We address the problem of the classical deterministic dynamics of a particle in a periodic asymmetric potential of the ratchet type. We take into account the inertial term in order to understand the role of the chaotic dynamics in the transport properties. By a comparison between the bifurcation diagram and the current, we identify the origin of the current reversal as a bifurcation from a chaotic to a periodic regime. Close to this bifurcation, we observed trajectories revealing intermittent chaos and anomalous deterministic diffusion.

Scale-free foraging patterns are widespread among animals. These may be the outcome of an optimal searching strategy to find scarce, randomly distributed resources, but a less explored alternative is that this behaviour may result from the interaction of foraging animals with a particular distribution of resources. We introduce a simple foraging model where individual primates follow mental maps and choose their displacements according to a maximum efficiency criterion, in a spatially disordered environment containing many trees with a heterogeneous size distribution. We show that a particular tree-size frequency distribution induces non-Gaussian movement patterns with multiple spatial scales (Lévy walks). These results are consistent with field observations of tree-size variation and spider monkey (Ateles geoffroyi) foraging patterns. We discuss the consequences that our results may have for the patterns of seed dispersal by foraging primates.

We introduce a formalism of fractional diffusion on networks based on a fractional Laplacian matrix that can be constructed directly from the eigenvalues and eigenvectors of the Laplacian matrix. This fractional approach allows random walks with long-range dynamics providing a general framework for anomalous diffusion and navigation, and inducing dynamically the small-world property on any network. We obtained exact results for the stationary probability distribution, the average fractional return probability, and a global time, showing that the efficiency to navigate the network is greater if we use a fractional random walk in comparison to a normal random walk. For the case of a ring, we obtain exact analytical results showing that the fractional transition and return probabilities follow a long-range power-law decay, leading to the emergence of Lévy flights on networks. Our general fractional diffusion formalism applies to regular, random, and complex networks and can be implemented from the spectral properties of the Laplacian matrix, providing an important tool to analyze anomalous diffusion on networks.

We introduce a strategy of navigation in undirected networks, including regular, random, and complex networks, that is inspired by Lévy random walks, generalizing previous navigation rules. We obtained exact expressions for the stationary probability distribution, the occupation probability, the mean first passage time, and the average time to reach a node on the network. We found that the long-range navigation using the Lévy random walk strategy, compared with the normal random walk strategy, is more efficient at reducing the time to cover the network. The dynamical effect of using the Lévy walk strategy is to transform a large-world network into a small world. Our exact results provide a general framework that connects two important fields: Lévy navigation strategies and dynamics on complex networks.

García-Calderón, Mateos, and Moshinsky Reply: van Dijk and Nogami [1] argue that their result for Pt t ÿ3 follows from a novel analytical expansion of r; t in the stationary solutions uk; r of the Schrö dinger equation; see Eq. (7) of Ref. [2]. They claim that the difference with our approach arises because our expansion in terms of the resonant states u n r and Moshinsky functions Mk n ; t, see Eqs. (12) and (14) of Ref. [3], is unsuitable to describe the long-time regime due to the outgoing character of resonant states at r R. They affirm, instead, that in their expansion a complex pole k n can have a positive or a negative real part, leading, respectively, to an outgoing or an incoming wave at r R. They express r; t as a linear combination involving both types of poles and the Moshinsky functions Mr ÿ R; k n ; t and obtain that at long times r; t t ÿ3=2 . By substitution of this quantity into the definition of Pt, they get the behavior mentioned above.These authors seem to ignore that poles with a positive real part, k n a n ÿ ib n , a n > 0 are related to poles with a negative real part, k ÿn ÿa n ÿ ib n , by time reversal considerations, i.e., k ÿn ÿk n , and, consequently, that both types of poles follow from a purely outgoing boundary condition [4]. Indeed, a careful reading of our Letter [3] shows that our expansion of Pt runs over the full set of poles, k n and k ÿn , as it does also the expansion of the time dependent Green's function, Eq. (9), that when substituted in Eq. (3), noting that Mk n ; t M0; k n ; t [5], leads to the expression,