Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems
For many years, I have been interested in global analysis of nonlinear systems. The original interest stemmed from the study of snap-through stability and jump phenomena in structures. For systems of this kind, where there exist multiple stable equilibrium states or periodic motions, it is important to examine the domains of attraction of these responses in the state space. It was through work in this direction that the cell-to-cell mapping methods were introduced. These methods have received considerable development in the last few years, and have also been applied to some concrete problems. The results look very encouraging and promising. However, up to now, the effort of developing these methods has been by a very small number of people. There was, therefore, a suggestion that the published material, scattered now in various journal articles, could perhaps be pulled together into book form, thus making it more readily available to the general audience in the field of nonlinear oscillations and nonlinear dynamical systems. Conceivably, this might facilitate getting more people interested in working on this topic. On the other hand, there is always a question as to whether a topic (a) holds enough promise for the future, and (b) has gained enough maturity to be put into book form. With regard to (a), only the future will tell. With regard to (b), I believe that, from the point of view of both foundation and methodology, the methods are far from mature.
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1)-multiplet absorption probability acyclic acyclic group affine function algorithm asymptotically stable barycentric coordinates bifurcation boundary cell function cell mapping method cell space cell state space Chapter compute Consider def1ned denoted determined differential equation discussed domains of attraction domiciles dynamical systems eigenvalues expected absorption Figure given global analysis global behavior gradient vectors group number hypersurface image cells in-phase index theory integers iterative L-neighborhood limit cycle limiting probability linear logistic map mapping step Markov chains nondegenerate nonlinear systems number of cells out-of-phase P-1 cell P-K point P-K solution parameter periodic cells periodic solutions persistent cell persistent group point of F probability vector properties regular cells Section shown in Fig simple cell mapping simplex singular cell singular doublet singular entities singular multiplets singular point sink cell strange attractor subgroup Theorem total number trajectories transient cells transient groups transition probability matrix unstable values vector field zero zeroth level