Resonance in Defect Turbulence under Periodic External Force

The periodically forced spatially extended Brusselator is investigated in the chaotic regime. We explore resonant or non-resonant patterns generated under various forcing frequencies and forcing amplitudes. Resonant spatially uniform oscillation and irregular structures are found. Furthermore two types of regular spatial patterns are generated under appropriate parameters. Our results of numerical simulations demonstrate that periodic force can give rise to resonant patterns in forced systems of spatiotemporal chaos similar to the situation of forced systems of regular oscillations.     

Hexagonal Standing-Wave Patterns in Periodically Forced Reaction-Diffusion Systems

The periodically forced spatially extended Brusselator is investigated in the oscillating regime. The temporal response and pattern formation within the 2：1 frequency-locking band where the system oscillates at one half of the forcing frequency are examined. An hexagonal standing-wave pattern and other resonant patterns are observed. The detailed phase diagram of resonance structure in the forcing frequency and forcing amplitude parameter space is calculated. The transitions between the resonant standing-wave patterns are of hysteresis when control parameters are varied, and the presence of multiplicity is demonstrated. Analysis in the framework of amplitude equation reveals that the spatial patterns of the standing waves come out as a result of Turing bifurcation in the amplitude equation.     

Dynamics of a pair of coupled nonlinear oscillators

A pair of coupled classical oscillators with a general potential and general form of coupling is investigated. For general potentials, the single-frequency solution is shown to be stable for small excitations. For special potentials, such system remains stable for an arbitrary excitation. In both cases, the stability does not depend on the form of coupling. Transition to the instability regime follows from the way how nonlinear potential entrains the energy transfer between the oscillators. Relation between the existence of multi-frequency quasi-periodic or periodic solutions and the instability of single-frequency ones is discussed.     

Wave Grouping of a Drifting Spiral Wave in the Presence of an External Field

The phenomenon of wave grouping, in which the dense waves and the sparse waves can form groups in front of the spiral tip when the spiral wave is meandering, has been reported in a chemical reaction system recently. We present a method to realize the phenomenon of wave grouping by applying an external field to the system. The numerical simulations are carried out on the basis of the FitzHugh-Nagumo equations.     

Synchronization defect lines in complex-oscillatory target waves

It is well known that one of key features of spiral waves in complex-oscillatory media is the appearance of synchronization defect lines, across which the phase of the oscillation changes by multiplies of 2π. In this Letter, we report the appearance of synchronization defect lines in target waves in complex-oscillatory media by studying a model of two-dimensional Rössler reaction-diffusion system subject to an appropriate periodic force in a small region of the center of domain. The geometric structure and stability of the defect lines are studied.     

Dynamics Analysis and Transition Mechanism of Bursting Calcium Oscillations in Non-Excitable Cells

A one-pool model with Ca^2＋-activated inositol-trisphosphate-concentration degradation is considered. For complex bursting Ca^2＋ oscillation, point-cycle bursting of subHopf-subHopf type is found to be in the intermediate state from quasi-periodic bursting to point-point bursting of subHopf-subHopf type. The fast-slow burster analysis is used to study the transition mechanisms among simple periodic oscillation, quasi-periodic bursting, point-point and point-cycle burstings. The dynamics analysis of different oscillations provides better insight into the generation and transition mechanisms of complex intra- and inter-cellular Ca^2＋ signalling.     

Controlling Spiral Dynamics in Excitable Media by a Weakly Localized Pacing

Spiral dynamics controlled by a weakly localized pacing around the spiral tip is investigated. Numerical simulations show two distinct characteristics when the pacing is applied with the weak amplitude for suitable frequencies： for a rigidly .rotating spiral, a transition from rigid rotation to meandering motion is observed, and for unstable spiral waves, spiral breakup can be prevented. Successfully preventing spiral breakup is relevant to the modulation of the tip trajectory induced by a localized pacing.     

Kinematic model of propagating arc-like segments with feedback

Using a kinematic approach, we propose a model of arc-like wave segments in which the free ends are stabilized by using a feedback algorithm. The model can demonstrate the experimental results and numerical computations of a reaction-diffusion system. This model also reveals some aspects of spiral wave dynamics with the free ends including not only the stabilization of wave segments using feedback, but also a critical behavior with respect to the initial wave size in media with fixed excitability.     

The boundary-induced spiral wave drift in the complex Ginzburg–Landau equation

We numerically investigate the boundary-induced spiral wave drift in the complex Ginzburg–Landau equation. We find some novel phenomena for the spiral drifting dynamics such as the chaotic behaviors, the transient chaos and asymmetrical attractors.     

