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Extracting and Representing Qualitative Behaviors of Complex Systems in Phase Spaces
, 1991
"... This paper describes research done at the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for the Laboratory's artificial intelligence research is provided in part by the Advanced Research Projects Agency of the Department of Defense under Office of Naval Res ..."
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Cited by 45 (16 self)
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This paper describes research done at the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for the Laboratory's artificial intelligence research is provided in part by the Advanced Research Projects Agency of the Department of Defense under Office of Naval Research contract N0001489 J3202, and in part by the National Science Foundation grant MIP9001651. The author is also supported by a G.Y. Chu Fellowship
HighOrder Compact Finite Difference Schemes for Computational Mechanics
, 1995
"... A class of highorder compact (HOC) finite difference schemes is developed that exhibits higherorder accuracy at the grid points yet utilizes only a compact stencil. This is achieved by using the governing differential equation to approximate leading truncation error terms in the central difference ..."
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Cited by 13 (0 self)
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A class of highorder compact (HOC) finite difference schemes is developed that exhibits higherorder accuracy at the grid points yet utilizes only a compact stencil. This is achieved by using the governing differential equation to approximate leading truncation error terms in the central difference scheme. The method is first developed for steady convection diffusion problems on uniform grids. The highorder accuracy of HOC schemes is demonstrated, as well as their tendency to suppress false oscillations. Second, this class of approximations is used to solve the streamfunction vorticity form of the 2D, steady, incompressible NavierStokes equations. Numerical results for this application compare favorably with previously published results in the literature, despite use of coarser grids with the HOC scheme. Third, HOC iterative performance is analyzed, revealing that even though HOC condition numbers are higher than those of more standard schemes, certain gradienttype algorithms actually converge slightly faster for HOC formulations than for standard formulations on problems of equivalent size. HOC theory is then extended to nonuniform grids in 1D and 2D by mapping to a computational domain with a uniform mesh. It is found that the derivatives of the mapping functions can degrade the overall accuracy of the HOC formulation if they are not approximated to sufficient order. A method is proposed for computing highorder compact grid metrics for the case where the grid is obtained by solving a differential equation, as happens with a Helmholtz grid generator. In the absence of such a differential equation, more standard noncompact differencing can be used to obtain highorder metrics. In 2D, the nonuniform grids must be orthogonal to maintain fourthorder accuracy. Finally, other restrictions to previous HOC schemes are removed by extending the theory to transient problems, nonlinear Poisson problems, and 3D linear Poisson problems. Transient HOC formulas are found to have slightly stricter stability requirements on the time step for forward Euler than for more standard schemes, although this is alleviated by the fact that coarser grids may be used. However, HOC forward Euler is an implicit formulation, making backward Euler and CrankNicolson more attractive because they are unconditionally stable.
Dynamics of two coupled van der Pol oscillators with delay coupling
 In Proceedings of ASME Design Engineering Technical Conference
, 1997
"... In this work, we investigate the dynamics of two weakly coupled van der Pol oscillators in which the coupling terms have time delay τ. Our work is motivated by applications to laser dynamics and the coupling of microwave oscillators. The governing equations are ¨x1 + x1 − ɛ (1 − x 2 1)˙x1 = ɛα ˙x2 ( ..."
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Cited by 6 (1 self)
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In this work, we investigate the dynamics of two weakly coupled van der Pol oscillators in which the coupling terms have time delay τ. Our work is motivated by applications to laser dynamics and the coupling of microwave oscillators. The governing equations are ¨x1 + x1 − ɛ (1 − x 2 1)˙x1 = ɛα ˙x2 (t − τ), ¨x2 + x2 − ɛ (1 − x 2 2)˙x2 = ɛα ˙x1 (t − τ), where the coupling is chosen to be through the damping terms because this form of coupling occurs in radiatively coupled microwave oscillator arrays. We use the method of averaging to obtain the approximate simplified system of three slowflow equations ˙R1 = 1 2 ˙R2 = 1
A Visual Programming Environment for Functional Languages
, 2002
"... I declare that this thesis is my own account of my research and contains as its main content work which has not previously been submitted for a degree at any tertiary education institution. Joel Kelso ii The purported advantages of Visual Programming, as applied to general purpose programming langua ..."
