In this paper, a new **modified** **Chua** **oscillator** is introduced. The original **Chua** **oscillator** is well known for **its** simple implementation and mathemati- cal modeling. A modification of the **oscillator** is proposed in order to facilitate the synchronization and the encryption and decryption scheme. The modifi- cation consists in changing the nonlinear term of the original **oscillator** to a smooth and bounded nonlinear function. A bifurcation diagram, a Poincar´e map and the Lyapunov exponents are presented as proofs of chaoticity of the newly **modified** **oscillator**. An **application** to **secure** **communications** is pro- posed in which two channels are used. Numerical simulations are performed in order to analyze the communication system.

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Received: 26 September 2019; Accepted: 22 October 2019; Published: 24 October 2019 Abstract: In this work a novel **chaotic** system with a line equilibrium is presented. First, a dynamical analysis on the system is performed, by computing **its** bifurcation diagram, continuation diagram, phase portraits and Lyapunov exponents. Then, the system is applied to the problem of **secure** communication. We assume that the transmitted signal is an additional state. For this reason, the nonlinear system is rewritten in a rectangular descriptor form and then an observer is constructed for achieving synchronization and input reconstruction. If we assume some rank conditions (on the nonlinearities and the solvability of a linear matrix inequality (LMI)) on the system matrices then the observer synchronization can be feasible. We evaluate and demonstrate our approach with specific numerical results.

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is down that the message signal hidden in **chaotic** carrier of the systems can be perfectly reconstructed, and the bandwidth, the characteristic of distortion, and the level of security are improved. The chaos masking method based on multistage **chaotic** synchronization system for **secure** **communications**, to some extent, enhances the degree of security, as the message is not only just added to the **chaotic** carrier signal twice but is also decrypted twice. For example, in [42], a new method of global chaos synchronization among three different structures of **chaotic** systems is proposed under the framework of drive-response systems based on Lyapunov stabilization theorem; then, this method is applied to **secure** com- munication through **chaotic** masking. At last, it ana- lyzes the characteristics of noise and then presents a de-noising method using wavelet transform. With the MATLAB, the effects of signal de-noising could be demonstrated.

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function is used to replace the discontinuous sign function in the final design process step, can eliminate chattering in the control input to ensure feasibility for an actual physical system. However, to our knowledge, no previously developed scheme can obtain such a robust continuous controller for chaos synchronization control in a **secure** **communications** system, necessitating the development of a sliding mode control scheme. Therefore, based on a sliding mode controller [22], this work addresses the synchronization issue of a **chaotic** system, while devising a sliding mode control criterion. While implemented with electronic components, system synchronization is validated using the **chaotic** system and the controller. Finally, by using LabView, the **chaotic** synchronization system, integrated with cryptography, is applied to **secure** communication i.e. the encryption and decryption of audio and image signals.

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scheme for anti-synchronization of two identi- cal hyperchaotic complex T-systems, with three unknown parameters. We use the result of anti-synchronization for **secure** **communications** via the **chaotic** masking method. By apply- ing this process, we can achieve chaos anti- synchronization of master and slave systems, identify the unknown parameters, and mask and unmask the message signals simultaneously. In **secure** communication, a parameter and a state of master system are used for masking the informa- tion, such that a **chaotic** signal is added to the in- formation. The anti-synchronized states and es- timation of unknown parameters of the slave sys- tem are used to unmask the information. Based on the Barbalate’s lemma and Hamilton-Jacobi- Bellman(HJB) technique, an optimal adaptive sliding-mod controller with parameters estima- tion rules is designed to anti-synchronize complex **chaotic** T-systems asymptotically.

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Physical layer security has been recently recognized as a promising new design paradigm to provide security in wireless networks. In addition to the existing conventional cryptographic methods, physical layer security exploits the dynamics of fading channels to enhance secured wireless links. In this approach, jamming plays a key role by generating noise signals to confuse the potential eavesdroppers, and significantly improves quality and reliability of **secure** **communications** between legitimate terminals [13]. Jamming attack is a serious threat to the wireless **communications**. Reactive jamming maximizes the attack efficiency by jamming only when the targets are communicating, which can be readily implemented using software-defined radios. In this paper, explore the use of the multi-input multi-output (MIMO) technology to achieve jamming resilient orthogonal frequency-division multiplexing (OFDM) communication[14] This is an unfavorable scenario for secrecy performance as the system is interference-limited. In the literature, assuming that the receiver operates in half duplex (HD) mode, the aforementioned problem has been addressed via use of cooperating nodes who act as jammers to confound the eavesdropper [15]. Cooperative jamming is an approach that has been recently proposed for improving physical layer based security for wireless networks in the presence of an eavesdropper. While the source transmits **its** message to **its** destination, a relay node transmits a jamming signal to create interference at the eavesdropper. In this paper, a

