Since early 90's, much attention has been paid to dynamic dissipative patterns in laboratories, especially, self-replicating pattern (SRP) is one of the most exotic phenomena.Employing model system such as the Gray-Scott model, it is confirmed also by numerics that SRP can be obtained via destabilization of standing or traveling spots. SRP is a typical example of transient dynamics, and hence it is not a priori clear that what kind of mathematical framework is appropriate to describe the dynamics. A framework in this direction is proposed by Nishiura-Ueyama [16], i.e., hierarchy structure of saddle-node points, which gives a basis for rigorous analysis. One of the interesting observation is that when there occurs self-replication, then only spots (or pulses) located at the boundary (or edge) are able to split. Internal ones do not duplicate at all. For 1D-case, this means that the number of newly born pulses increases like 2k after k-th splitting, not 2''1 -splitting where all pulses split simultaneously. The main objective in this article is two-fold: One is to construct a local invariant manifold near the onset of self-replication, and derive the nonlinear ODE on it. The other is to study the manner of splitting by analysing the resulting ODE, and answer the question "2'-splitting or edge-splitting?" starting from a single pulse. It turns out that only the edge-splitting occurs, which seems a natural consequence from a physical point of view, because the pulses at edge are easier to access fresh chemical resources than internal ones.
The unidirectional motion of a camphor boat along an annular water channel is observable. When camphor boats are placed in a water channel, both homogeneous and inhomogeneous states occur as collective motions, depending on the number of boats. The inhomogeneous state is a type of congestion, that is, the velocities of the boats change with temporal oscillation, and the shock wave appears along the line of travel of the boats. The unidirectional motion of a single camphor boat and the homogeneous state can be represented by traveling wave solutions in a mathematical model. Because the experimental results described above are thought of as a type of bifurcation phenomena, the destabilization of traveling wave solutions may indicate the emergence of congestion. We previously attempted studying a linearized eigenvalue problem associated with a traveling wave solution. However, the problem is too difficult to analyze rigorously, even for just two camphor boats. Therefore we developed a center manifold theory and derived a reduced model in our previous work. In the present paper, we study the reduced model, and show that the original model and our reduced model qualitatively exhibit the same properties by applying numerical techniques. Moreover, we demonstrate that the numerical results obtained in our models for camphor boats are quite similar to those in a car-following model, OV model, but there are some different features between our reduced model and a typical OV model.
In this paper, two component reaction-diffusion systems with a specific bistable nonlinearity are concerned. The systems have the bifurcation structure of pitch-fork type of traveling front solutions with opposite velocities, which is rigorously proved and the ordinary differential equations describing the dynamics of such traveling front solutions are also derived explicitly. It enables us to know rigorously precise information on the dynamics of traveling front solutions. As an application of this result, the imperfection structure under small perturbations and the dynamics of traveling front solutions on heterogeneous media are discussed.
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