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  • 1
    Electronic Resource
    Electronic Resource
    Bifurcation of the Hodgkin and Huxley equations: A new twist (1993)
    Springer
    Bulletin of mathematical biology 55 (1993), S. 937-952 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The Hodgkin and Huxley equations model action potentials in squid giant axons. Variants of these equations are used in most models for electrial activity of excitable membranes. Computational tools based upon the theory of nonlinear dynamical systems are used here to illustrate how the dynamical behavior of the Hodgkin Huxley model changes as functions of two of the system parameters.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02460693
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    On a four parameter family of planar vector fields (1987)
    Springer
    Archive for rational mechanics and analysis 97 (1987), S. 321-352 
    Details
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00280410
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  • 3
    Electronic Resource
    Electronic Resource
    Isochrons and phaseless sets (1975)
    Springer
    Journal of mathematical biology 1 (1975), S. 259-273 
    Details
    ISSN: 1432-1416
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary Winfree has developed mathematical models for his phase resetting experiments on biological clocks. These models lead him to ask a number of mathematical questions concerning dynamical systems. This paper deals with these mathematical questions. In Winfree's terminology we show the existence of isochrons and establish some of their properties.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01273747
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  • 4
    Electronic Resource
    Electronic Resource
    The dynamics of density dependent population models (1977)
    Springer
    Journal of mathematical biology 4 (1977), S. 101-147 
    Details
    ISSN: 1432-1416
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary The dynamics of density-dependent population models can be extraordinarily complex as numerous authors have displayed in numerical simulations. Here we commence a theoretical analysis of the mathematical mechanisms underlying this complexity from the viewpoint of modern dynamical systems theory. After discussing the chaotic behavior of one-dimensional difference equations we proceed to illustrate the general theory on a density-dependent Leslie model with two age classes. The pattern of bifurcations away from the equilibrium point is investigated and the existence of a “strange attractor” is demonstrated — i.e. an attracting limit set which is neither an equilibrium nor a limit cycle. Near the strange attractor the system exhibits essentially random behavior. An approach to the statistical analysis of the dynamics in the chaotic regime is suggested. We then generalize our conclusions to higher dimensions and continuous models (e.g. the nonlinear von Foerster equation).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275980
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