Function Projective Synchronization of Discrete-Time Chaotic and Hyperchaotic Systems Using Backstepping Method

In this paper, a function projective synchronization scheme is developed to investigate the function projective synchronization between the discrete-time driven chaotic system and the discrete-time response chaotic system. With the aid of symbolic-numeric computation, we use the scheme to study the function projective synchronization between 2D Lorenz discrete-time system and Hdnon discrete-time system, as well as that between 3D discrete-time hyperchaotic system and Henon-like map via three scalar controllers, respectively. Moreover numerical simulations are used to verify the effectiveness of the proposed scheme.     

A new kind of nonlinear fractional-order chaotic phenomenon in coupled systems： coexistence of anti-phase and complete synchronization

In this paper, we have found a kind of interesting nonlinear phenomenon hybrid synchronization in linearly coupled fractional-order chaotic systems. This new synchronization mechanism, i.e., part of state variables are anti- phase synchronized and part completely synchronized, can be achieved using a single linear controller with only one drive variable. Based on the stability theory of the fractional-order system, we investigated the possible existence of this new synchronization mechanism. Moreover, a helpful theorem, serving as a determinant for the gain of the controller, is also presented. Solutions of coupled systems are obtained numerically by an improved Adams Bashforth-Moulton algorithm. To support our theoretical analysis, simulation results are given.     

Hopf bifurcation analysis and circuit implementation for a novelfour-wing hyper-chaotic system

In the paper, a novel four-wing hyper-chaotic system is proposed and analyzed. A rare dynamic phenomenon is found that this new system with one equilibrium generates a four-wing-hyper-chaotic attractor as parameter varies. The system has rich and complex dynamical behaviors, and it is investigated in terms of Lyapunov exponents, bifurcation diagrams, Poincare′ maps, frequency spectrum, and numerical simulations. In addition, the theoretical analysis shows that the system undergoes a Hopf bifurcation as one parameter varies, which is illustrated by the numerical simulation. Finally, an analog circuit is designed to implement this hyper-chaotic system.     

Chaos Control in a New Three-Dimensional Chaotic T System

In this paper, we study chaos control of the new 3D chaotic system. We use three feedback methods （the linear, speed, doubly-periodic function controller） to suppress the chaos to unstable equilibrium. As a result, some controllers are obtained. Moreover, numerical simulations are used to verify the effectiveness of the obtained controllers.     

Heteroclinic Orbit Existence on a Type of Chaotic System with Delays

In this paper, we discuss a type of chaotic system with delays. We study the equilibrium points and the existence of heteroclinic orbit of the system. Heteroclinic orbit existence theorem is proposed and proved by applying the undetermined coefficient method, which shows the complex dynamical properties of this system.     

Exponential synchronization of chaotic Lur’e systems with time-varying delay via sampled-data control

In this paper, we study the exponential synchronization of chaotic Lur＇e systems with time-varying delays via sampled-data control by using sector nonlinearties. In order to make full use of information about sampling intervals and interval time-varying delays, new Lyapunov-Krasovskii functionals with triple integral terms are introduced. Based on the convex combination technique, two kinds of synchronization criteria are derived in terms of linear matrix inequal- ities, which can be efficiently solved via standard numerical software. Finally, three numerical examples are provided to demonstrate the less conservatism and effectiveness of the proposed results.     

Function Projective Synchronization in Discrete-Time Chaotic System with Uncertain Parameters

The function projective synchronization of discrete-time chaotic systems is presented. Based on backstepping design with three controllers, a systematic, concrete and automatic scheme is developed to investigate function projective synchronization （FPS） of discrete-time chaotic systems with uncertain parameters. With the aid of symbolic-numeric computation, we use the proposed scheme to illustrate FPS between two identical 3D Henon-like maps with uncertain parameters. Numeric simulations are used to verify the effectiveness of our scheme.     

Existence of heteroclinic orbits in a novel three-order dynamical system

In this paper,we design a novel three-order autonomous system.Numerical simulations reveal the complex chaotic behaviors of the system.By applying the undetermined coefficient method,we find a heteroclinic orbit in the system.As a result,the Si’lnikov criterion along with some other given conditions guarantees that the system has both Smale horseshoes and chaos of horseshoe type.     

Implementation of a multi-scroll novel two-attractor grid chaotic system

This paper proposed a method of generating two attractors in a novel grid multi-scroll chaotic system. Based on a newly generated three-dimensional system, a two-attractor grid multi-scroll attractor system can be generated by adding two triangular waves and a sign function. Some basic dynamical properties, such as equilibrium points, bifurcations, and phase diagrams, were studied. Furthermore, the system was experimentally confirmed by an electronic circuit. The circuit simulation results and numerical simulation results verified the feasibility of this method.     

Generalized Synchronization Between Different Fractional-Order Chaotic Systems

In this paper, using scalar feedback controller and stability theory of fractional-order systems, a generalized synchronization method for different fractional-order chaotic systems is established. Simulation results show the effectiveness of the theoretical results.     