Effects of Periodic Forcing Amplitude on the Spiral Wave Resonance Drift

We study dynamics of spiral waves under a uniform periodic temporal forcing in an excitable medium. With a specific combination of frequency and amplitude of the external periodic forcing, a resonance drift of a spiral wave occurs along a straight line, and it is accompanied by a complicated ‘flower-like＇ motion on each side of this bifurcate boundary line. It is confirmed that the straight-line drift frequency of spiral waves is not locked to the nature rotation frequency as the forcing amplitude expends are further verified numerically for a simplified kinematical the range of the spiral wave frequency. These results model.     

Pattern Synchronization in a Two-Layer Neuronal Network

Pattern synchronization in a two-layer neuronal network is studied. For a single-layer network of Rulkov map neurons, there are three kinds of patterns induced by noise. Additive noise can induce ordered patterns at some intermediate noise intensities in a resonant way; however, for small and large noise intensities there exist excitable patterns and disordered patterns, respectively. For a neuronal network coupled by two single-layer networks with noise intensity differences between layers, we find that the two-layer network can achieve synchrony as the interlayer coupling strength increases. The synchronous states strongly depend on the interlayer coupling strength and the noise intensity difference between layers.     

Firing Patterns and Transitions in Coupled Neurons Controlled by a Pacemaker

To reveal the dynamics of neuronal networks with pacemakers, the firing patterns and their transitions are investigated in a ring HR neuronal network with gap junctions under the control of a pacemaker. Compared with the situation without pacemaker, the neurons in the network can exhibit wrious firing patterns as the externed current is applied or the coupling strength of pacemaker varies. The results are beneficial for understanding the complex cooperative behaviour of large neural assemblies with pacemaker control.     

Average Synchronization and Temporal Order in a Noisy Neuronal Network with Coupling Delay

Average synchronization and temporal order characterized by the rate of firing are studied in a spatially extended network system with the coupling time delay, which is locally modelled by a two-dimensional Rulkov map neuron. It is shown that there exists an optimal noise level, where average synchronization and temporal order are maximum irrespective of the coupling time delay. Furthermore, it is found that temporal order is weakened when the coupling time delay appears. However, the coupling time delay has a twofold effect on average synchronization, one associated with its increase, the other with its decrease. This clearly manifests that random perturbations and time delay play a complementary role in synchronization and temporal order.     

Preserving synchronization under matrix product modifications

In this article we present a methodology under which stability and synchronization of a dynamical master/slave system configuration are preserved under modification through matrix multiplication. The objective is to show that under a defined multiplicative group, hyperbolic critical points are preserved along the stable and unstable manifolds. The properties of this multiplicative group were determined through the use of simultaneous Jordan decomposition. It is also shown that a consequence of this approach is the preservation of the signature of the Jacobian matrix associated with the dynamical system. To illustrate the results we present several examples of different modified systems.     

Synchronization of Complex Network with Drivingly Coupled Scheme

We present a network model with a new coupled scheme which is the generalization of drive-response systems called a drivingly coupled network. The synchronization of the network is investigated by numerical simulations based on Lorenz systems. By calculating the largest transversal Lyapunov exponents of such network, the stable and unstable regions of synchronous state for eigenvalues in such network can be obtained and many kinds of drivingly coupled arrays based on Lorenz systems such as all-to-all, star-shape, ring-shape and chain-shape networks are considered.     

Nonlinear Dynamics of a Sliding Chain in a Periodic Potential

Nonlinear dynamics of the sliding process of a chain driven with a constant velocity at one end in a periodic substrate potential is investigated. The driven chain exhibits distinctly different dynamical characteristics at different velocities. In the low velocity region, the chain moves in a stick-slip manner. When the driving velocity is increased, the stick-slip behaviour is replaced by complicated and regular oscillatory motions. The dependence of the dynamics on the coupling strength is studied and the step-like behaviour is found, where different steps correspond to different dynamical phases.     

Stability analysis and design of amplitude death induced by a time-varying delay connection

The present Letter considers amplitude death in a pair of oscillators coupled by a time-varying delay connection. A linear stability analysis is used to derive the boundary curves for amplitude death in a connection parameters space. The delay time can be arbitrarily long for certain amplitude of delay variation and coupling strength. A simple systematic procedure for designing such variation and strength is provided. The theoretical results are verified by a numerical simulation.     

Projective cluster synchronization in drive-response dynamical networks

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.     

Stochastic synchronization of coupled neural networks with intermittent control

In this Letter, we study the exponential stochastic synchronization problem for coupled neural networks with stochastic noise perturbations. Based on Lyapunov stability theory, inequality techniques, the properties of Weiner process, and adding different intermittent controllers, several sufficient conditions are obtained to ensure exponential stochastic synchronization of coupled neural networks with or without coupling delays under stochastic perturbations. These stochastic synchronization criteria are expressed in terms of several lower-dimensional linear matrix inequalities (LMIs) and can be easily verified. Moreover, the results of this Letter are applicable to both directed and undirected weighted networks. A numerical example and its simulations are offered to show the effectiveness of our new results.