On the dynamical mechanism of cross-over from chaotic to turbulent states

The Portevin-Le Chatelier e®ect is one of the few examples of organization of defects. Here the spatio-temporal dynamics emerges from the cooperative behavior of the constituent defects, namely dislocations and point defects. Recent dynamical approach to the study of experimental time series reports an intriguing cross-over phenomenon from a low dimensional chaotic to an infinite dimensional scale invariant power-law regime of stress drops in experiments on CuAl single crystals and AlMg polycrystals, as a function of strain rate. We show that an extension of a dynamical model due to Ananthakrishna and coworkers for the Portevin-Le Chatelier effect reproduces this cross-over. At low and medium strain rates, the model shows chaos with the structure of the attractor resembling the reconstructed experimental attractor. At high strain rates, the model exhibits a power-law statistics for the magnitudes and durations of the stress drops as in experiments. Concomitantly, the largest Lyapunov exponent is zero. In this regime, there is a finite density of null exponents which itself follows a power law. This feature is similar to the Lyapunov spectrum of a shell model for turbulence. The marginal nature of this state is visualized through slow manifold approach