arXiv subjects
Publications and source records attributed to Ricardo Lima.
We review a method of control for Hamiltonian systems which is able to create smooth invariant tori. This method of control is based on an apt modification of the perturbation which is small and localized in phase space.
Chaos often represents a severe obstacle for the set-up of many-body experiments, e.g., in fusion plasmas or turbulent flows. We propose a strategy to control chaotic diffusion in conservative systems. The core of our approach is a small apt modification of the system which channels chaos by building barriers to diffusion. It leads to practical prescriptions for an experimental apparatus to operate in a regular regime (drastic enhancement of confinement). The experimental realization of this control on a Travelling Wave Tube opens the possibility to practically achieve the control of a wide range of systems at a low additional cost of energy.
In this paper, we present a model describing the time evolution of two dimensional surface waves in gravity and infinite depth. The model of six interacting modes derives from the normal form of the system describing the dynamics of surface waves and is governed by a Hamiltonian system of equations of cubic order in the amplitudes of the waves. We derive a Hamiltonian system with two degrees of freedom from this Hamiltonian using conserved quantities. The interactions are those of two coupled Benjamin-Feir resonances. The temporal evolution of the amplitude of the different modes is described according to the parameters of the system. In particular, we study the energy exchange produced by the modulations of the amplitudes of the modes. The evolution of the modes reveals a chaotic dynamics.
We present a method of localised control of chaos in Hamiltonian systems. The aim is to modify the perturbation locally by a small control term which makes the controlled Hamiltonian more regular. We provide an explicit expression for the control term which is able to recreate invariant (KAM) tori without modifying other parts of phase space. We apply this method of localised control to a forced pendulum model, to the delta-kicked rotor (standard map) and to a non-twist Hamiltonian.
We present a method to control transport in Hamiltonian systems. We provide an algorithm - based on a perturbation of the original Hamiltonian localized in phase space - to design small control terms that are able to create isolated barriers of transport without modifying other parts of phase space. We apply this method of localized control to a forced pendulum model and to a system describing the motion of charged particles in a model of turbulent electric field.
We present a technique to control chaos in Hamiltonian systems which are close to integrable. By adding a small and simple control term to the perturbation, the system becomes more regular than the original one. We apply this technique to a model that reproduces turbulent ExB drift and show numerically that the control is able to drastically reduce chaotic transport.
It is shown that a relevant control of Hamiltonian chaos is possible through suitable small perturbations whose form can be explicitly computed. In particular, it is possible to control (reduce) the chaotic diffusion in the phase space of a Hamiltonian system with 1.5 degrees of freedom which models the diffusion of charged test particles in a ``turbulent'' electric field across the confining magnetic field in controlled thermonuclear fusion devices. Though still far from practical applications, this result suggests that some strategy to control turbulent transport in magnetized plasmas, in particular tokamaks, is conceivable.
The statistics of wrinkling flame front is invetigated by the quantum filed theory methods. We dwell on the WKB approximation in the functional integral which is analogous to the Wyld functional integral in turbulence. The main contribution to statistics is due to a coupled field-force configuration. This configuration is related to a kink between metastable exact pole solutions of the Syvashinsky equation. These kinks are responsible for both the formation of new cusps and the rapid power-law acceleration of the mean flame-front. The problem of asymptotic stability of the solutions is discussed.
The response of inviscid incompressible unbounded fluid subject to a localized external perturbation is studed. The physically relevant hypotheses on the mode coupling mechanisma is justified by renormalization group method. The scaling laws for the scalar and vector velocity potentials are derived. The spectrum of energy of perturbed fluid versus the distance apart from perturbation is computed.
A flux of ideal fluid coupled to perturbation is investigated by nonperturbative methods of the quantum field theory. Asymptotic behavior of the flux coupled to perturbation turns out to be similiar to that of superfluids.
We have calculated the asymptotics of Green's function of the differential equation of nonlinear diffusion in the microscopic range with strong porosity fluctuations in the problem of flow through irregular packed beds for arbitrary space dimension and arbitrary covariance of porosity fluctuations.
We construct a coupled set of nonlinear reaction-diffusion equations which are exactly solvable. The model generalizes both the Burger equation and a Boltzman reaction equation recently introduced by Th. W. Ruijgrok and T. T. Wu.