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  • 1
    Digitale Medien
    Digitale Medien
    TWO TYPES OF HOPF BIFURCATION POINTS: SOURCES AND SINKS OF FAMILIES OF PERIODIC ORBITS (1980)
    Oxford, UK : Blackwell Publishing Ltd
    Annals of the New York Academy of Sciences 357 (1980), S. 0 
    Details
    ISSN: 1749-6632
    Quelle: Blackwell Publishing Journal Backfiles 1879-2005
    Thema: Allgemeine Naturwissenschaft
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1111/j.1749-6632.1980.tb29695.x
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  • 2
    Digitale Medien
    Digitale Medien
    Modeling reflex asymmetries with implicit delay differential equations (1998)
    Springer
    Bulletin of mathematical biology 60 (1998), S. 999-1015 
    Details
    ISSN: 1522-9602
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Biologie , Mathematik
    Notizen: Abstract Neuromuscular reflexes with time-delayed negative feedback, such as the pupil light reflex, have different rates depending on the direction of movement. This asymmetry is modeled by an implicit first-order delay differential equation in which the value of the rate constant depends on the direction of movement. Stability analyses are presented for the cases when the rate is: (1) an increasing and (2) a decreasing function of the direction of movement. It is shown that the stability of equilibria in these dynamical systems depends on whether the rate constant is a decreasing or increasing function. In particular, when the asymmetry has the shape of an increasing step function, it is possible to have stability which is independent of the value of the time delay or the steepness (i.e., gain) of the negative feedback.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1006/S0092-8240(98)90000-3
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  • 3
    Digitale Medien
    Digitale Medien
    The Fredholm Alternative for Functional Differential Equations of Mixed Type (1999)
    Springer
    Journal of dynamics and differential equations 11 (1999), S. 1-47 
    Details
    ISSN: 1572-9222
    Schlagwort(e): Fredholm alternative ; exponential dichotomy ; functional differential equations
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract We prove a Fredholm alternative theorem for a class of asymptotically hyperbolic linear differential difference equations of mixed type. We also establish the cocycle property and the spectral flow property for such equations, providing an effective means of calculating the Fredholm index. Such systems can arise from equations which describe traveling waves in a spatial lattice.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1021889401235
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  • 4
    Digitale Medien
    Digitale Medien
    Transition layers for singularly perturbed delay differential equations with monotone nonlinearities (1989)
    Springer
    Journal of dynamics and differential equations 1 (1989), S. 3-43 
    Details
    ISSN: 1572-9222
    Schlagwort(e): Differential delay equations ; singular perturbations ; transition layers
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract Transition layers arising from square-wave-like periodic solutions of a singularly perturbed delay differential equation are studied. Such transition layers correspond to heteroclinic orbits connecting a pair of equilibria of an associated system of transition layer equations. Assuming a monotonicity condition in the nonlinearity, we prove these transition layer equations possess a unique heteroclinic orbit, and that this orbit is monotone. The proof involves a global continuation for heteroclinic orbits.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01048789
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  • 5
    Digitale Medien
    Digitale Medien
    The Global Structure of Traveling Waves in Spatially Discrete Dynamical Systems (1999)
    Springer
    Journal of dynamics and differential equations 11 (1999), S. 49-127 
    Details
    ISSN: 1572-9222
    Schlagwort(e): Traveling waves ; spatially discrete systems ; lattice differential equations ; continuation methods ; heteroclinic orbits ; Lin's method ; Mel'nikov method
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract We obtain existence of traveling wave solutions for a class of spatially discrete systems, namely, lattice differential equations. Uniqueness of the wave speed c, and uniqueness of the solution with c≠0, are also shown. More generally, the global structure of the set of all traveling wave solutions is shown to be a smooth manifold where c≠0. Convergence results for solutions are obtained at the singular perturbation limit c → 0.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1021841618074
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  • 6
    Digitale Medien
    Digitale Medien
    The Poincare-Bendixson theorem for monotone cyclic feedback systems (1990)
    Springer
    Journal of dynamics and differential equations 2 (1990), S. 367-421 
    Details
    ISSN: 1572-9222
    Schlagwort(e): Cellular control system ; cyclic system ; monotonicity ; negative feedback ; Poincare-Bendixson theorem
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract We prove the Poincare-Bendixson theorem for monotone cyclic feedback systems; that is, systems inR n of the form $$x_i = f_i (x_i , x_{i - 1} ), i = 1, 2, ..., n (\bmod n).$$ We apply our results to a variety of models of biological systems.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01054041
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  • 7
    Digitale Medien
    Digitale Medien
    A Poincaré-Bendixson theorem for scalar reaction diffusion equations (1989)
    Springer
    Archive for rational mechanics and analysis 107 (1989), S. 325-345 
    Details
    ISSN: 1432-0673
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract For scalar equations $$u_t = u_{xx} + f(x, u, u_x )$$ with x ε S 1 and f ε C 2 we show that the classical theorem of Poincaré and Bendixson holds: the ω-limit set of any bounded solution satisfies exactly one of the following alternatives: - it consists in precisely one periodic solution, or - it consists of solutions tending to equilibrium as $$t \to \pm \infty $$ This is surprising, because the system is genuinely infinite-dimensional.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00251553
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  • 8
    Digitale Medien
    Digitale Medien
    Boundary layer phenomena for differential-delay equations with state-dependent time lags, I. (1992)
    Springer
    Archive for rational mechanics and analysis 120 (1992), S. 99-146 
    Details
    ISSN: 1432-0673
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract In this paper we begin a study of the differential-delay equation $$\varepsilon x'(t) = - x(t) + f(x(t - r)), r = r(x(t))$$ . We prove the existence of periodic solutions for 0〈ɛ〈ɛ 0, where ɛ 0 is an optimal positive number. We investigate regularity and monotonicity properties of solutions x(t) which are defined for all t and of associated functions like η(t)=t−r(x(t)). We begin the development of a Poincaré-Bendixson theory and phase-plane analysis for such equations. In a companion paper these results will be used to investigate the limiting profile and corresponding boundary layer phenomena for periodic solutions as ɛ approaches zero.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00418497
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  • 9
    Digitale Medien
    Digitale Medien
    Floquet bundles for scalar parabolic equations (1995)
    Springer
    Archive for rational mechanics and analysis 129 (1995), S. 245-304 
    Details
    ISSN: 1432-0673
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract For linear scalar parabolic equations such as $$u_t = u_{xx} + a(t,x)u_x + b(t,x)u$$ on a finite interval 0≦x≦π, with various boundary conditions, we obtain canonical Floquet solutions u n (t, x). These solutions are characterized by the property that z(u n (t, x))=n for all tεℝ, where z(·) denotes the zero crossing (lap) number of Matano. The coefficients a(t, x) and b(t, x) are not assumed to be periodic in t, but if they are, the solutions u n (t, x) reduce to the standard Floquet solutions. Our results may naturally be expressed in the language of linear skew product flows. In this context, we obtain for each N≧1 an exponential dichotomy between the bundles span {u n (·,·)} n =1/N and $$\overline {span} \{ u_n ( \cdot , \cdot )\} _{n = N + 1}^\infty $$ .
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00383675
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  • 10
    Digitale Medien
    Digitale Medien
    Applications of generic bifurcation. I (1975)
    Springer
    Archive for rational mechanics and analysis 59 (1975), S. 159-188 
    Details
    ISSN: 1432-0673
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00249688
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