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  • 1
    Electronic Resource
    Electronic Resource
    Front initiation on microdesigned composite catalysts (2002)
    Woodbury, NY : American Institute of Physics (AIP)
    Chaos 12 (2002), S. 190-203 
    Details
    ISSN: 1089-7682
    Source: AIP Digital Archive
    Topics: Physics
    Notes: We first briefly review the subject of spatiotemporal pattern formation on microdesigned composite catalysts. One of the most significant interaction mechanisms between different reacting domains (consisting of different metal catalysts such as Pt and Rh, coupled through surface diffusion) is the initiation of reaction fronts at the interface between them. We then explore in some detail the effect of two-dimensional composite geometry on this basic building block of composite catalyst dynamics. © 2002 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.1452739
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  • 2
    Electronic Resource
    Electronic Resource
    A dynamical systems approach to spiral wave dynamics (1994)
    Woodbury, NY : American Institute of Physics (AIP)
    Chaos 4 (1994), S. 453-460 
    Details
    ISSN: 1089-7682
    Source: AIP Digital Archive
    Topics: Physics
    Notes: A simple system of five nonlinear ordinary differential equations is shown to reproduce many dynamical features of spiral waves in two-dimensional excitable media.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.166023
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  • 3
    Electronic Resource
    Electronic Resource
    Chaotic advection in a complex annular geometry (1991)
    New York, NY : American Institute of Physics (AIP)
    Physics of Fluids 3 (1991), S. 1063-1067 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: The dynamics of Lagrangian particles in a complex geometry is studied, both experimentally and through a full numerical simulation of the Navier–Stokes equations. The geometry is an annulus whose walls can be rotated independently. Stationary cylindrical rods can be positioned within the annulus in several arrangements. A variety of heteroclinic orbits are found at low Reynolds numbers, where the fluid flow is steady. As the flow becomes unsteady to a time-periodic (two-dimensional) state, it spontaneously gives rise to heteroclinic tangles that provide the organizing structure for the chaotic motion of fluid particles.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.858086
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  • 4
    Electronic Resource
    Electronic Resource
    OPTICAL IMAGING AND CONTROL OF GENETICALLY DESIGNATED NEURONS IN FUNCTIONING CIRCUITS (2005)
    Palo Alto, Calif. : Annual Reviews
    Annual Review of Neuroscience 28 (2005), S. 533-563 
    Details
    ISSN: 0147-006X
    Source: Annual Reviews Electronic Back Volume Collection 1932-2001ff
    Topics: Biology , Medicine
    Notes: Proteins with engineered sensitivities to light are infiltrating the biological mechanisms by which neurons generate and detect electrochemical signals. Encoded in DNA and active only in genetically specified target cells, these proteins provide selective optical interfaces for observing and controlling signaling by defined groups of neurons in functioning circuits, in vitro and in vivo. Light-emitting sensors of neuronal activity (reporting calcium increase, neurotransmitter release, or membrane depolarization) have begun to reveal how information is represented by neuronal assemblies, and how these representations are transformed during the computations that inform behavior. Light-driven actuators control the electrical activities of central neurons in freely moving animals and establish causal connections between the activation of specific neurons and the expression of particular behaviors. Anchored within mathematical systems and control theory, the combination of finely resolved optical field sensing and finely resolved optical field actuation will open new dimensions for the analysis of the connectivity, dynamics, and plasticity of neuronal circuits, and perhaps even for replacing lostĐ??or designing novelĐ??functionalities.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1146/annurev.neuro.28.051804.101610
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  • 5
    Electronic Resource
    Electronic Resource
    Immune network behavior—I. From stationary states to limit cycle oscilations (1993)
    Springer
    Bulletin of mathematical biology 55 (1993), S. 745-780 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We develop a model for the idiotypic interaction between two B cell clones. This model takes into account B cell proliferation, B cell maturation, antibody production, the formation and subsequent elimination of antibody-antibody complexes and recirculation of antibodies between the spleen and the blood. Here we investigate, by means of stability and bifurcation analysis, how each of the processes influences the model's behavior. After appropriate nondimensinalization, the model consists of eight ordinary differential equations and a number of parameters. We estimate the parameters from experimental sources. Using a coordinate system that exploits the pairwise symmetry of the interactions between two clones, we analyse two simplified forms of the model and obtain bifurcation diagrams showing how their five equilibrium states are related. We show that the so-called immune states lose stability if B cell and antibody concentrations change on different time scales. Additionally, we derive the structure of stable and unstable manifolds of saddle-tye equilibria, pinpoint their (global) bifurcations and show that these bifurcations play a crucial role in determining the parameter regimes in which the model exhibits oscillatory behavior.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02460672
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  • 6
    Electronic Resource
    Electronic Resource
