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  • 1
    Electronic Resource
    Electronic Resource
    Structure of the global attractor of cyclic feedback systems (1995)
    Springer
    Journal of dynamics and differential equations 7 (1995), S. 141-190 
    Details
    ISSN: 1572-9222
    Keywords: Cyclic feedback system ; Conley index theory ; Lyapunov function ; global dynamics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We characterize the dynamics on global attractors of cyclic feedback systems. Under mild restrictions the description is given in terms of a semiconjugacy to a simple model system which possesses Morse-Smale dynamics. However, for the completely general case, no simple model system is feasible and hence we introduce a weaker notion of equivalence, namely, topological semiequivalency. We then prove that the global attractor of a cyclic feedback system is topologically semiequivalent to the original model flow. Main ingredients in the proof are the discrete Lyapunov function introduced by Mallet-Paret and Smith and the Conley index theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02218817
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Singular Index Pairs (1999)
    Springer
    Journal of dynamics and differential equations 11 (1999), S. 399-425 
    Details
    ISSN: 1572-9222
    Keywords: Singular perturbation ; isolating neighborhood ; index pair ; Conley index ; Nagumo equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Using Conley's idea of slow exit points, we construct index pairs for singularity perturbed families of lows. Some applications are also presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1021909803014
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    The Conley Index for Fast-Slow Systems I. One-Dimensional Slow Variable (1999)
    Springer
    Journal of dynamics and differential equations 11 (1999), S. 427-470 
    Details
    ISSN: 1572-9222
    Keywords: Fast-slow systems ; Conley index ; heteroclinic connection ; periodic orbit
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We develop a qualitative theory for fast-slow systems with a one-dimensional slow variable. Using Conley index theory for singularity perturbed systems, conditions are given which imply that if one can construct heteroclinic connections and periodic orbits in systems with the derivative of the slow variable set to 0, these orbits persist when the derivative of the slow variable is small and nonzero.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1021961819853
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    The evolution of slow dispersal rates: a reaction diffusion model (1998)
    Springer
    Journal of mathematical biology 37 (1998), S. 61-83 
    Details
    ISSN: 1432-1416
    Keywords: Key words: Evolution of dispersal ; Migration modification ; Reaction ; diffusion ; Montone systems ; Perturbation of Morse decomposition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract.  We consider n phenotypes of a species in a continuous but heterogeneous environment. It is assumed that the phenotypes differ only in their diffusion rates. With haploid genetics and a small rate of mutation, it is shown that the only nontrivial equilibrium is a population dominated by the slowest diffusing phenotype. We also prove that if there are only two possible phenotypes, then this equilibrium is a global attractor and conjecture that this is true in general. Numerical simulations supporting this conjecture and suggesting that this is a robust phenomenon are also discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002850050120
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    Paper (German National Licenses)
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