Quasi-periodic motions in dynamical systems. Review of a renormalisation group approach

Power series expansions naturally arise whenever solutions of ordinary differential equations are studied in the regime of perturbation theory. In the case of quasi-periodic solutions the issue of convergence of the series is plagued of the so-called small divisor problem. In this paper we review a method recently introduced to deal with such a problem, based on renormalisation group ideas and multiscale techniques. Applications to both quasi-integrable Hamiltonian systems (KAM theory) and non-Hamiltonian dissipative systems are discussed. The method is also suited to situations in which the perturbation series diverges and a resummation procedure can be envisaged, leading to a solution which is not analytic in the perturbation parameter: we consider explicitly examples of solutions which are only infinitely differentiable in the perturbation parameter, or even defined on a Cantor set.Comment: 36 pages, 8 figures, review articl

Large deviation rule for Anosov flows

The volume contraction in dissipative reversible transitive Anosov flows obeys a large deviation rule (fluctuation theorem).Comment: See instruction at the beginning of the tex file, in order to obatin the (two) postscript figures. The file is in Plain Te

Rawls’s inclusivism and the case of ‘religious militants for peace’: A reply to Weithman’s restrictive inclusivism

Across almost a decade, Desmond Tutu, Anglican cleric and chairman of South Africa’s Truth and Reconciliation Commission, supported a model of civil resistance against the apartheid regime based solely on religious argument. Tutu is one of what Appleby (2000) calls the “religious militants for peace”: people of faith who use religious arguments to buttress resistance against unjust regimes and to support vital political change with regard to rights and justice. Yet the employment of religious arguments to justify political action seems to contradict the liberal democratic requirements of public reason, particularly the duty of liberal citizens to provide reasons that others could reasonably endorse. If “religious militants” violate their duty of civility by appealing to their comprehensive doctrines, should liberal democracy exclude this form of religiously founded dissent as being unreasonable? Or, rather, should liberal democracy embrace and support the efforts of “religious militants” to enhance and/or restore political justice

Using the general link transmission model in a dynamic traffic assignment to simulate congestion on urban networks

This article presents two new models of Dynamic User Equilibrium that are particularly suited for ITS applications, where the evolution of vehicle flows and travel times must be simulated on large road networks, possibly in real-time. The key feature of the proposed models is the detail representation of the main congestion phenomena occurring at nodes of urban networks, such as vehicle queues and their spillback, as well as flow conflicts in mergins and diversions. Compared to the simple word of static assignment, where only the congestion along the arc is typically reproduced through a separable relation between vehicle flow and travel time, this type of DTA models are much more complex, as the above relation becomes non-separable, both in time and space. Traffic simulation is here attained through a macroscopic flow model, that extends the theory of kinematic waves to urban networks and non-linear fundamental diagrams: the General Link Transmission Model. The sub-models of the GLTM, namely the Node Intersection Model, the Forward Propagation Model of vehicles and the Backward Propagation Model of spaces, can be combined in two different ways to produce arc travel times starting from turn flows. The first approach is to consider short time intervals of a few seconds and process all nodes for each temporal layer in chronological order. The second approach allows to consider long time intervals of a few minutes and for each sub-model requires to process the whole temporal profile of involved variables. The two resulting DTA models are here analyzed and compared with the aim of identifying their possible use cases. A rigorous mathematical formulation is out of the scope of this paper, as well as a detailed explanation of the solution algorithm. The dynamic equilibrium is anyhow sought through a new method based on Gradient Projection, which is capable to solve both proposed models with any desired precision in a reasonable number of iterations. Its fast convergence is essential to show that the two proposed models for network congestion actually converge at equilibrium to nearly identical solutions in terms of arc flows and travel times, despite their two diametrical approaches wrt the dynamic nature of the problem, as shown in the numerical tests presented here