Tyson [Ann. NY Acad. Sci. 316 (1979) 279] conjectured that the stable homogeneous positive steady state may coexist with stable echo wave (meaning anti-phase wave) in linearly coupled Oregonators (and thus gave a conjecture on the bifurcation diagram of this system). In this paper, we rigorously prove stability of the in-phase wave and existence of the anti-phase wave. Our proof procedure actually gives a general method (or line) to deal with the analogous problem. For instance, to prove stability of the in-phase wave, following our line one may decompose the corresponding variational equations (a four-dimensional system) into two independent planar systems; also for instance, existence of the anti-phase wave can be concluded as existence and uniqueness of limit cycle of the associate oscillator. In addition, according to parameter regimes of existence of the anti-phase wave and the stable homogeneous positive steady state, we give their coexistence regime and specify it, and in particular give the regime of the coupled coefficient. The specified results show that the theoretical results are in good accord with Tyson's numerical results.

The responses of nonlinear dynamics of two classes to coupling are investigated. It is shown both analytically and numerically that coupling has an excitation ability in a network of the linearly coupled systems. That is, when an uncoupled system is degenerated to a stable steady state from a limit cycle but in the "marginal" state due to the system parameter, an appropriate coupling strength can excite the limit cycle such that the coupled systems exhibit synchronous oscillation; when the uncoupled system is in a stable limit cycle but close to a chaotic attractor, a certain coupling strength can induce the chaotic attractor such that the coupled systems reach chaotic synchronization. Such excitation functions of coupling are different from its traditional role where coupling mainly synchronizes the coupled systems with the original dynamics of the uncoupled system.

In-phase waves and echo waves are commonly existing in coupled systems. In Physica D 151 (2001) 199 we proved existence of echo waves in linearly coupled oregonators and showed that the echo waves may coexist with the stable homogeneous positive steady state. This Letter develops a different approach (so-called "reducing dimension") compared with our "mode decomposition" method of stability of in-phase waves in Phys. Lett. A 301 (2002) 231, and rigorously proves stability of the echo waves. The method can be easily applied to other similar systems.

This Letter introduces a new method-mode decomposition-for stability analysis of periodic orbits. Using this method, the stability of a periodic solution of an autonomous system, as well as the stability of synchronization within three chaotic systems with linear coupling, can be analyzed. As an example, a rigorous sufficient condition on the coupling coefficients for achieving chaos synchronization is obtained, for the case of three-coupled identical Lorenz systems. Numerical simulations are shown for demonstration.

All cell components exhibit intracellular noise due to random births and deaths of individual molecules, and extracellular noise due to environment perturbations. In particular, gene regulation is an inherently noisy process with transcriptional control, alternative splicing, translation, diffusion and chemical modification reactions, which all involve stochastic fluctuations. Such stochastic noises may not only affect the dynamics of the entire system but may also be exploited by living organisms to actively facilitate certain functions, such as cooperative behavior and communication. Results: We have provided a general model and an analytic tool to examine the cooperative behavior of a multi-cell system with both intracellular and extracellular stochastic fluctuations. A multi-cell system with a synthetic gene network is adopted to demonstrate the effects of noises and coupling on collective dynamics. These results establish not only a theoretical foundation but also a quantitative basis for understanding essential roles of noises on cooperative dynamics, such as synchronization and communication among cells.