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

Reducing or enhancing chaos using periodic orbits

A method to reduce or enhance chaos in Hamiltonian flows with two degrees of freedom is discussed. This method is based on finding a suitable perturbation of the system such that the stability of a set of periodic orbits changes (local bifurcations). Depending on the values of the residues, reflecting their linear stability properties, a set of invariant tori is destroyed or created in the neighborhood of the chosen periodic orbits. An application on a paradigmatic system, a forced pendulum, illustrates the method

An Algorithm to Predict E-Bike Power Consumption Based on Planned Routes

E-bikes, i.e., bikes equipped with a small electrical engine, are becoming increasingly widespread, thanks to their positive contribution to mobility and sustainability. A key component of an e-bike is the battery that feeds the drive unit: clearly, the higher the capacity of the battery, the longer the distances that the biker will cover under engine support. On the negative side, the additional weight incurred by the electric components is likely to ruin the riding experience in case the battery runs out of power. For this reason, an integrated hardware-software system that provides accurate information about the remaining range is essential, especially for older or “not-in-shape” bikers. Many e-bikes systems are already equipped with a small control unit that displays useful information, such as speed, instantaneous power consumption, and estimated range as well. Existing approaches rely on machine learning techniques applied to collected data, or even on the remaining battery capacity and the assistance level required by the drive unit. They do not consider crucial aspects of the planned route, in particular the difference in altitude, the combined weight of bike and biker, and road conditions. In this paper, we propose a mathematical model implemented in an application to compute battery consumption, and hence the presumed remaining range, in a more accurate way. Our application relies on external sources to compute the route and the elevation data of a number of intermediate points. We present the mathematical model on which our application is based, we show the implemented application in shape of an app, and we report the results of the experiments

Emergence of a non trivial fluctuating phase in the XY model on regular networks

We study an XY-rotor model on regular one dimensional lattices by varying the number of neighbours. The parameter $2\ge\gamma\ge1$ is defined. $\gamma=2$ corresponds to mean field and $\gamma=1$ to nearest neighbours coupling. We find that for $\gamma<1.5$ the system does not exhibit a phase transition, while for $\gamma > 1.5$ the mean field second order transition is recovered. For the critical value $\gamma=\gamma_c=1.5$, the systems can be in a non trivial fluctuating phase for whichthe magnetisation shows important fluctuations in a given temperature range, implying an infinite susceptibility. For all values of $\gamma$ the magnetisation is computed analytically in the low temperatures range and the magnetised versus non-magnetised state which depends on the value of $\gamma$ is recovered, confirming the critical value $\gamma_{c}=1.5$

Phase Ordering Dynamics of ϕ4\phi^4 Theory with Hamiltonian Equations of Motion

Phase ordering dynamics of the (2+1)- and (3+1)-dimensional $\phi^4$ theory with Hamiltonian equations of motion is investigated numerically. Dynamic scaling is confirmed. The dynamic exponent $z$ is different from that of the Ising model with dynamics of model A, while the exponent $\lambda$ is the same.Comment: to appear in Int. J. Mod. Phys.

Unveiling the nature of out-of-equilibrium phase transitions in a system with long-range interactions

Recently, there has been some vigorous interest in the out-of-equilibrium quasistationary states (QSSs), with lifetimes diverging with the number N of degrees of freedom, emerging from numerical simulations of the ferromagnetic XY Hamiltonian Mean Field (HMF) starting from some special initial conditions. Phase transitions have been reported between low-energy magnetized QSSs and large-energy unexpected, antiferromagnetic-like, QSSs with low magnetization. This issue is addressed here in the Vlasov N \rightarrow \infty limit. It is argued that the time-asymptotic states emerging in the Vlasov limit can be related to simple generic time-asymptotic forms for the force field. The proposed picture unveils the nature of the out-of-equilibrium phase transitions reported for the ferromagnetic HMF: this is a bifurcation point connecting an effective integrable Vlasov one-particle time-asymptotic dynamics to a partly ergodic one which means a brutal open-up of the Vlasov one-particle phase space. Illustration is given by investigating the time-asymptotic value of the magnetization at the phase transition, under the assumption of a sufficiently rapid time-asymptotic decay of the transient force field

Anomalous transport in Charney-Hasegawa-Mima flows

Transport properties of particles evolving in a system governed by the Charney-Hasegawa-Mima equation are investigated. Transport is found to be anomalous with a non linear evolution of the second moments with time. The origin of this anomaly is traced back to the presence of chaotic jets within the flow. All characteristic transport exponents have a similar value around $\mu=1.75$, which is also the one found for simple point vortex flows in the literature, indicating some kind of universality. Moreover the law $\gamma=\mu+1$ linking the trapping time exponent within jets to the transport exponent is confirmed and an accumulation towards zero of the spectrum of finite time Lyapunov exponent is observed. The localization of a jet is performed, and its structure is analyzed. It is clearly shown that despite a regular coarse grained picture of the jet, motion within the jet appears as chaotic but chaos is bounded on successive small scales.Comment: revised versio

Targeted mixing in an array of alternating vortices

Transport and mixing properties of passive particles advected by an array of vortices are investigated. Starting from the integrable case, it is shown that a special class of perturbations allows one to preserve separatrices which act as effective transport barriers, while triggering chaotic advection. In this setting, mixing within the two dynamical barriers is enhanced while long range transport is prevented. A numerical analysis of mixing properties depending on parameter values is performed; regions for which optimal mixing is achieved are proposed. Robustness of the targeted mixing properties regarding errors in the applied perturbation are considered, as well as slip/no-slip boundary conditions for the flow

Passive Tracer Dynamics in 4 Point-Vortex Flow

The advection of passive tracers in a system of 4 identical point vortices is studied when the motion of the vortices is chaotic. The phenomenon of vortex-pairing has been observed and statistics of the pairing time is computed. The distribution exhibits a power-law tail with exponent $\sim 3.6$ implying finite average pairing time. This exponents is in agreement with its computed analytical estimate of 3.5. Tracer motion is studied for a chosen initial condition of the vortex system. Accessible phase space is investigated. The size of the cores around the vortices is well approximated by the minimum inter-vortex distance and stickiness to these cores is observed. We investigate the origin of stickiness which we link to the phenomenon of vortex pairing and jumps of tracers between cores. Motion within the core is considered and fluctuations are shown to scale with tracer-vortex distance $r$ as $r^{6}$. No outward or inward diffusion of tracers are observed. This investigation allows the separation of the accessible phase space in four distinct regions, each with its own specific properties: the region within the cores, the reunion of the periphery of all cores, the region where vortex motion is restricted and finally the far-field region. We speculate that the stickiness to the cores induced by vortex-pairings influences the long-time behavior of tracers and their anomalous diffusion.Comment: 18 pages, 15 figure