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It is crucial to determine the optimal number of clusters for the clustering quality in cluster analysis. From the standpoint of sample geometry, two concepts, i.e., the sample clustering dispersion degree and the sample clustering synthesis degree, are defined, and a new clustering validity index is designed. Moreover, a method for determining the optimal number of clusters based on an agglomerative hierarchical clustering (AHC) algorithm is proposed. The new index and the method can evaluate the clustering results produced by the AHC and determine the optimal number of clusters for multiple types of datasets, such as linear, manifold, annular, and convex structures. Theoretical research and experimental results indicate the validity and good performance of the proposed index and the method.

Based on the modified system approach the generalized synchronization (GS) in two bidirectionally coupled discrete dynamical systems is classified into several types, and under some conditions, the existence, Lipschitz smoothness and Holder continuity of two kinds of GS therein are derived and theoretically proved. In addition, numerical simulations validate the present theory.

This paper studies the existence of Holder continuity of the generalized synchronization (GS) manifold. When the modified response system has an asymptotically stable equilibrium, periodic or quasi-periodic orbit, and chaotic attractor, GS is classified into four types accordingly. The first three types of GS are considered, and based on the Schauder fixed point theorem, sufficient conditions for Holder continuous GS in the coupled non-autonomous systems are derived and theoretically proved.

This paper analyzes the approximation of a general long-memory ARFIMA ( p , d , q ) process by a short-memory ARMA(1, 1) process. To validate this approximation, a mean square error forecast criterion is considered, and the calculation of the mean square error between the observation X t + l of an ARFIMA process and the l -step-ahead forecast of the ARMA(1, 1) process is presented. The performance of the ARMA(1, 1) approximation to an ARFIMA model is illustrated by using an application to a DNA sequence of orf virus. This paper gives some theoretical justification on the mean square error forecast criterion based on a selected ARMA(1, 1) model when compared to the general ARFIMA( p , d , q ) process. The paper also provides a DNA sequence of orf virus analysis when the time series is originated from the tangent values of the Chaos Game Representation coordinates for each nucleotide. The conclusions of this paper work well because the estimator value of d is small ( d ˆ = 0.16 ). The paper also gives the other parameter estimate of the fitted ARFIMA (0, d , 1) model and one-step predictions using ARMA(1, 1) model.

Hybrid projective synchronization (HPS), in which the different state variables can synchronize up to different scaling factors, is numerically observed in coupled partially linear chaotic complex nonlinear systems without adding any control term in the present paper. The scaling factors of HPS are hardly predictable. Linear feedback control method is thus adopted to control them onto any desired values based on Lyapunov stability theory. Moreover, numerical simulations are given to illustrate and verify the analytical results.

This paper demonstrates theoretically that two completely different systems can implement GS via impulsive control, moreover by using impulsive control, for a given manifold y = H ( x ) we construct a response system to achieve GS with drive system and the synchronization manifold is y = H ( x ). Our theoretical results are supported by numerical examples.

This paper addresses the Q-S synchronization between chaotic and/or hyper-chaotic systems. Based on the Lyapunov stability theorem, a general scheme for Q-S synchronization of chaotic and/or hyper-chaotic systems is proposed. The drive and response systems discussed in this paper can be strictly different dynamical systems (including different dimensional systems). Four illustrative numerical simulations are also given to demonstrate the effectiveness of the proposed general scheme for Q-S synchronization.

A systematic and concrete scheme is proposed to study the full state hybrid projective synchronization (FSHPS) of a general class of chaotic maps based on the active control idea. The scheme is accessible to the FSHPS of two identical or different chaotic maps. The 3D generalized Henon map and 3D discrete-time Grassi–Miller map are chosen to illustrate the proposed scheme, and numerical simulations are given to show the effectiveness of the proposed chaos synchronization method.

