Abstract. A 3-D **new** **chaotic** **attractor** with **two** **quadratic** **nonlinearities** is proposed in this paper. The dynamical properties of the **new** **chaotic** system are described in terms of phase portraits, equilibrium points, Lyapunov exponents, Kaplan-Yorke dimension, dissipativity, etc. We show that the **new** **chaotic** system has three unstable equilibrium points. The **new** **chaotic** **attractor** is dissipative in nature. As an engineering application, adaptive **synchronization** of identical **new** **chaotic** attractors is designed via nonlinear control and Lyapunov stability theory. Furthermore, an electronic **circuit** realization of the **new** **chaotic** **attractor** is presented in detail to confirm the feasibility of the theoretical **chaotic** **attractor** model.

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For practical **implementation** of **chaotic** systems, it is important to design suitable electronic **circuit** design of **chaotic** systems [31-35]. Sambas et al. [31] discussed the **circuit** design of a six-term novel **chaotic** system with hidden **attractor**. Sambas et al. [32] derived a **circuit** design for a **new** 4-D **chaotic** system with hidden **attractor**. Sambas et al. [33] discussed the numerical simulation and **circuit** **implementation** for a Sprott **chaotic** system with one hyperbolic sinusoidal nonlinearity. Vaidyanathan et al. [34] presented a **new** 4-D **chaotic** hyperjerk system, and discussed **its** **synchronization**, **circuit** design and applications in RNG, image encryption and chaos-based steganography. Vaidyanathan et al. [35] reported a **new** **chaotic** **attractor** with **two** **quadratic** **nonlinearities** and discussed **its** **synchronization** via adaptive control and **circuit** **implementation**.

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Abstract—This paper reports the finding of a **new** 2-scroll **chaotic** system with four **quadratic** nonlinear terms. We also discover interesting properties of the **new** **chaotic** system such as equilibrium points, bifurcation and Lyapunov exponents. For a comparative study, we apply four different control methods for the global **synchronization** of the **new** **chaotic** system with itself and present a comparative analysis of the results obtained with the four control methods, viz. (A) Adaptive Control, (B) Backstepping Control, (C) Passive Control and (D) Integral Sliding Mode Control. Our results show that integral sliding mode control gives the best results for achieving **synchronization** of the **new** **chaotic** system with itself. An electronic **circuit** simulation of the **new** 2-scroll **chaotic** system is shown using MultiSIM to check the feasibility of the model.

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Abstract. A 3-D novel double-convection **chaotic** system with three **nonlinearities** is proposed in this research work. The dynamical properties of the **new** **chaotic** system are described in terms of phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, dissipativity, stability analysis of equilibria, etc. Adaptive control and **synchronization** of the **new** **chaotic** system with unknown parameters are achieved via nonlinear controllers and the results are established using Lyapunov stability theory. Furthermore, an electronic **circuit** realization of the **new** 3-D novel **chaotic** system is presented in detail. Finally, the **circuit** experimental results of the 3-D novel **chaotic** **attractor** show agreement with the numerical simulations.

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In the last few decades, **chaotic** and hyperchaotic systems have been applied in several areas of science and engineering [1-2]. Some important applications of **chaotic** systems can be listed out such as chemical reactors [3-5], oscillators [6-8], neural networks [9-10], memristors [11-12], ecology [13-14], robotics [15-16], Tokamak reactors [17-18], finance [19-20], etc.

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Chaos **synchronization** is an important problem in nonlinear science. During the last three decades, **synchronization** has received a great interest among various scientists [1–7]. The **synchronization** can be seen as the property shared by some objects to express a uniform rate of coexistence. For example, **two** harmonic oscillators can be synchronized if their periods are equal. However, for the case of **chaotic** oscillators, the concepts of frequency and phase are not well defined and, therefore, **two** **chaotic** oscillators can be synchronized if eventually, after a transitional time (a long or short time span), the oscillations coincide exactly at all times despite both oscillators started at different initial conditions. The idea of synchronizing **two** identical **chaotic** systems from different initial conditions was introduced in the seminal work in [1]. After that, several **synchronization** schemes were introduced in [50,58–66]. Besides, the practical applications of **chaotic** **synchronization** has some limitations to accomplish identical **synchronization**. For example, parameter mismatch will probably destroy the manifold of a **synchronization**. To deal with this issue, generalized **synchronization** approaches were introduced [3,67]. In this manner, we perform the **synchronization** of **two** **chaotic** oscillators following the approach given in [3]. Therefore, the chaos generator model from Equation (3) in Generalized Hamiltonian form, is given by

