hidden attractor”. Systems with hidden attractors, especially chaotic systems, have attracted significant attention. Some examples of such systems are systems with a line equilibrium, systems without equilibrium or systems with stable equilibria etc. In some interesting new research, systems in which equilibrium points are located on different special curves are reported. This chapter introduces a three-dimensional autonomous system with a square-shaped equilibrium and without equilibrium points. Therefore, such system belongs to a class of systems with hidden attractors. The fundamental dynamics properties of such system are studied through phase portraits, Poincare map, bifurcation diagram, and Lyapunov exponents. Anti-synchronization scheme for our systems is proposed and confirmed by the Lyapunov stability. Moreover, an electronic circuit is implemented to show the feasibility of the mathematical model. Finally, we introduce the fractional order form of such system.

A 3-D new chaotic system with three nonlinearities is proposed in this paper. We show that the new chaotic system consists of equilibrium points on the z-axis (line equilibrium) as well as two equilibrium points on the (x, y)-plane. Thus, the new chaotic system has hidden attractors. The dynamical properties of the new chaotic system are explored in terms of 2-D phase plots, bifurcation diagram, Lyapunov exponents, and offset-boosting control. Numerical results show that the proposed scheme has an excellent performance

In this work, a secure second-order discrete time delta sigma (ΔΣM) modulator is used to securely modulate two physiological signals for e-health applications. The chaotic system used is a new 3D chaotic system that is used to generate random keys for encryption. Quintic nonlinearity is integrated into the existing sprott chaotic system and then studied via nonlinear dynamics tools, which reveal complex dynamics, including the coexistence of three distinct chaotic attractors and the phenomenon of intermittency. A microcontroller implementation of the system is performed to demonstrate its real-life applicability. The bits generated from the chaotic regime of the proposed oscillator are tested successfully via the NIST-800.22 test. The pseudo random bit streams generated from the chaoticity of the oscillator using the system parameters and initial conditions as secret keys are used in a 2-stage confusion process to encrypt two physiological signals. At the receiving end, the physiological signals are successfully recovered, and some standard metrics, such as key space analysis, key sensitivity, and cross-correlation, are applied to verify the strength of the encryption scheme. A comparison with recent and good standing contributions in the literature is also provide at the end of this work, which shows that the proposed scheme is robust against common attacks. The results obtained in this study open the possibility of applying chaos in digital communication systems for the conception of secure e-health services.p. 2Highlights• A new random chaotic system applicable in information protection is conceived and experimented.• Random encryption data are successfully generated from the chaotic system in its confused state and tested statistically. • Patient's ECG signal is securely transferred via a secure and simple communication scheme.

By introducing a sine function, a non-autonomous multi-wing chaotic system is proposed. The system has an infinite number of equilibrium points and produces symmetrical attractors. The complex dynamical behaviors of the system are demonstrated by phase portraits, Lyapunov exponents spectrum and bifurcation diagram. The effect of driving amplitudes and initial conditions on the resulting system dynamics is then thoroughly investigated. The resulting attractors will enter different oscillatory states or have topological changes. The rotational coexisting attractors depend on the initial conditions and external amplitudes. Besides, a variety of interesting symmetrical transient behaviors and initial-offset boosting behaviors are also found. The driving amplitudes of the system affect the number of attractor wings, a weak signal detection circuit is accordingly designed to estimate the amplitude of periodic weak signals at diverse frequencies. The circuit operates near the switching threshold between the four-wing and two-wing chaotic attractor. Finally, an experimental investigation of the proposed design is performed that demonstrates the theoretical and simulation results.

The employment of chaotic maps in a variety of applications such as cryptosecurity, image encryption schemes, communication schemes, and secure communication has been made possible thanks to their properties of high levels of complexity, ergodicity, and high sensitivity to the initial conditions, mainly. Of considerable interest is the implementation of these dynamical systems in electronic devices such as field programmable gate arrays (FPGAs) with the intention of experimentally reproducing their dynamics, leading to exploiting their chaotic properties in real phenomena. In this work, the implementation of a one-dimensional chaotic map that has no fixed points is performed on an FPGA device with the objective of being able to reproduce its chaotic behavior as well as possible. The chaotic behavior of the introduced system is determined by estimating the Lyapunov exponents and its chaotic behavior is also analyzed using bifurcation diagrams. Simulations of the system are realized via Matlab, as well as in C and the very high-speed integrated circuit (VHSIC) hardware description language (VHDL). Experimental results on FPGA show that they are like those obtained in the simulations; therefore, this chaotic dynamical system could be used as an element in some encryption schemes such as in the generation of cryptographically secure pseudorandom numbers.

We construct a three-dimensional chaotic system based on TANG chaotic system and analyze the system dynamics based on the equilibrium points, phase diagram, Lyapunov exponential spectrum, bifurcation diagram, and complexity. The more complexly dynamic behaviors indicate that our designed system has different topology, bifurcation anti-monotonic property and attractor coexistence phenomenon from TANG system. The system has the coexistence of chaotic attractor and point attractor and can select the initial value that is suitable for chaotic secure communication by analyzing the coexistence attractor. The actual digital circuit is implemented with field programmable gate array (FPGA). The system is extremely sensitive to the selections of initial values and system parameters and has good and rich dynamic behavior, which is suitable for the field of chaotic confidential communication. The linear feedback method is used to control the system synchronous and the simulation of synchronous circuit is realized by Multisim software. The simulation results are consistent with the numerical analysis, which lays a theoretical foundation for the application of the system in the field of synchronized confidential communication.