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ItemShortest Paths Through 3-Dimensional Cluttered Environments(Georgia Institute of Technology, 2014-06) Lu, Jun ; Diaz-Mercado, Yancy ; Egerstedt, Magnus B. ; Zhou, Haomin ; Chow, Shui-NeeThis paper investigates the problem of finding shortest paths through 3-dimensional cluttered environments. In particular, an algorithm is presented that determines the shortest path between two points in an environment with obstacles which can be implemented on robots with capabilities of detecting obstacles in the environment. As knowledge of the environment is increasing while the vehicle moves around, the algorithm provides not only the global minimizer – or shortest path – with increasing probability as time goes by, but also provides a series of local minimizers. The feasibility of the algorithm is demonstrated on a quadrotor robot flying in an environment with obstacles.
ItemWiener chaos expansion and simulation of electromagnetic wave propagation excited by a spatially incoherent source(Georgia Institute of Technology, 2010) Badieirostami, Majid ; Adibi, Ali ; Zhou, Hao-Min ; Chow, Shui-NeeFirst, we propose a new stochastic model for a spatially incoherent source in optical phenomena. The model naturally incorporates the incoherent property into the electromagnetic wave equation through a random source term. Then we propose a new numerical method based on Wiener chaos expansion (WCE) and apply it to solve the resulting stochastic wave equation. The main advantage of the WCE method is that it separates random and deterministic effects and allows the random effects to be factored out of the primary partial differential equation (PDE) very effectively. Therefore, the stochastic PDE is reduced to a set of deterministic PDEs for the coefficients of the WCE method which can be solved by conventional numerical algorithms. We solve these secondary deterministic PDEs by a finite-difference time domain (FDTD) method and demonstrate that the numerical computations based on the WCE method are considerably more efficient than the brute-force simulations. Moreover, the WCE approach does not require generation of random numbers and results in less computational errors compared to Monte Carlo simulations.
ItemModel for efficient simulation of spatially incoherent Right using the Wiener chaos expansion method(Georgia Institute of Technology, 2007-11) Badieirostami, Majid ; Adibi, Ali ; Zhou, Hao-Min ; Chow, Shui-NeeWe demonstrate a new and efficient technique for modeling and simulation of spatially incoherent sources using the Wiener chaos expansion method. By implementing this new model, we show that a practical-size photonic structure with a spatially incoherent input source can be analyzed more than 2 orders of magnitude faster compared with the conventional models without sacrificing the accuracy.
ItemQuasiperiodic and Chaotic Dynamics in Bose-Einstein Condensates in Periodic Lattices and Superlattices(Georgia Institute of Technology, 2005-07-23) van Noort, Martijn ; Porter, Mason ; Yi, Yingfei ; Chow, Shui-NeeWe employ KAM theory to rigorously investigate the transition between quasiperiodic and chaotic dynamics in cigar-shaped Bose-Einstein condensates (BEC) in periodic lattices and superlattices. Toward this end, we apply a coherent structure ansatz to the Gross-Pitaevskii equation to obtain a parametrically forced Duffing equation describing the spatial dynamics of the condensate. For shallow-well, intermediate-well, and deep-well potentials, we find KAM tori and Aubry-Mather sets to prove that one obtains mostly quasiperiodic dynamics for condensate wave functions of sufficiently large amplitude, where the minimal amplitude depends on the experimentally adjustable BEC parameters. We show that this threshold scales with the square root of the inverse of the scattering length, whereas the rotation number of tori above this threshold is proportional to the amplitude. As a consequence, one obtains the same dynamical picture for lattices of all depths, as an increase in its amplitude essentially only affects scaling in phase space. Our approach is applicable to periodic superlattices with an arbitrary number of rationally dependent wave numbers.
ItemQuasiperiodic Dynamics in Hamiltonian 1 1/2 Degree of Freedom Systems Far from Integrability(Georgia Institute of Technology, 2005-01-21) Chow, Shui-Nee ; van Noort, Martijn ; Yi, YingfeiThe subject of this paper is two-quasiperiodicity in a large class of one-and-a-half degree of freedom Hamiltonian systems. The main result is that such systems have invariant tori for any internal frequency that is of constant type and sufficiently large, relative to the forcing frequency. An explicit bound on the minimum value of the internal frequency is presented. The systems under consideration are not required to be small perturbations of integrable ones.
ItemTravelling Wave Solutions in a Tissue Interaction Model for Skin Pattern Formation(Georgia Institute of Technology, 2003) Ai, Shangbing ; Chow, Shui-Nee ; Yi, YingfeiWe discuss the existence and the uniqueness of travelling wave solutions for a tissue interaction model on skin pattern formation proposed by Cruywagen and Murray. The geometric theory of singular perturbations is employed.
ItemThe Cyclicity of Period Annulus of Degenerate Quadratic Hamiltonian System with Elliptic Segment(Georgia Institute of Technology, 2000) Chow, Shui-Nee ; Li, Chengzhi ; Yi, YingfeiWe study the cyclicity of period annuli (or annulus) for general degenerate quadratic Hamiltonian systems with an elliptic segment or a saddle loop, under quadratic perturbations. By using geometrical arguments and studying the respective Abelian integral based on the Picard-Fuchs equation, it is shown that the cyclicity of period annuli or annulus for such systems equals two. This result, together with those of ,,,,, gives a complete solution to the infinitesimal Hilbert 16th problem in the case of degenerate quadratic Hamiltonian systems under quadratic perturbations.
ItemCenter Manifolds for Invariant Sets(Georgia Institute of Technology, 1999) Chow, Shui-Nee ; Liu, Weishi ; Yi, YingfeiWe derive a general center manifolds theory for a class of compact invariant sets of flows generated by a smooth vector fields in R^n. By applying the Hadamard graph transform technique, it is shown that, associated to certain dynamical characteristics of the linearized flow along the invariant set, there exists an invariant manifold (called a center manifold) of the invariant set which contains every locally bounded solution (in particular, contains the invariant set) and is persistent under small perturbations.
ItemPersistence of Invariant Tori on Submanifolds in Hamiltonian Systems(Georgia Institute of Technology, 1999) Chow, Shui-Nee ; Li, Yong ; Yi, YingfeiGeneralizing the degenerate KAM theorem under the Rüssmann non-degeneracy and the isoenergetic KAM theorem, we employ a quasi-linear iterative scheme to study the persistence and frequency preservation of invariant tori on a smooth sub-manifold for a real analytic, nearly integrable Hamiltonian system. Under a nondegenerate condition of Rüssmann type on the sub-manifold, we shall show the following: a) the majority of the unperturbed tori on the sub-manifold will persist; b) the perturbed toral frequencies can be partially preserved according to the maximal degeneracy of the Hessian of the unperturbed system and be fully preserved if the Hessian is nondegenerate; c) the Hamiltonian admits normal forms near the perturbed tori of arbitrarily prescribed high order. Under a sub-isoenergetic nondegenerate condition on an energy surface, we shall show that the majority of unperturbed tori give rise to invariant tori of the perturbed system of the same energy which preserve the ratio of certain components of the respective frequencies.
ItemCenter Manifolds for Invariant Manifolds(Georgia Institute of Technology, 1997) Chow, Shui-Nee ; Liu, Weishi ; Yi, YingfeiWe study dynamics of flows generated from smooth vector fields in R^n in the vicinity of an invariant and closed smooth manifold Y. By applying the Hadamard graph transform technique, we show that there exists an invariant manifold (called a center manifold of Y) based on the information of the linearization along Y, which contains every locally bounded solution and is persistent under small perturbations.