Image of the Week: Convergent chaos

By Phil Brown on August 28, 2017

For our image this week we’ve taken an image from Covergent chaos, a recent paper published in Journal of Physics A. Marc Pradas, Alain Pumir, Greg Huber and Michael Wilkinson study the instabilities and convergence of trajectories revealing and commenting on the criteria for chaos. They use large-deviation and extreme-value statistics to explain the effect. Their results show that the interpretation of the ‘butterfly effect’ needs to be carefully qualified and they argue that the combination of mixing and clustering processes makes their specific model relevant to understanding the evolution of simple organisms.

Trajectories for a dynamical system. The colormap is chosen so that blue and yellow correspond to sparse and highly dense regions, respectively.

Figure 1: Trajectories for a dynamical system. The colormap is chosen so that blue and yellow correspond to sparse and highly dense regions, respectively. From Marc Pradas et al 2017 J. Phys. A: Math. Theor. 50 275101, © IOP Publishing, All Rights Reserved.

The image describes trajectories, many of which show strong and long lasting convergence, despite the fact that they must ultimately diverge.

We emphasise that the clustering displayed in figure 1 is different from the fractal patterns which are seen in illustrations of strange attractors (such as the Lorenz equations or the Henon map) in phase space. These systems have contraction in some directions in phase space, and the paths of the trajectories cluster together into bundles. However, the formation of these bundles does not imply that phase points converge, because they can separate along the line tangent to the trajectory bundle. The definition of chaos implies that this separation is expected to increase exponentially. Our figure 1, however, shows how trajectory separations do behave as a function of time, revealing that the eventual exponential separation may be achieved only after episodes of strong convergence.

Marc Pradas and Michael Wilkinson work at the Open University. Greg Huber works at the Kavli Institute for Theoretical Physics, University of Santa Barbara. Alain Pumir works at ENS Lyon, and is also a member of the editorial board of Journal of Physics A.

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