Is this scaling nonlinear?

One of the most celebrated findings in complex systems in the last decade is that different indexes y (e.g. patents) scale nonlinearly with the population x of the cities in which they appear, i.e. y∼x (ß) ,ß≠1. More recently, the generality of this finding has been questioned in studies that used new databases and different definitions of city boundaries. In this paper, we investigate the existence of nonlinear scaling, using a probabilistic framework in which fluctuations are accounted for explicitly. In particular, we show that this allows not only to (i) estimate ß and confidence intervals, but also to (ii) quantify the evidence in favour of ß≠1 and (iii) test the hypothesis that the observations are compatible with the nonlinear scaling. We employ this framework to compare five different models to 15 different datasets and we find that the answers to points (i)-(iii) crucially depend on the fluctuations contained in the data, on how they are modelled, and on the fact that the city sizes are heavy-tailed distributed.

Searching chaotic saddles in high dimensions.

We propose new methods to numerically approximate non-attracting sets governing transiently chaotic systems. Trajectories starting in a vicinity Ω of these sets escape Ω in a finite time τ and the problem is to find initial conditions x∈Ω with increasingly large τ=τ(x). We search points x' with τ(x')>τ(x) in a search domain in Ω. Our first method considers a search domain with size that decreases exponentially in τ, with an exponent proportional to the largest Lyapunov exponent λ1. Our second method considers anisotropic search domains in the tangent unstable manifold, where each direction scales as the inverse of the corresponding expanding singular value of the Jacobian matrix of the iterated map. We show that both methods outperform the state-of-the-art Stagger-and-Step method [Sweet et al., Phys. Rev. Lett. 86, 2261 (2001)] but that only the anisotropic method achieves an efficiency independent of τ for the case of high-dimensional systems with multiple positive Lyapunov exponents. We perform simulations in a chain of coupled Hénon maps in up to 24 dimensions (12 positive Lyapunov exponents). This suggests the possibility of characterizing also non-attracting sets in spatio-temporal systems.

Characterizing weak chaos using time series of Lyapunov exponents.

We investigate chaos in mixed-phase-space Hamiltonian systems using time series of the finite-time Lyapunov exponents. The methodology we propose uses the number of Lyapunov exponents close to zero to define regimes of ordered (stickiness), semiordered (or semichaotic), and strongly chaotic motion. The dynamics is then investigated looking at the consecutive time spent in each regime, the transition between different regimes, and the regions in the phase space associated to them. Applying our methodology to a chain of coupled standard maps we obtain (i) that it allows for an improved numerical characterization of stickiness in high-dimensional Hamiltonian systems, when compared to the previous analyses based on the distribution of recurrence times; (ii) that the transition probabilities between different regimes are determined by the phase-space volume associated to the corresponding regions; and (iii) the dependence of the Lyapunov exponents with the coupling strength.

Prevalence of marginally unstable periodic orbits in chaotic billiards.

The dynamics of chaotic billiards is significantly influenced by coexisting regions of regular motion. Here we investigate the prevalence of a different fundamental structure, which is formed by marginally unstable periodic orbits and stands apart from the regular regions. We show that these structures both exist and strongly influence the dynamics of locally perturbed billiards, which include a large class of widely studied systems. We demonstrate the impact of these structures in the quantum regime using microwave experiments in annular billiards.

Nontwist non-Hamiltonian systems.

We show that the nontwist phenomena previously observed in Hamiltonian systems exist also in time-reversible non-Hamiltonian systems. In particular, we study the two standard collision-reconnection scenarios and we compute the parameter space breakup diagram of the shearless torus. Besides the Hamiltonian routes, the breakup may occur due to the onset of attractors. We study these phenomena in coupled phase oscillators and in non-area-preserving maps.