Meng Zhan

Error function attack of chaos synchronization based encryption schemes

Different chaos synchronization based encryption schemes are reviewed and compared from the practical point of view. As an efficient cryptanalysis tool for chaos encryption, a proposal based on the error function attack is presented systematically and used to evaluate system security. We define a quantitative measure (quality factor) of the effective applicability of a chaos encryption scheme, which takes into account the security, the encryption speed, and the robustness against channel noise. A comparison is made of several encryption schemes and it is found that a scheme based on one-way coupled chaotic map lattices performs outstandingly well, as judged from quality factor.

Effects of frequency-degree correlation on synchronization transition in scale-free networks

Explosive synchronization in the scale-free network with a positive frequency-degree correlation has been reported recently (GOMEZ G. J. et al., Phys. Rev. Lett., 106 (2011) 128701). In this article, we generalize this study and find that the explosive synchronization is replaced by a kind of hierarchical synchronization if the microscopic correlation between the frequency and the interacting topology of the network becomes negative. A star network model is set to prove this novel behavior. We also find that the degree assortativity has significant influence on the explosive synchronization but slight impact on the hierarchical synchronization. These findings are meaningful for revealing unusual effects of correlations between dynamics and structure of complex networks. Copyright (c) EPLA, 2013

Synchronizing large number of nonidentical oscillators with small coupling

The topic of synchronization of oscillators has attracted great and persistent interest, and all previous conclusions and intuitions have convinced that large coupling is required for synchronizing a large number of coupled nonidentical oscillators. Here the influences of different spatial frequency distributions on the efficiency of frequency synchronization are investigated by studying arrays of coupled oscillators with diverse natural frequency distributions. A universal log-normal distribution of critical coupling strength K-c for synchronization irrespective of the initial natural frequency is found. In particular, a physical quantity roughness R of spatial frequency configuration is defined, and it is found that the efficiency of synchronization increases monotonously with R. For large R we can reach full synchronization of arrays with a large number of oscillators at finite K-c. Two typical kinds of synchronization, the multiple-clustering one and the single-center-clustering one, are identified for small and large Rs, respectively. The mechanism of the latter type is the key reason for synchronizing long arrays with finite K-c. Copyright (C) EPLA, 2012

Insensitive dependence of delay-induced oscillation death on complex networks

Oscillation death (also called amplitude death), a phenomenon of coupling induced stabilization of an unstable equilibrium, is studied for an arbitrary symmetric complex network with delay-coupled oscillators, and the critical conditions for its linear stability are explicitly obtained. All cases including one oscillator, a pair of oscillators, regular oscillator networks, and complex oscillator networks with delay feedback coupling, can be treated in a unified form. For an arbitrary symmetric network, we find that the corresponding smallest eigenvalue of the Laplacian λ(N) (0 >λ(N) ≥ -1) completely determines the death island, and as λ(N) is located within the insensitive parameter region for nearly all complex networks, the death island keeps nearly the largest and does not sensitively depend on the complex network structures. This insensitivity effect has been tested for many typical complex networks including Watts-Strogatz (WS) and Newman-Watts (NW) small world networks, general scale-free (SF) networks, Erdos-Renyi (ER) random networks, geographical networks, and networks with community structures and is expected to be helpful for our understanding of dynamics on complex networks.

Inhomogeneous stationary and oscillatory regimes in coupled chaotic oscillators

The dynamics of linearly coupled identical Lorenz and Pikovsky-Rabinovich oscillators are explored numerically and theoretically. We concentrate on the study of inhomogeneous stable steady states (oscillation death (OD) phenomenon) and accompanying periodic and chaotic regimes that emerge at an appropriate choice of the coupling matrix. The parameters, for which OD occurs, are determined by stability analysis of the chosen steady state. Three model-specific types of transitions to and from OD are observed: (1) a sharp transition to OD from a nonsymmetric chaotic attractor containing random intervals of synchronous chaos; (2) transition to OD from the symmetry-breaking chaotic regime created by negative coupling; (3) supercritical bifurcation of OD into inhomogeneous limit cycles and further evolution of the system to inhomogeneous chaotic regimes that coexist with complete synchronous chaos. These results may fill a gap in the understanding of the mechanism of OD in coupled chaotic systems.