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Cited by 3 (0 self)
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I declare that this thesis is my own account of my research and contains as its main content work which has not previously been submitted for a degree at any tertiary education institution. Joel Kelso ii The purported advantages of Visual Programming, as applied to general purpose programming languages, have remained largely unfulfilled. The essence of this thesis is that functional programming languages have at least one natural visual representation, and that a useful programming environment can be based upon this representation. This thesis describes the implementation of a Visual Functional Programming Environment (VFPE). The programming environment has several significant features. • The environment includes a program editor that is inherently
Transition Curves For The QuasiPeriodic Mathieu Equation
, 1998
"... In this work we investigate an extension of Mathieu's equation, the quasiperiodic (QP) Mathieu equation given by # +[# + # (cos t + cos #t)] # =0 for small # and irrational #. Of interest is the generation of stability diagrams that identify the points or regions in the ## parameter plane (for ..."
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Cited by 3 (0 self)
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In this work we investigate an extension of Mathieu's equation, the quasiperiodic (QP) Mathieu equation given by # +[# + # (cos t + cos #t)] # =0 for small # and irrational #. Of interest is the generation of stability diagrams that identify the points or regions in the ## parameter plane (for fixed #) for which all solutions of the QP Mathieu equation are bounded. Numerical integration is employed to produce approximations to the true stability diagrams both directly and through contour plots of Lyapunov exponents. In addition, we derive approximate analytic expressions for transition curves using two distinct techniques: (1) a regular perturbation method under which transition curves # = #(#; #) are each expanded in powers of #, and (2) the method of harmonic balance utilizing Hill's determinants. Both analytic methods deliver results in good agreement with those generated numerically in the sense that predominant regions of instability are clearly coincident. And, both analytic techniques enable us to gain insight into the structure of the corresponding numerical plots. However, the perturbation method fails in the neighborhood of resonant values of # due to the problem of small divisors; the results obtained by harmonic balance do not display this undesirable feature. Key words. quasiperiodic, Floquet theory, Hill's equation, Mathieu equation, perturbations, stability AMS subject classifications. 34, 34D, 34E, 34D08, 34D10, 34E10 PII. S0036139996303877 1.
Design of Interactive Environment for Numerically Intensive Parallel Linear Algebra Calculations
, 2004
"... Problem Solving Environments have a well established position as an essential tool for computational science. We focus our attention in this article on how to provide parallel computation capabilities to such environments that would allow for seamless access to hardware and software resources for nu ..."
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Problem Solving Environments have a well established position as an essential tool for computational science. We focus our attention in this article on how to provide parallel computation capabilities to such environments that would allow for seamless access to hardware and software resources for numerical linear algebra. Instead of focusing on a particular implementation, we present an exploration of the design space of such an interactive environment and consequences of particular design choices. We also show tests of a prototype implementation of our ideas with emphasis on the performance perceived by the end user.
52 LIVING WITH A NEW MATHEMATICAL SPECIES
"... Computers are both the creature and the creator of mathematics. They are, in the apt phrase of Seymour Papert, "mathematicsspeaking beings". More recently J. David Bolter in his stimulating book Turing's Man [4] calls computers "embodied mathematics". Computers shape and enhance the power of mathem ..."
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Computers are both the creature and the creator of mathematics. They are, in the apt phrase of Seymour Papert, "mathematicsspeaking beings". More recently J. David Bolter in his stimulating book Turing's Man [4] calls computers "embodied mathematics". Computers shape and enhance the power of mathematics, while mathematics shapes and enhances the power of computers. Each forces the other to grow and change, creating, in Thomas Kuhn's language, a new mathematical paradigm. Until recently, mathematics was a strictly human endeavor. But suddenly, in a brief instant on the time scale of mathematics, a new species has entered the mathematical ecosystem. Computers speak mathematics, but in a dialect that is difficult for some humans to understand. Their number systems are finite rather than infinite; their addition is not commutative; and they don't really understand "zero", not to speak of "infinity". Nonetheless, they do embody
McKoon. Correspondence concerning this article should be addressed
"... Implementation of global memory models with software that does symbolic computation ..."
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Implementation of global memory models with software that does symbolic computation