Development of a Hamiltonian Formulation for the Anomalous Particle Transport in C...
Abstract
ln this proposal we describe our experimental and theoretical research plans in nonlinear dynamics applied to hamiltonian and dissipative systems, turbulence and biological systems. Hamiltonian chaos and turbulence will be studied by applying the theory of nonlinear hamiltonian dynamical systems to investigate several problems in the magnetic confinement of plasma. The main restriction to the application of the magnetic confinement of plasma is the unexpected loss of particles that leave the plasma. Several recent experiments confirmed that this anomalous transport of particles depends on the configurations of the electric and magnetic fields in the edge of the plasma. We intend to investigate the effect of these configurations on this transport. The electric field equilibrium is changed by the modification of its radial profile in the edge of the plasma, as already experimentally observed in some tokamaks. The magnetic field is modified by resonant external electric currents, as the ones given by a magnetic limiter or a divertor. We will also investigate the origin of the turbulence of the drift waves in tokamaks. ln particular, the appearance of spatial and temporal chaotic waves will be analyzed as consequence of dominant waves, as well as the experimental control of this kind of turbulence. ln several experiments with dissipative systems we will address basic properties of classical chaos theory that can be experimentally verified by studying the behavior of chaotic systems of interest in applied physics, engineering and medicine. The dynamical properties are identified in the parameter space, where we can represent common properties of the attractors and the bifurcations between these attractors. Some methods of control are analyzed, considering both ideal systems (with infinite power) as well as non ideal (with finite powers). Systems described by either differential equations or analytical maps will be considered. The main chaotic systems to be investigated are: electromechanical devices, electric circuits, particle advection in fluids and biological rhythms. The phenomena of higher interest are the bifurcations, the synchronization and the instabilities in the parameter space. The changes of these properties with the appearance of stable and unstable manifolds and chaotic saddles in the phase space of the systems will be also investigated. We will investigate the properties of the attractors by applying metric (Lyapunov spectrum) and topological (parameter spaces, isoperiodic diagrams, topological planes, etc.) techniques. We will also study: the phase synchronization of complex chaotic attractors found in electrical circuits, such as the double scroll attractor of the Chua circuit. With new experimental and analysis techniques, we will study the formation of drops of water in regimen of low dripping rate with a high speed video camera. The coupling of bubbles forming from two close nozzles will be studied by measuring the bubbling pressure waves and the data will be analyzed with the technique of bi-spectral analysis that also will be applied to characterize the coupling of oscillating flames. A more extensive study of acoustic resonators will be carried out to verify if certain systems are in fact of intermediate statistics or if the distributions of nearest neighbors of the eigenvalues are affected by missing levels... (AU)
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