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Abstract. The **secure** communication using synchronization between identical **chaotic** systems have been introduced in literature for a long time. A well-known practical **application** of **chaotic** synchronized systems is the Pecora and Carroll (P-C) **secure** communication method. In this paper, the P-C **secure** communication algorithm is applied to a novel three dimensional, autonomous **chaotic** attractor. Having a 45 ○ slope between sub-driver and sub- receiver circuits of a novel **chaotic** attractor clearly demonstrates that it can be used for the purpose of **secure** **communications**.

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In this paper, the existence of **chaotic** behavior in the single-well Duffing Os- cillator was examined under parametric excitations using Melnikov method and Lyapunov exponents. The minimum and maximum values were obtained and the dynamical behaviors showed the intersections of manifold which was illustrated using the MATCAD software. This extends some results in the li- terature. Simulation results indicate that the single-well **oscillator** is sensitive to sinusoidal signals in high frequency cases and with high damping factor, the amplitude of the **oscillator** was reduced.

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In this work we have considered the dynamics of multi-degree of freedom impact oscillators, using the example of a two degree of freedom system. From a detailed numerical study of this system we have observed a range of dynamical behaviour, including periodic motion, chatter (both periodic and **chaotic**), sticking and chaos. Of particular interest are phenomena which do not occur in single degree of freedom system, such as non-symmetric changes in periodicity, multiple resonance peaks and the existence of chatter in a system without preloading. From a modelling perspective, if such phenomena occur in a physical system, then using a multi-degree of freedom model may be necessary to capture the dynamics of the system.

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Spread spectrum signals for digital **communications** were originally invented for military communication, but nowadays they are widely used to provide reliable communication in a variety of commercial applications including mobile and wireless **communications**, which provide resistance to hostile jamming, hide the signal by transmitting it at low power, or make it possible for multiple users to communicate through the same channel. Conventionally binary pseudo-random (PN) sequence codes were used. It is a deterministic, periodic signal that is known to both transmitter and receiver which appears to an unauthorized listener, to be similar to white noise. DS-CDMA is a multiple access technique in which multiple users can transmit their data on the same channel using orthogonal spreading sequences [4].

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Abstract In this paper, the **modified** simplest equation method is successfully imple- mented to find travelling wave solutions of the generalized forms B (n, 1) and B(−n, 1) of Burgers equation. This method is direct, effective and easy to calculate, and it is a powerful mathematical tool for obtaining exact travelling wave solutions of the generalized forms B(n, 1) and B(−n, 1) of Burgers equa- tion and can be used to solve other nonlinear partial differential equations in mathematical physics.

Recently, increasing attention has been paid to discrete time models. The reasons are as follows: The numerical simulations of continuous-time models are obtained by the discretizing models. It is common practice to discretize the continuous-time model for experimental or computational purposes. The discrete-time model inherits the dynamic characteristics of the continuous-time model, and it also retains functional similarity to the continuous-time system and any physical or biological reality that the continuous- time model has []. At last, the discrete-time models have rich dynamical behaviors as compared to continuous-time models. We can get more accurate numerical simulations results from discrete time models. Herein, we will consider an **oscillator** model described by diﬀerence equations.

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the sensitivity and roundoff error, different methods are suggested [Ibrahiml986], one of.. them is to add an error feedback circuit into the structure. Or we can increase the number.. o[r]

