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Invariant Manifolds (invariant + manifold)

Distribution by Scientific Domains

Physics and Astronomy	50%

Selected Abstracts

Invariant manifolds, phase correlations of chaotic orbits and the spiral structure of galaxies

MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 1 2006
N. Voglis
ABSTRACT In the presence of a strong m= 2 component in a rotating galaxy, the phase-space structure near corotation is shaped to a large extent by the invariant manifolds of the short-period family of unstable periodic orbits terminating at L1 or L2. The main effect of these manifolds is to create robust phase correlations among a number of chaotic orbits large enough to support a spiral density wave outside corotation. The phenomenon is described theoretically by soliton-like solutions of a Sine,Gordon equation. Numerical examples are given in an N -body simulation of a barred spiral galaxy. In these examples, we demonstrate how the projection of unstable manifolds in configuration space reproduces essentially the entire observed bar,spiral pattern. [source]

Symbolic methods for invariant manifolds in chemical kinetics

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 1 2006
Simon J. Fraser
Abstract Chemical reactions show a separation of time scales in transient decay due to the stiffness of the ordinary differential equations (ODEs) that describe their evolution. This evolution can be represented as motion in the phase space spanned by the concentration variables of the chemical reaction. Transient decay corresponds to a collapse of the "compressible fluid" representing the continuum of possible dynamical states of the system. Collapse occurs sequentially through a hierarchy of nested, attracting, slow invariant manifolds (SIMs), i.e., sets that map into themselves under the action of the phase flow, eventually reaching the asymptotic attractor of the system. Using a symbolic manipulative language, explicit formulas for the SIMs can be found by iterating functional equations obtained from the system's ODEs. Iteration converges geometrically fast to a SIM at large concentrations and, if necessary, can be stabilized at small concentrations. Three different chemical models are examined in order to show how finding the SIM for a model depends on its underlying dynamics. For every model the iterative method provides a global SIM formula; however, formal series expansions for the SIM diverge in some models. Repelling SIMs can be also found by iterative methods because of the invariance of trajectory geometry under time reversal. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006 [source]

Application of nonlinear time,scaling for robust controller design of reaction systems

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 1 2002
P. Moya
Abstract Even though the basic mechanisms of operation of reaction systems are relatively simple the dynamical models obtained from first principles are complex and contain highly uncertain terms. To develop reliable model-based controllers it is therefore necessary to simplify the system dynamics preserving the features which are essential for control. Towards this end, co-ordinate transformations identifying the states which are dependent/independent of the reactions and flows have been reported in the literature. This has allowed, for instance, the design of observers which are insensitive to the (usually unknown) reaction functions. The main contribution of this paper is to show the utility of nonlinear state-dependent time-scaling to simplify the system dynamics, and consequently the controller design. In particular, we show that with time-scaling and an input transformation we can reveal the existence of attractive invariant manifolds, which allows us to reduce the dimension of the system. As an application we study the well-known fourth order baker's yeast fed-batch fermentation process model, whose essential dynamics is captured by a planar system perturbed by an exponentially decaying term. We then exploit this particular structure to design, with reduced control authority, a nonlinear asymptotically stabilizing control law which is robust with respect to the reaction function. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Invariant manifolds, phase correlations of chaotic orbits and the spiral structure of galaxies

Ordered and chaotic spiral arms

ASTRONOMISCHE NACHRICHTEN, Issue 9-10 2008
P.A. Patsis
Abstract The stellar flow at the arms of spiral galaxies is qualitatively different among different morphological types. The stars that reinforce the spiral arms can be either participating in an ordered or in a chaotic flow. Ordered flows are associated with normal (non-barred) spiral galaxies. Typically they are described with precessing ellipses corresponding to stable periodic orbits at successive energies (Jacobi constants). On the contrary, the spiral arms in barred-spiral systems may be supported by stars in chaotic motion. The trajectories of these stars are associated with the invariant manifolds of the unstable Lagrangian points (L1,2). Response and orbital models indicate that this kind of spirals either stop at an azimuth smaller than , /2, or present large gaps at about this angle. Chaotic spirals appear in strong bars having (L1,2) close to the ends of the bar. The arms of barred-spiral systems with corotation away from the end of the bar can be either as in the case of normal spirals, or supported by banana-like orbits surrounding the stable Lagrangian points (L4,5). We find also models combining ordered and chaotic flows. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]