In this paper, we study the projective cluster synchronization in a drive-response dynamical network with 1 + N coupled partially linear chaotic systems. Because the scaling factors characterizing the dynamics of projective synchronization remain unpredictable, pinning control ideas are adopted to direct the different scaling factors onto the desired values. It is also shown that the projection cluster synchronization can be realized by controlling only one node in each cluster. Numerical simulations on the chaotic Lorenz system are illustrated to verify the theoretical results.

Chaos synchronization, as an important topic, has become an active research subject in nonlinear science. Over the past two decades, chaos synchronization between nonlinear systems has been extensively studied, and many types of synchronization have been announced. This paper introduces another novel type of chaos synchronization – full state hybrid projective synchronization (FSHPS), which includes complete synchronization, anti-synchronization and projective synchronization as its special item. Based on the Lyapunov’s direct method, the general FSHPS scheme is given and illustrated with Lorenz chaotic system and hyperchaotic Chen system as examples. Numerical simulations are used to verify the effectiveness of the proposed scheme.

Projective synchronization (characterized by a scaling factor α ) and its control have recently been well investigated in two coupled chaotic oscillators, but it has not been realized yet in the case of more than two chaotic systems. In this paper, we study the projective synchronization in 1 + N coupled chaotic systems by introducing a drive-response dynamical networks model. Furthermore, pinning control techniques are also adopted to direct the scaling factor onto the desired value. It is also shown that the projection synchronization of the drive-response dynamical networks can be realized by controlling only one node. Numerical simulations on the chaotic Lorenz system are illustrated to verify the theoretical results.

Using periodic parametric perturbation to control chaos in Lorenz system will be finished in this paper. Based on the chaotic perturbation criteria—Melnikov method, a appropriate parametric condition for controlling chaos will be given out, which guide the system from chaotic motion to low-periodic motion. Some results of the numerical simulation are also explained clearly by rigorous analysis.

The problem of suppressing chaos in the time-delayed feedback control (TDFC) system is considered. The validity of Melnikov's technique in the planar Hamiltonian system perturbed by weak periodic terms and time-delay terms is discussed. Melnikov's technique is used to analyze a TDFC chaotic system that is nonautonomous. It is revealed with analytical results that the delayed time can be independent of the period of the inherent unstable orbit in the chaotic attractor. Numerical simulations demonstrate the effectiveness of Melnikov's analysis.

A simple control method is presented to convert chaos into periodic motion by using linear state feedback of an available system variable. The low-periodic orbits can be derived from the chaotic attractor with slight change of the dynamical structure in the original dynamics. Melnikov's perturbation method in generalized Hamiltonian systems is introduced to explain the control mechanism of directing chaotic motion towards low-periodic motion in the Lorenz equations. Moreover, the existence of periodic orbits is examined in the perturbed Lorenz system. Simulation results verify the effectiveness of Melnikov's analysis.

In this paper boundary value problems of Riemann-Hilbert type for monogenic functions in the up half space are investigated in both the classical sense and L p sense. The explicit formulae of the solutions are obtained as well.

The paper studies the existence of generalized synchronization in complex networks, which consist of chaotic systems. When a part of modified nodes are chaotic, and the others have asymptotically stable equilibriums or orbital asymptotically stable periodic solutions, under certain conditions, the existence of generalized synchronization can be turned to the problem of contractive fixed point in the family of Lipschitz functions. In addition, theoretical proofs are proposed to the exponential attractive property of generalized synchronization manifold. Numerical simulations validate the theory.

This paper studies the existence of Hölder continuity of generalized synchronization (GS). Based on the modified system approach, GS is classified into three types: equilibrium GS, periodic GS, and C-GS, when the modified system has an asymptotically stable equilibrium, asymptotically stable limit cycles, and chaotic attractors, respectively. The existence of the first two types of Hölder continuous GS inertial manifolds are strictly theoretically proved.

We study the projective synchronization with different scaling factors in 1+N coupled chaotic systems in a drive-response dynamical networks(DRDNs) model. By using a simple linear control, some sufficient criteria for projective synchronization with different scaling factors in the response networks are given based on Lyapunov function method. Numerical simulations on chaotic Lorenz system verify the rightness of the theoretical analysis scheme.