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not only in the context of lasers [26, 27], but also in hy- drodynamics [28] and in nanomechanical resonators [29]. It is thus natural to ask whether an excitable response is still possible in the bounded-phase case. In the present work, we provide theoretical and experimental evidence of a mechanism leading to an excitable-like response, oc- curring at the transition from the phase-locking to the bounded-phase regime. Our system bifurcates from a phase-locked to a **chaotic**, self-pulsating state in which the pulses are not accompanied by phase slips. In “stan- dard” excitable systems, the self-pulsating state is asso- ciated to a simple attractor such as a limit cycle, and as a consequence the system always follows the same, unique path in phase space, and produces identical pulses. This property is not verified here. Nevertheless, albeit **chaotic**, the self-pulsating state is quite regular, because it con- sists of pulses of similar amplitudes, almost periodic in time (see inset of Fig. 2(b) and Fig. 3 (b)).Thus, we found that the response associated to this particular **chaotic** at- tractor presents features that are similar to excitability: Existence of a threshold i.e. need of a finite perturbation in order to trigger a response (which is fairly indepen- dent of the amplitude of the perturbation), and a well- defined refractory time during which the system cannot be excited again, after a first stimulus. The paper is organized as follows. In the next section, we describe the model equations and present numerical bifurcation curves and diagrams. We show that a **chaotic**, bounded phase attractor exists, and that close to the bifurcation point, intensity pulses with excitable-like characteristics can be found. In section III, we describe the experimen- tal setup and results, and compare them to the numerical predictions. Section IV is devoted to the conclusions.

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There is a separate class of problems called the problem of initial conditions for the Hot Big Bang Theory. Since the homogeneous Universe may form spherical or hyperbolic 3-dimensional manifold, **its** flatness at present is mysterious. Both at radiation and at matter dominated stages of the Universe expansion the size of causally connected region, particle horizon, grows faster than a physical spatial size does, so the Universe must be highly inhomogeneous at large scales, that is quite contrary to all observations. The Universe expansion is adiabatic, keeps the entropy conserved in a comoving volume, hence the entropy density in the very early Universe was enormously high s ≫ M 3

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and y k are the true and observed log-transformed intensities at the kth voxel, respectively, and β k is the bias field at the kth voxel. If the gain field is known, then it is relatively easy to estimate the tissue class by applying a conventional intensity-based segmentation to the corrected data. The following discussion is based the model of (2) and estimation of the gain field β k . **Modified** FCM algorithm(M-FCM)

This new generation now has the role of keeping a balance between maintain- ing the Italian quality in the fast-paces realm of design. Understanding the context in which products are made into is an important factor in achieving this goal, something that Irvine is a prime example of. With the new generation increasingly becoming non-Italian, Italian Design as we know it will continue to change, with **its** new definition dependent on the innovation that occur within the design capital that is Milan.

The seminal paper of Baptista [9] on **chaotic** cryptography inspires us to proposea one-way hash function based on **chaotic** maps. A thorough analysis of Baptista’sscheme was carried out by Alvarez et al [10] and they show that Baptista’salgorithm is vulnerable to all the four of cipher text only, known plain text, chosenplain text and chosen cipher text attacks. On the other hand, it was shown inthe literature that Baptista’s scheme has a lot of potential and could be modifiedto build a hash function. K.W.Wong **modified** Baptista’s algorithm by adoptinga dynamic look-up table to avoid collisions and preimage attack [11] and thencame up with a hashing scheme in 2003 [12]. X.Yi [13] in 2005 proposed a hashfunction based on **chaotic** tent maps which is claimed to be better than Wong’sscheme in **its** computational complexity. More recently H.Yang et al [14] havepublished another hash function based on a **chaotic** map network and Q.Yanget al [15] have published a hash function based on cell neural network.These approaches use a multitude of **chaotic** maps [16–19] and we show inthis paper that using two **chaotic** maps judiciously achieves a **secure** hash function. A **Secure** Keyed One-Way HashFunction{h k : k K}

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Genetic Algorithms (GAs) and Particles Swarm Optimization (PSO) are both population based algorithms that have proven to be successful in solving a variety of difficult problems. However, both models have strengths and weaknesses. Comparisons between GAs and PSOs have been made by Eberhart and Angeline, and both conclude that a hybrid of the standard GA and PSO models could lead to further advances [1,2]. Recently, a hybrid GA/PSO algorithm, Breeding Swarms (BSs), combining the strengths of GA with those of PSO, has been proposed by Matthew and Terence [3]. The performance of BS is competitive with both GA and PSO. BS was able to locate an optimum significantly faster than either GA or PSO. Inspired by the idea of breeding swarms [3], this paper proposes a GA/PSO hybrid algorithm, which combines the standard velocity and position update rules of PSO with the ideas of selection, crossover and mutation from GA. The operations inherited from GA facilitate a search globally but not exactly, while the interactions of PSO effectuate a search for an optimal. In order to improve the whole performance and to enhance the GA’s operations in terms of the searching ability, the notion of chaos is introduced to the initialization and replaced the ordinary GA mutation, and then a novel **Chaotic** Hybrid Algorithm (CHA) is proposed.

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