1,409 research outputs found

### 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

### 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

### 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

### Bryuno Function and the Standard Map

For the standard map the homotopically non-trivial invariant curves of
rotation number satisfying the Bryuno condition are shown to be analytic in the
perturbative parameter, provided the latter is small enough. The radius of
convergence of the Lindstedt series -- sometimes called critical function of
the standard map -- is studied and the relation with the Bryuno function is
derived: the logarithm of the radius of convergence plus twice the Bryuno
function is proved to be bounded (from below and from above) uniformily in the
rotation number.Comment: 120 K, Latex, 33 page

### Conservation of resonant periodic solutions for the one-dimensional nonlinear Schroedinger equation

We consider the one-dimensional nonlinear Schr\"odinger equation with
Dirichlet boundary conditions in the fully resonant case (absence of the
zero-mass term). We investigate conservation of small amplitude
periodic-solutions for a large set measure of frequencies. In particular we
show that there are infinitely many periodic solutions which continue the
linear ones involving an arbitrary number of resonant modes, provided the
corresponding frequencies are large enough and close enough to each other (wave
packets with large wave number)

### Periodic solutions for the Schroedinger equation with nonlocal smoothing nonlinearities in higher dimension

We consider the nonlinear Schroedinger equation in higher dimension with
Dirichlet boundary conditions and with a non-local smoothing nonlinearity. We
prove the existence of small amplitude periodic solutions. In the fully
resonant case we find solutions which at leading order are wave packets, in the
sense that they continue linear solutions with an arbitrarily large number of
resonant modes. The main difficulty in the proof consists in solving a "small
divisor problem" which we do by using a renormalisation group approach.Comment: 60 pages 8 figure

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