MIB Galerkin method for elliptic interface problems

Material interfaces are omnipresent in the real-world structures and devices. Mathematical modeling of material interfaces often leads to elliptic partial differential equations (PDEs) with discontinuous coefficients and singular sources, which are commonly called elliptic interface problems. The development of high-order numerical schemes for elliptic interface problems has become a well defined field in applied and computational mathematics and attracted much attention in the past decades. Despite of significant advances, challenges remain in the construction of high-order schemes for nonsmooth interfaces, i.e., interfaces with geometric singularities, such as tips, cusps and sharp edges. The challenge of geometric singularities is amplified when they are associated with low solution regularities, e.g., tip-geometry effects in many fields. The present work introduces a matched interface and boundary (MIB) Galerkin method for solving two-dimensional (2D) elliptic PDEs with complex interfaces, geometric singularities and low solution regularities. The Cartesian grid based triangular elements are employed to avoid the time consuming mesh generation procedure. Consequently, the interface cuts through elements. To ensure the continuity of classic basis functions across the interface, two sets of overlapping elements, called MIB elements, are defined near the interface. As a result, differentiation can be computed near the interface as if there is no interface. Interpolation functions are constructed on MIB element spaces to smoothly extend function values across the interface. A set of lowest order interface jump conditions is enforced on the interface, which in turn, determines the interpolation functions. The performance of the proposed MIB Galerkin finite element method is validated by numerical experiments with a wide range of interface geometries, geometric singularities, low regularity solutions and grid resolutions. Extensive numerical studies confirm the designed second order convergence of the MIB Galerkin method in the L∞ and L2 errors. Some of the best results are obtained in the present work when the interface is C1 or Lipschitz continuous and the solution is C2 continuous.

MIB method for elliptic equations with multi-material interfaces

Elliptic partial differential equations (PDEs) are widely used to model real-world problems. Due to the heterogeneous characteristics of many naturally occurring materials and man-made structures, devices, and equipments, one frequently needs to solve elliptic PDEs with discontinuous coefficients and singular sources. The development of high-order elliptic interface schemes has been an active research field for decades. However, challenges remain in the construction of high-order schemes and particularly, for nonsmooth interfaces, i.e., interfaces with geometric singularities. The challenge of geometric singularities is amplified when they are originated from two or more material interfaces joining together or crossing each other. High-order methods for elliptic equations with multi-material interfaces have not been reported in the literature to our knowledge. The present work develops matched interface and boundary (MIB) method based schemes for solving two-dimensional (2D) elliptic PDEs with geometric singularities of multi-material interfaces. A number of new MIB schemes are constructed to account for all possible topological variations due to two-material interfaces. The geometric singularities of three-material interfaces are significantly more difficult to handle. Three new MIB schemes are designed to handle a variety of geometric situations and topological variations, although not all of them. The performance of the proposed new MIB schemes is validated by numerical experiments with a wide range of coefficient contrasts, geometric singularities, and solution types. Extensive numerical studies confirm the designed second order accuracy of the MIB method for multi-material interfaces, including a case where the derivative of the solution diverges.

Complete synchronization in coupled limit-cycle systems

Most of the previous studies on the synchronization of coupled limit-cycle systems focused on synchronization in the context of phase or frequency locking. In this letter, we study complete synchronization in coupled limit-cycle systems in terms of complete trajectory locking, by generalizing the master stability function method, which is originally developed in the study of synchronization in coupled chaotic systems. Four typical types of transverse Lyapunov exponent of standard mode, which lead to different synchronization-desynchronization patterns, are classified. In many respects, the behaviors in such coupled systems are distinct from those in coupled chaotic systems, such as the size effect and the desynchronous behavior.

Periodic states with functional phase relation in weakly coupled chaotic Hindmarsh-Rose neurons

Abstract Periodic states with functional phase relation are found for extremely weak coupling in the coupled Hindmarsh–Rose (HR) neurons model and the coupled Pikovsky circuit model where all oscillators take a same orbit while the phases of different oscillators have functional relation. By analyzing a periodically forced chaotic neuron, the phase distribution of various oscillators of the coupled HR neurons system is obtained. The response curves of phase distribution correspond to different spike lock. For the Pikovsky circuit model, similar phase distribution is also obtained.

Splay States in a Ring of Coupled Oscillators: From Local to Global Coupling

In this work, we study dynamical behavior in a ring of coupled nonlinear oscillators and focus on the stability of homogeneous steady state and splay states by performing linear stability analysis. Different cases from local (nearest-neighbor) coupling to global (all-to-all) coupling by increasing the range of coupling are studied. Some explicit stability results for each possible splay state are analytically obtained. It is valuable to compare their similarities and differences for these different cases. All these results could shed improved light on our understanding of the organization of splay states in coupled oscillators.