论文信息 - From Time Series to Symbolic Dynamics: an Algebraic Topological Approach

From Time Series to Symbolic Dynamics: an Algebraic Topological Approach

A new approach to constructing a dynamical systems model from experimental time series data is presented. Using the the ideas of delay reconstruction a multivalued dynamical system is constructed. The multivalued approach is taken to allow for bounded experimental error. This system is then analyzed and algebraic invariants based on the Conley index are computed. These invariants have implications concerning the dynamics which must occur, e.g. symbolic dynamics and horseshoes. Theorems indicating conditions under which these invariants can be lifted back to the original physical system are then proven.

Instytut InformatiykiUniwersytet Jagiello

[1] Andrzej Szymczak. The Conley index for decompositions of isolated invariant sets , 1995 .

[2] Gabriel B. Mindlin,et al. Topological analysis of chaotic time series data from the Belousov-Zhabotinskii reaction , 1991 .

[3] K. Mischaikow,et al. Chaos in the Lorenz equations: a computer-assisted proof , 1995, math/9501230.

[4] M. C. Carbinatto,et al. Horseshoes and the Conley Index Spectrum , 1999 .

[5] Konstantin Mischaikow,et al. Horseshoes and the conley index spectrum-II: the theorem is sharp , 1999 .

[6] A. Szymczak. The Conley index and symbolic dynamics , 1996 .

[7] P. Holmes,et al. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[8] Philip Holmes,et al. A magnetoelastic strange attractor , 1979 .

[9] A. Szymczak. A combinatorial procedure for finding isolating neighbourhoods and index pairs , 1997, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[10] C. Conley. Isolated Invariant Sets and the Morse Index , 1978 .

[11] Mark R. Muldoon,et al. Topology from time series , 1993 .

[12] Marian Mrozek,et al. Shape index and other indices of Conley type for local maps on locally compact Hausdorff spaces , 1994 .

[13] Konstantin Mischaikow,et al. Isolating neighborhoods and chaos , 1995 .

[14] Konstantin Mischaikow,et al. Zeta functions, periodic trajectories, and the conley index , 1995 .