ar X iv : c ha o - dy n / 96 06 01 6 v 1 1 J ul 1 99 6 Hopf ’ s last hope : spatiotemporal chaos in terms of unstable recurrent patterns

ar X iv : c ha o - dy n / 96 01 01 7 v 1 24 J an 1 99 6 Tools for Nonlinear Analysis : I . Unfolding of Dynamical Systems

Automated algorithms for derivation of amplitude equations in the vicinity of monotonic and Hopf bifurcation manifolds are presented. The implementation is based on Mathematica programming, and is illustrated by several examples.

ar X iv : c ha o - dy n / 98 12 01 6 v 1 9 D ec 1 99 8 Queues , Stacks , and Transcendentality at the Transition to Chaos

We examine the one-humped map at the period-doubling transition to chaos, and ask whether its long-term memory is stack-like (last-in, first-out) or queue-like (first-in, first-out). We show that it can be recognized by a real-time automaton with one queue, or two stacks, and give several new grammatical characterizations of it. We argue that its memory has a queue-like character, since a singl...

ar X iv : c ha o - dy n / 99 10 00 5 v 1 6 O ct 1 99 9 The Finite - Difference Analysis and Time Flow

X iv : c ha o - dy n / 99 10 02 5 v 1 1 8 O ct 1 99 9 Asymptotic expansions of unstable ( stable ) manifolds in time - discrete systems

By means of an updated renormalization method, we construct asymptotic expansions for unstable manifolds of hyperbolic fixed points in the double-well map and the dissipative Hénon map, both of which exhibit the strong homoclinic chaos. In terms of the asymptotic expansion , a simple formulation is presented to give the first homoclinic point in the double-well map. Even a truncated expansion o...

ar X iv : c ha o - dy n / 99 11 02 5 v 1 1 8 N ov 1 99 9 General properties of propagation in chaotic systems

We conjecture that in one-dimensional spatially extended systems the propagation velocity of correlations coincides with a zero of the convective Lyapunov spectrum. This conjecture is successfully tested in three different contexts: (i) a Hamiltonian system (a Fermi-Pasta-Ulam chain of oscillators); (ii) a general model for spatio-temporal chaos (the complex Ginzburg-Landau equation); (iii) exp...