Global Bifurcational Approach to Complex Spatio^temporal Patterns in Dissipative Systems

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In a regime of far-from equilibrium there appears a diversity of complex patterns such as self-replication, spatio-temporal patterns, and collisions among particle-like patterns. One of the powerful tools to understand these things is dynamical system theory, however its naive application in general does not work partly due to the high dimensionality of phase space and large deformation of solutions. What should be the clue for us to start with in understanding such behaviors? We need to alter our way of thinking, namely "Let us think about the geometric structures that guide solution orbits creating such a chaotic dynamism, rather than keeping track of the deformations of solutions in detail". In other words, we should try to characterize geometric structures of the infinite dimensional phase space in which behaviors of solution orbits become easily detectable. Taking this viewpoint, we accomplished the following two main things. Please refer to the published papers for other aspect o … More f achievements.1.Unfolding of generalized heteroclinic cycle implies spatio-temporal chaos.Chimerical methods, such as AUTO, give us a great amount of information on an unstable solution, as well as on the behavior of its unstable manifold. Heteroclinic cycle connecting several stationary patterns was identified as a key to understand the complex behaviors like spatio-temporal chaos for the Gray-Scott model. The mechanism itself has much wider applicability to other model systems.2.Role of "Scattors" for collision process among particle-like patterns.Scattering of particle-like patterns in dissipative systems has much attention from various fields. We focused on the issue how the input-output relation is controlled at a head-on collision where traveling pulses or spots interact strongly.It had remained an open problem due to the large deformation of patterns at a colliding point. We found that special type of unstable steady or time-periodic solutions called scattors and their stable and unstable manifolds direct the traffic flow of orbits.Such scattors are in general highly unstable even in ID case which causes a variety of input-output relations through the scattering process. We illustrate the ubiquity of scattors by using the complex Ginzburg-Landau equation, the Gray-Scott model and a three-component reaction diffusion model arising in gas-discharge phenomena. Less