Local Activity Is The Origin Of Complexity

Nature abounds with complex pattern and structures emerging from homogeneous media operating far from thermodynamic equilibrium. It is well known that a vast majority of active homogeneous media can exhibit complexity in the form of dissipative structures are modeled by a reaction-diffusion partial differential equation(PDE). On the other hand, given any nonlinear PDE, we can induce many associated CNN equation. Although it is not a exactly statement, extensive computer experiments have shown that for the vast majority of cases, the respective solutions can be made virtually indistinguishable by choosing a sufficiently large array size and by optimizing the CNN cell and coupling parameters. In particularly, on 1998 Chua introduced the idea that "Local activity is the origin of complexity". It has provided a powerful tool for studying the emergence of complex pattern in a homogeneous lattice formed by coupled cells.In this thesis, under the theory of local activity, we study the relation between the complexity matrix Y_Q(S) and characteristic polynomial A_Q(S). Besides, the fundamental local activity theory asserts that a wide spectrum of complex behaviors may exist if the corresponding cell parameters of a CNN are chosen at, or nearby the edge of chaos. Thus, the theory of edge of chaos is applied to study the Oregonator CNN equation, which implies that an uncoupled cell on the edge of chaos may cause a reaction-diffusion equation to oscillate under appropriate choice of diffusion coefficients.In Chapter 1 we introduce some background of complexity and describe the pattern which formed by CNN paradigm. In Chapter 2, to illustrate the complexity can be described by CNN equation, we discuss the relation between the PDE and CNN equation. Also, we outline Chua' idea of "Local activity is the origin of complexity". In Chapter 3, we establish the relation between the complexity matrix Y_Q(S) and characteristic polynomial A_Q(S). Furthermore, we taking the Oregonator CNN equation as a example to illustrate that an uncoupled cell of a reaction-diffusion equation on the edge of chaos is potentially unstable. That's to say, it will be arise a Hopf-link. At last, we analyze the stability of bifurcating periodic solutions and the direction of Hopf bifurcation by applying the normal form theory and the center manifold theorem. Again, a numerical simulation is given to demonstrate the effectiveness of the theoretical analysis.