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The remainder of this paper is organized as follows. The model description and preliminaries are proposed in Section 2. Based on impulsive control theory and Lyapunov method, we shall try to propose a **new** and practical impulsive strategy to realize the **synchronization** for a class of **chaotic** systems in Section 3. Finally, some typical examples will be included to show the correctness of the theoretical results in Section 4, and the paper will be concluded in Section 5.

to secure communications. The transmission link is established via the Internet and National Instruments LabView, with the information encrypted and decrypted in the transmitter and the receiver, respectively, by computers. The audio signal is transmitted as follows. First, the initial conditions of the system and the sampling amount are determined. An identical number of keys are then generated, with which the audio signal is mixed accordingly. Figures 5 to 7 display the audio processing system, as well as summarize the experimental results. The image counterpart is described as follows. The encryption process is divided into **two** phases. The first one is the image itself, with both the symmetric key cryptosystem (SKC) and the master/slave system as the key, through which the image signal is encoded despite the same set of initial values. The initial values subsequently vary with the master/slave system status. The second one is the encryption of the master/slave system signal, with a Lorenz system as the key, as a means of removing the interception likelihood during data transmission. Figure 8 illustrates the image encryption/decryption system, where the Lorenz system is of invariant initial conditions; meanwhile, the master **chaotic** system randomly generates the initial conditions. Figures 9 to 11 summarize the experimental results. An entity photo of a Sprott master/slave system and a sliding mode controller are shown in Fig. 12.

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In this paper, we provide a **new** approach to study the geometry of **attractor**. By applying category, we investigate the relationship between **attractor** and **its** attraction basin. In a complete metric space, we prove that the categories of **attractor** and **its** attraction basin are always equal. Then we apply this result to both autonomous and non-autonomous systems, and obtain a number of cor- responding results.

Isochronal function projective **synchronization** between **chaotic** and time-delayed **chaotic** systems with unknown parameters is investigated in this article. Based on Lyapunov stability theory, adaptive controllers and parameter updating laws are designed to achieve the isochronal function projective **synchronization** between **chaotic** and time-delayed **chaotic** systems. The scheme is applied to realize the **synchronization** between time-delayed Lorenz systems and time-delayed hyper- **chaotic** Chen systems, respectively. Numerical simulations are also presented to show the effectiveness of the proposed method.

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Abstract: The problem of estimating reachable sets of nonlinear impulsive control systems with **quadratic** nonlinearity and with uncertainty in initial states and in the matrix of system is studied. The problem is studied under uncertainty conditions with set-membership description of uncertain variables, which are taken to be unknown but bounded with given bounds. We study the case when the system nonlinearity is generated by the combination of **two** types of functions in related differential equations, one of which is bilinear and the other one is **quadratic**. The problem may be reformulated as the problem of describing the motion of set-valued states in the state space under nonlinear dynamics with state velocities having bilinear-**quadratic** kind. Basing on the techniques of approximation of the generalized trajectory tubes by the solutions of control systems without measure terms and using the techniques of ellipsoidal calculus we present here a state estimation algorithms for the studied nonlinear impulsive control problem bilinear-**quadratic** type.

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recurrently investigated in **synchronization** of **chaotic** systems are: 1) under which conditions systems synchronize (values of physical parameters, initial conditions, etc.) and consequently, 2) under which conditions one is able to design **chaotic** sys- tems that synchronize with others? To answer the first question Lyapunov stability theory may be used—cf. [17]–[19], for the second, distinct control approaches have been put to test—cf. [6], [7], [12]–[14]. Some papers rely on analytic study –[8], [20], [21] and others on numerical methods and validation in simulation—cf. [16], [22], [23]. Beyond stability theory, work on **synchronization** analysis includes the study of synchroniza- tion in networks of oscillators—cf. [24]–[29], relying on tools from e.g., graph theory. Fundamental work on pinning-control- lability just appeared in [24]. In [26] conditions are established in terms of the average coupling path lengths among network nodes; see also [25]. The recent paper [28] establishes synchro-

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Chaos **synchronization** has attracted a great deal of attention since Pecora and Carroll [] established a chaos **synchronization** scheme for **two** identical **chaotic** systems with dif- ferent initial conditions. Various eﬀective methods such as robust control [], the sliding method control [], linear and nonlinear feedback control [], function projective [–], adaptive control [], active control [], backstepping control [], generalized backstep- ping method control [] and anti-**synchronization** [] have been presented to synchro- nize various **chaotic** systems.

