Crafting networks to achieve, or not achieve, chaotic states

Abstract

The influence of networks topology on collective properties of dynamical systems defined upon it is studied in the thermodynamic limit. A network model construction scheme is proposed where the number of links, the average eccentricity and the clustering coefficient are controlled. This is done by rewiring links of a regular one dimensional chain according to a probability $p$ within a specific range $r$, that can depend on the number of vertices $N$. We compute the thermodynamic behavior of a system defined on the network, the $XY-$rotors model, and monitor how it is affected by the topological changes. We identify the network dimension $d$ as a crucial parameter: topologies with $d\textless{}2$ exhibit no phase transitions while ones with $d\textgreater{}2$ display a second order phase transition. Topologies with $d=2$ exhibit states characterized by infinite susceptibility and macroscopic chaotic/turbulent dynamical behavior. These features are also captured by $d$ in the finite size context

Slowing Down of So-called Chaotic States: “Freezing” the Initial State

“…The direct consequences of these results go in two directions. First we confirmed the chaotic states observed and discussed in [12,13,16] indeed are not a transient state like a QSS and, because of the presence of a large continuous spectrum, we can as well confirm the chaotic nature of the macroscopic behavior in these states, much like a turbulent one. Second, when performing our theoretical analysis using the formalism developed in [18], we were able to show for the first time that it is possible to uncover some dynamical information from this formalism, and the successful prediction of the scaling law shows that the formalism is adequate to predict some finite size dynamical features of systems with many degrees of freedom with underlying Hamiltonian microscopic dynamics.…”

“…1 the evolution of the order parameter at a fixed density of energy ε for three different values of γ and a system size of N = 2 13 . Simulations have been performed using the optimal fifth order symplectic integrator described in [17], and the fast-Fourier transform made use of the FFTW package.…”

“…They were first spotted on systems of rotators evolving on a regular lattice, with a critical range of interaction and number of neighbors [12]. Further studies have shown that this behavior occurred as well on so-called lace networks, when the effective network dimension was around d = 2 [13]. Studying these systems for different number of constituents N and a fixed density of energy ε, the chaotic behavior of the order parameter was persistent, and it appeared that the width of the fluctuations around its mean value was not changing with N, implying an infinite susceptibility over a given range of values of ε.…”

Networks of planar Hamiltonian systems

“…There are no precise equivalents of these problems for any related networks that have appeared in the literature previously (e.g. [6,9,11,12,15,13,14,28,29]). We have made significant progress in some special cases, but obtaining general results remains an open challenge.…”

“…First is the class of systems introduced by Smereka [10] in the search for a Hamiltonian version of the Kuramoto model [5]. Together with those considered by Zanette-Hampton-Mikhailov [11,12], De Nigris-Leoncini [13,14], and Virkar-Restrepo-Meiss [15], these consist of planar…”