Crafting networks to achieve, or not achieve, chaotic states

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

Emergence of a collective crystal in a classical system with long-range interactions

A one-dimensional long-range model of classical rotators with an extended degree of complexity, as compared to paradigmatic long-range systems, is introduced and studied. Working at constant density, in the thermodynamic limit one can prove the statistical equivalence with the Hamiltonian Mean Field model (HMF) and $\alpha$-HMF: a second order phase transition is indeed observed at the critical energy threshold $\varepsilon_c=0.75$. Conversely, when the thermodynamic limit is performed at infinite density (while keeping the length of the hosting interval $L$ constant), the critical energy $\varepsilon_c$ is modulated as a function of $L$. At low energy, a self-organized collective crystal phase is reported to emerge, which converges to a perfect crystal in the limit $\epsilon \rightarrow 0$. To analyze the phenomenon, the equilibrium one particle density function is analytically computed by maximizing the entropy. The transition and the associated critical energy between the gaseous and the crystal phase is computed. Molecular dynamics show that the crystal phase is apparently split into two distinct regimes, depending on the the energy per particle $\varepsilon$. For small $\varepsilon$, particles are exactly located on the lattice sites; above an energy threshold $\varepsilon{*}$, particles can travel from one site to another. However, $\varepsilon{*}$ does not signal a phase transition but reflects the finite time of observation: the perfect crystal observed for $\varepsilon >0$ corresponds to a long lasting dynamical transient, whose life time increases when the $\varepsilon >0$ approaches zero.Comment: 6 pages, 4 figure

Mixing properties in the advection of passive tracers via recurrences and extreme value theory

In this paper we characterize the mixing properties in the advection of passive tracers by exploiting the extreme value theory for dynamical systems. With respect to classical techniques directly related to the Poincar\'e recurrences analysis, our method provides reliable estimations of the characteristic mixing times and distinguishes between barriers and unstable fixed points. The method is based on a check of convergence for extreme value laws on finite datasets. We define the mixing times in terms of the shortest time intervals such that extremes converge to the asymptotic (known) parameters of the Generalized Extreme Value distribution. Our technique is suitable for applications in the analysis of other systems where mixing time scales need to be determined and limited datasets are available.Comment: arXiv admin note: text overlap with arXiv:1107.597

Existence of Quasi-stationary states at the Long Range threshold

In this paper the lifetime of quasi-stationary states (QSS) in the $\alpha-$HMF model are investigated at the long range threshold ($\alpha=1$). It is found that QSS exist and have a diverging lifetime $\tau(N)$ with system size which scales as \mbox{\ensuremath{\tau}(N)\ensuremath{\sim}}\log N, which contrast to the exhibited power law for $\alpha<1$ and the observed finite lifetime for $\alpha>1$. Another feature of the long range nature of the system beyond the threshold ($\alpha>1$) namely a phase transition is displayed for $\alpha=1.5$. The definition of a long range system is as well discussed

Growth of a tree with allocations rules: Part 2 Dynamics

Following up on a previous work we examine a model of transportation network in some source-sink flow paradigm subjected to growth and resource allocation. The model is inspired from plants, and we add rules and factors that are analogous to what plants are subjected to. We study how different resource allocation schemes affect the tree and how the schemes interact with additional factors such as embedding the network into a 3D space and applying gravity or shading. The different outcomes are discussed. PACS. 05.45.-a Nonlinear dynamics and chaos-05.65.+b Self-organized systemsComment: European Physical Journal B: Condensed Matter and Complex Systems, Springer-Verlag, In pres

Growth of a tree with allocations rules: Part 1 Kinematics

A non-local model describing the growth of a tree-like transportation network with given allocation rules is proposed. In this model we focus on tree like networks, and the network transports the very resource it needs to build itself. Some general results are given on the viability tree-like networks that produce an amount of resource based on its amount of leaves while having a maintenance cost for each node. Some analytical studies and numerical surveys of the model in "simple" situations are made. The different outcomes are discussed and possible extensions of the model are then discussed

Chaotic Jets

The problem of characterizing the origin of the non-Gaussian properties of transport resulting from Hamiltonian dynamics is addressed. For this purpose the notion of chaotic jet is revisited and leads to the definition of a diagnostic able to capture some singular properties of the dynamics. This diagnostic is applied successfully to the problem of advection of passive tracers in a flow generated by point vortices. We present and discuss this diagnostic as a result of which clues on the origin of anomalous transport in these systems emerge.Comment: Proceedings of the workshop Chaotic transport and complexity in classical and quantum dynamics, Carry le rouet France (2002

A signal processing method: Detection of Lévy flights in chaotic trajectories

International audienceTransport in low dimensional Hamiltonian chaos can be anomalous due to stickiness and rise of Lévy flights. We suggest a signal processing method to detect these flights in signals, in order to characterize the nature of transport (diffusive or anomalous). We use time-frequency techniques such as Fractional Fourier transform and matching pursuit in order to be robust to noise. The method is tested on data obtained from chaotic advection

Ergodicit\'e, collage et transport anomal

Nous nous int\'eressons \`a la convergence vers sa moyenne spatiale ergodique de la moyenne temporelle d'une observable d'un flow hamiltonien \`a un degr\'e et demi de libert\'e avec espace des phases mixte. L'analyse est faite au travers de l'\'evolution de la distribution des moyennes en temps fini d'un ensemble de conditions initiales sur la m\^eme composante ergodique. Un exposant caract\'erisant la vitesse de convergence est d\'efini. Les r\'esultats indiquent que pour le syst\`eme consid\'er\'e la convergence \'evolue en $t^{\alpha}$, avec $\alpha=0.45$ pour alors qu'elle \'evolue en $t^{1/2}$ lorsque la dynamique est globalement chaotique dans l'espace des phases. De m\^eme une loi $\alpha=1-\beta/2$ reliant cet exposant $\alpha$ \`a l'exposant caract\'eristique du deuxi\`eme moment associ\'e aux propri\'et\'es de transport $\beta$ est propos\'ee et est v\'erifi\'ee pour les cas consid\'er\'es

Control of chaotic advection

A method of chaos reduction for Hamiltonian systems is applied to control chaotic advection. By adding a small and simple term to the stream function of the system, the construction of invariant tori has a stabilization effect in the sense that these tori act as barriers to diffusion in phase space and the controlled Hamiltonian system exhibits a more regular behaviour