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In order to observe the hybrid **synchronization** behav- ior of Chen hyper-**chaotic** system, assume that we have **two** Chen hyper-**chaotic** systems where the drive system with four state variables denoted by the subscript 1 and the response system with identical equations denoted by the subscript 2. Obviously, the initial condition on the drive system is different from that of response system. For the system (6), the drive and response systems are defined below, respectively:

oscillations in **two**-degree-of-freedom systems with qu adratic **nonlinearities** is investigated. The fundamental parametric resonance of the first mode and 3:1 internal resonance is considered, followed b y 1:2 in ternal and par ametr ic reso nance s of the second m ode. Th e meth od of mu ltiple time scales is app lied t o derive fo u r fir st-or der no n-lin ear or dinar y different ial equations that describe the modulation of the amplitudes and phases of both modes caused by resonance. These equ ations are u sed to determine steady state amplitu des. To determine stability of the steady state solu tions, small distu rbances in the amplitu des and phases are su pe rposed on the stea dy state solu tions and the resu lt ing equ atio ns are linearized. The eigenvalues of the corresponding system of first-order differential equ ations determine the stability of the steady state solutions. The instability modes are discussed and the amplitude and frequency response curves are presented by varying parameters of the system.

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The study of nonlinear behaviour, such as stability, control and **synchronization** of dynam- ical systems, has become an interdisciplinary research. As a result, several research works have been published on this subject by researchers of different disciplines [5, 6, 9, 34]. Due to the importance and interdisciplinary nature of nonlinear dynamical systems, **its** applications have been discovered in field studies like mathematics, physics, chemistry,

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Definition 2.4 [10] Let (X,τ) be a topological space and α an operator associated with τ. A subset A of X is said to be α-open if for each x є X there exists an open set U containing x such that α(U) ⊂ A. according to this definition every α-open is open and not conversely. Let us denote the collection of all α-open, semi-open sets in the topological space ( X , τ ) by τ α , SO(τ), respectively. We then have τ ⊆ τ α ⊆ SO ( τ ) . A subset B of X is said to be α- closed [7] if **its** complement is α-open.

This paper has been organized as follows. In Section 2, we derive results for the adaptive **synchronization** of identical uncertain Li systems (2009). In Section 3, we derive results for the adaptive **synchronization** of identical uncertain T systems (2008). In Section 4, we derive results for the adaptive **synchronization** of non- identical Li and uncertain T systems. In Section 5, we summarize the main results obtained in this paper.

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Control of complex irregular dynamics has evolved as one of the central issues in applied nonlinear science during the last decade. Nowadays the notion of chaos control involving stabilization of unstable periodic or stationary states in nonlinear dynamic systems has been extended to a much wider class of problems. Since the discovery of chaos **synchronization** introduced in (Carroll, 1990), there have been tremendous interests in studying the **synchronization** of **chaotic** systems. Recently, much research on the fuzzy model- based designs to stability and **synchronization** for **chaotic** systems have been carried out based on Takagi–Sugeno (T–S) fuzzy models (Park et al., 2002; Lian et al., 2001). In (Yan- Wu Wang et al., 2003), a fuzzy model-based designs for Chen’s **chaotic** stability and **synchronization** have been proposed. Based on the fuzzy hyper **chaotic** models, simpler fuzzy controllers have been designed for synchronizing hyper **chaotic** systems in (Hongbin Zhang, 2005). In this work, utilizing BELBIC model introduced in (Ali Reza Mehrabian et al., 2006; Saeed Jafarzadeh et al., 2008), we will design an intelligent controller for **synchronization** of **two** **new** 3D

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Abstract. In this paper, we study the **synchronization** control problem of a class of fractional-order **chaotic** financial systems with uncertain parameters. Based on the theory of fractional order stability and adaptive control method, a simple fractional-order synchronous controller of **two** fractional-order **chaotic** financial systems is designed, and a simple theoretical analysis is given to prove that the error system is stable. Then **two** fractional-order hybrid **synchronization** methods are proposed. Finally numerical simulation demonstrates the validity of methods.

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