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.

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.

An effective method of chaotification via time-delay feedback for a simple finite-dimensional continuous-time autonomous system is made rigorous in this paper. Some mathematical conditions are derived under which a nonchaotic system can be controlled to become chaotic, where the chaos so generated is in a rigorous mathematical sense of Li-Yorke in terms of the Marotto theorem. Numerical simulations are given to verify the theoretical analysis. Chaos has been found useful lately in various areas of science, engineering, and technology. Therefore, purposefully generating chaos (called chaotification, or anticontrol of chaos) has investigated rather intensively in the past few years. Recently, Wang, Chen, and Yu [Chaos 10, 771-779 (2000)] developed an anticontrol method via time-delay feedback for chaotifying a continuous-time dynamical system. The fundamental idea of this anticontrol method is correct and insightful, but a time-delay differential equation used therein is only approximated by a related discrete map, leaving some room for improvement. To present a mathematically rigorous approach, this paper adopts the same anticontrol idea but further improves its technical contents thereby deriving a similar yet rigorous design method for chaotificaiton. A rather general continuous-time system can be driven from nonchaotic to chaotic by using time-delay feedback perturbation on the system parameters or employing an exogenous time-delay state-feedback input, where the generated chaos is in a precise mathematical sense of Li-Yorke in terms of the Marotto theorem.