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  • 1
    Electronic Resource
    Electronic Resource
    Schwach Majorisierende Elemente und die Besonderen Monotonie-Eigenschaften von Randwertaufgaben Zweiter Ordnung (1979)
    Springer
    Manuscripta mathematica 28 (1979), S. 317-336 
    Details
    ISSN: 1432-1785
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It is a well known theorem for Sturmian boundary value problems Lx=r Rx=0 that the pair (L, R=0) is inverse monotone (i. e. Lx ≧0, Rx=0 ⇒ x≧0) if there exists a weak majorizing element, i. e. a function z≧0 satisfying Lz≧0, Rz=0. We show that this criterion carries over to ordinary boundary value problems of arbitrary order if in addition there exists an inverse monotone pair ‘larger” than (L, R) in a certain sense. This follows from a variant of Schröder's theorem [10] combined with a result on strict monotonicity of nonnegative Green's functions. However, it will also be shown that the additional condition can only be dispensed with, if the boundary value problem is at most of the second order. Furthermore an analogous result holds, for elliptic boundary value problems in arbitrary dimensions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01954612
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  • 2
    Electronic Resource
    Electronic Resource
    Zur Stabilität von Differenzenverfahren für Systeme linearer gewöhnlicher Randwertaufgaben (1978)
    Springer
    Numerische Mathematik 29 (1978), S. 209-226 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L 10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper we give a simple stability theory for finite difference approximations to linear ordinary boundary value problems. In particular we consider stability with respect to a maximum norm including all difference quotients up to the order of the differential equation. It is shown that stability in this sense holds if and only if the principal part of the differential equation is discretized in a “stable way”. This last property is characterized by root conditions which we prove to be satisfied for some classes of finite difference schemes. Our approach simplifies and generalizes some known results of the literature where Sobolev norms or merely the maximum norm are used.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390339
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Numerical Taylor expansions of invariant manifolds in large dynamical systems (1998)
    Springer
    Numerische Mathematik 80 (1998), S. 1-38 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 34C35, 58F14, 65F50, 65H17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. In this paper we develop a numerical method for computing higher order local approximations of invariant manifolds, such as stable, unstable or center manifolds near steady states of a dynamical system. The underlying system is assumed to be large in the sense that a large sparse Jacobian at the equilibrium occurs, for which only a linear (black box) solver and a low dimensional invariant subspace is available, but for which methods like the QR–Algorithm are considered to be too expensive. Our method is based on an analysis of the multilinear Sylvester equations for the higher derivatives which can be solved under certain nonresonance conditions. These conditions are weaker than the standard gap conditions on the spectrum which guarantee the existence of the invariant manifold. The final algorithm requires the solution of several large linear systems with a bordered Jacobian. To these systems we apply a block elimination method recently developed by Govaerts and Pryce [12, 14].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050357
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  • 4
    Electronic Resource
    Electronic Resource
    On invariant closed curves for one-step methods (1987)
    Springer
    Numerische Mathematik 51 (1987), S. 103-122 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) 65L05, 58F08, 58F22 ; CR: G1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We show that a one-step method as applied to a dynamical system with a hyperbolic periodic orbit, exhibits an invariant closed curve for sufficiently small step size. This invariant curve converges to the periodic orbit with the order of the method and it inherits the stability of the periodic orbit. The dynamics of the one-step method on the invariant curve can be described by the rotation number for which we derive an asymptotic expression. Our results complement those of [2, 3] where one-step methods were shown to create invariant curves if the dynamical system has a periodic orbit which is stable in either time direction or if the system undergoes a Hopf bifurcation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01399697
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  • 5
    Electronic Resource
    Electronic Resource
    A Direct Approach to Homoclinic Orbits in the Fast Dynamics of Singularly Perturbed Systems (1999)
    Springer
    Journal of dynamics and differential equations 11 (1999), S. 671-709 
    Details
    ISSN: 1572-9222
    Keywords: Homoclinic orbits ; singular perturbations ; bifurcation ; FitzHugh–Nagumo system
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Homoclinic orbits in the fast dynamics of singular perturbation problems are usually analyzed by a combination of Fenichel's invariant manifold theory with general transversality arguments (see Ref. 29 and the Exchange Lemma in Ref. 16). In this paper an alternative direct approach is developed which uses a two-time scaling and a contraction argument in exponentially weighted spaces. Homoclinic orbits with one last transition are treated and it is shown how ε-expansions can be extracted rigorously from this approach. The result is applied to a singularity perturbed Bogdanov point in the FitzHugh–Nagumo system.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022663512855
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    Simulating the motion of the leech: A biomechanical application of DAEs (1998)
    Springer
    Numerical algorithms 19 (1998), S. 1-12 
    Details
    ISSN: 1572-9265
    Keywords: Lagrange's equations of motion ; viscoelasticity ; semi-explicit methods ; 92C10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper a mathematical model is developed for the dynamical behaviour of a hydrostatic skeleton. The basic configuration is taken from the worm-like shape of the medicinal leech. It consists of a sequence of hexahedra with damped elastic springs as edges to model the various parts of the musculature. The system is stabilized by the constraint of constant volume either in the whole body or in prescribed compartments. We set up Lagrange's equations of motion with the Lagrange multipliers being the pressure values in the compartments. The equations of motion lead to a large differential-algebraic system which is solved by an application of semi-explicit numerical methods. Though the model has not yet been adapted to experimental data, first simulations with a simplified set of parameters show that it is capable of generating basic movements of the leech such as crawling and swimming.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019146221279
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    Paper (German National Licenses)
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