Periodic,quasiperiodic, and chaotic breathers in two-dimensional discrete β-Fermi-Pasta-Ulam lattice

Using numerical method, we investigate whether periodic, quasiperiodic, and chaotic breathers are supported by the two-dimensional discrete Fermi-Pasta-Ulam (FPU) lattice with linear dispersion term. The spatial profile and time evolution of the two-dimensional discrete β -FPU lattice are segregated by the method of separation of variables, and the numerical simulations suggest that the discrete breathers (DBs) are supported by the system. By introducing a periodic interaction into the linear interaction between the atoms, we achieve the coupling of two incommensurate frequencies for a single DB, and the numerical simulations suggest that the quasiperiodic and chaotic breathers are supported by the system, too.     

Generalized Synchronization of Time-Delayed Discrete Systems

This paper establishes two theorems for two time-delayed （chaotic） discrete systems to achieve timedelayed generalized synchronization （TDGS）. These two theorems uncover the genera/forms of two TDGS systems via a prescribed transformation. As examples, we convert the Lorenz three-dimensional chaotic map to an equal time-delayed system as the driving system, and construct the TDGS driven systems according to the Theorems 1 and 2. Numerical simulations demonstrate the effectiveness of the proposed theorems.     

Master equation describing the diffusion process for a coherent state

The evolution of a pure coherent state into a chaotic state is described very well by a master equation, as is validated via an examination of the coherent state＇s evolution during the diffusion process, fully utilizing the technique of integration within an ordered product （IWOP） of operators. The same equation also describes a limitation that maintains the coherence in a weak diffusion process, i.e., when the dissipation is very weak and the initial average photon number is large. This equation is dp/dt = -κ[a＋ap -a＋pa -apa＋ ＋ paa＋]. The physical difference between this diffusion equation and the better-known amplitude damping master equation is pointed out.     

PC synchronization of a class of chaotic systems via event-triggered control

The PC synchronization of a class of chaotic systems is investigated in this paper. The drive system is assumed to have only one state variable available. By constructing proper observers, some novel criteria for PC synchronization are proposed via event-triggered control scheme. The Lu¨ system and Chen system are taken as examples to demonstrate the efficiency of the proposed approach.     

Chaotic Ratchet Effect for Bose-Einstein Condensed Atoms in 1D Optical Lattices

The chaotic ratchet effect for Bos-Einstein condensed atoms in an optical lattice is investigated. By using the direct perturbation method we obtain the chaotic solution of the condensed system. Theoretical analysis reveals that the transport of the condensed atoms in the ratchet potential is a chaotic one, and corresponding numerical results agree well with the theoretical results.     

H∞ synchronization of chaotic neural networks with time-varying delays

In this paper,we investigate the problem of H∞ synchronization for chaotic neural networks with time-varying delays.A new model of the networks with disturbances in both master and slave systems is presented.By constructing a suitable Lyapunov–Krasovskii functional and using a reciprocally convex approach,a novel H∞ synchronization criterion for the networks concerned is established in terms of linear matrix inequalities(LMIs)which can be easily solved by various effective optimization algorithms.Two numerical examples are given to illustrate the effectiveness of the proposed method.     

Solution of Schrodinger Equation for Two-Dimensional Complex Ouartic Potentials

We investigate the quasi-exact solutions of the Schrodinger wave equation for two-dimensional non-hermitian complex Hamiltonian systems within the frame work of an extended complex phase space characterized by x = x1 ＋ ip3, y = x2 ＋ ip4, px= p1＋ ix3, py= p2 ＋ ix4. Explicit expressions of the energy eigenvalues and the eigenfunctions for ground and first excited states for a complex quartic potential are obtained. Eigenvalue spectra of some variants of the complex quartic potential, including PT-symmetrie one, are also worked out.     

Fusion,fission, and annihilation of complex waves for the （2＋l）-dimensional generalized Calogero-Bogoyavlenskii-Schiff system

With the help of the symbolic computation system, Maple and Riccati equation （ξ＇ = ao ＋ a1ξ＋ a2ξ2）, expansion method, and a linear variable separation approach, a new family of exact solutions with q = lx ＋ my ＋ nt ＋ Г（x,y, t） for the （2＋1）-dimensional generalized Calogero-Bogoyavlenskii-Schiff system （GCBS） are derived. Based on the derived solitary wave solution, some novel localized excitations such as fusion, fission, and annihilation of complex waves are investigated.     

Does the eigenratio synchronizability of λ2/λN represent the a complex network？

In the study of complex networks, it is commonly believed that the eigenratio λ2/λN of the Laplacian matrix of a network represents the network synchronizability, especially for symmetric networks. This paper gives two counterexamples to show that this is not true for the case where the network has a disconnected synchronized region. Consequently, a simple answer is presented to the question of when the eigenratio λ2/λN does represent the network synchronizability.     

Complex Wave Solutions for （2＋1）-Dimensional Modified Dispersive Water Wave System

The extended Riccati mapping approach^[1] is further improved by generalized Riccati equation, and combine it with variable separation method, abundant new exact complex solutions for the （2＋1）-dimensional modified dispersive water-wave （MDWW） system are obtained. Based on a derived periodic solitary wave solution and a rational solution, we study a type of phenomenon of complex wave.