    Immune network behavior—II. From oscillations to chaos and stationary states (1993)
    Springer
    Bulletin of mathematical biology 55 (1993), S. 781-816 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Two types of behavior have been previously reported in models of immune networks. The typical behavior of simple models, which involve B cells only, is stationary behavior involving several steady states. Finite amplitude perturbations may cause the model to switch between different equilibria. The typical behavior of more realistic models, which involve both B cells and antibody, consists of autonomous oscillations and/or chaos. While stationary behavior leads to easy interpretations in terms of idiotypic memory, oscillatory behavior seems to be in better agreement with experimental data obtained in unimmunized animals. Here we study a series of models of the idiotypic interaction between two B cell clones. The models differ with respect to the incorporation of antibodies, B cell maturation and compartmentalization. The most complicated model in the series has two realistic parameter regimes in which the behavior is respectively stationary and chaotic. The stability of the equilibrium states and the structure and interactions of the stable and unstable manifolds of the saddle-type equilibria turn out to be factors influencing the model's behavior. Whether or not the model is able to attain any form of sustained oscillatory behavior, i.e. limit cycles or chaos, seems to be determined by (global) bifurcations involving the stable and unstable manifolds of the equilibrium states. We attempt to determine whether such behavior should be expected to be attained from reasonable initial conditions by incorporating an immune response to an antigen in the model. A comparison of the behavior of the model with experimental data from the literature provides suggestions for the parameter regime in which the immune system is operating.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02460673
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  • 7
    Electronic Resource
    Electronic Resource
    Noninvertibility and the structure of basins of attraction in a model adaptive control system (1991)
    Springer
    Journal of nonlinear science 1 (1991), S. 95-105 
    Details
    ISSN: 1432-1467
    Keywords: adaptive control ; noninvertibility ; basins of attraction ; bifurcations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Summary We present a two-dimensional, nonlinear map, arising from a simple adaptive control problem, which exhibits disconnected boundaries separating the basins of attraction of its coexisting attractors. We perform a detailed study of the relation between this phenomenon and the noninvertible nature of the map and demonstrate how the complex basin structure is caused by a change in the number of preimages of points along a stable manifold.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01209149
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  • 8
    Electronic Resource
    Electronic Resource
    Two-dimensional invariant manifolds and global bifurcations: some approximation and visualization studies (1997)
    Springer
    Numerical algorithms 14 (1997), S. 125-140 
    Details
    ISSN: 1572-9265
    Keywords: computational dynamics ; global bifurcations ; invariant manifolds ; 34 ; 65
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We illustrate and discuss the computer-assisted study (approximation and visualization) of two-dimensional (un)stable manifolds of steady states and saddle-type limit cycles for flows in R n . Our investigation highlights a number of computational issues arising in this task, along with our solutions and “quick-fixes” for some of these problems. Two examples illustrative of both successes and shortcomings of our current approach are presented. Representative “snapshots” demonstrate the dependence of two-dimensional invariant manifolds on a bifurcation parameter as well as their interactions. Such approximation and visualization studies are a necessary component of the computer-assisted study and understanding of global bifurcations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019104828180
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  • 9
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    Reconstruction of normal forms by learning informed observation geometries from data (2017)
    National Academy of Sciences
    Details
    Publication Date: 2017-08-22
    Description: The discovery of physical laws consistent with empirical observations is at the heart of (applied) science and engineering. These laws typically take the form of nonlinear differential equations depending on parameters; dynamical systems theory provides, through the appropriate normal forms, an “intrinsic” prototypical characterization of the types of dynamical regimes accessible to a given model. Using an implementation of data-informed geometry learning, we directly reconstruct the relevant “normal forms”: a quantitative mapping from empirical observations to prototypical realizations of the underlying dynamics. Interestingly, the state variables and the parameters of these realizations are inferred from the empirical observations; without prior knowledge or understanding, they parametrize the dynamics intrinsically without explicit reference to fundamental physical quantities.
    Print ISSN: 0027-8424
    Electronic ISSN: 1091-6490
    Topics: Biology , Medicine , Natural Sciences in General
    Published by National Academy of Sciences
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  • 10
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    Adaptive Method for the Experimental Detection of Instabilities (1999)
    American Physical Society
    Details
    Publication Date: 1999-01-18
    Print ISSN: 0031-9007
    Electronic ISSN: 1079-7114
    Topics: Physics
    Published by American Physical Society
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