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  • 1985-1989  (5)
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Year
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  • 1
    Electronic Resource
    Electronic Resource
    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
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00251553
    Library Location Call Number Volume/Issue/Year Availability
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    Paper (German National Licenses)
    Fulltext
  • 2
    Electronic Resource
    Electronic Resource
    Global Hopf bifurcation of two-parameter flows (1986)
    Springer
    Archive for rational mechanics and analysis 94 (1986), S. 59-81 
    Details
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The behavior of center-indices, as introduced by J. Mallet-Paret & J. Yorke, is analyzed for two-parameter flows. The integer sum of center-indices along a one-dimensional curve in parameter space is called the H-index. A nonzero H-index implies global Hopf bifurcation. The index H is not a homotopy invariant. This fact is due to the occurrence of stationary points with an algebraically double eigenvalue zero, which we call B-points. To each B-point we assign an integer B-index, such that the H-index relates to the B-indices by a formula such as occurs in the calculus of residues. This formula is easily applied to study global bifurcation of periodic solutions in diffusively coupled two-cells of chemical oscillators and to treat spatially heterogeneous time-periodic oscillations in porous catalysts.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00278243
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    Paper (German National Licenses)
    Fulltext
  • 3
    Book
    Book
    Global bifurcation of periodic solutions with symmetry /; 1309 (1988)
    Berlin [u.a.] :Springer,
    Details
    Title: Global bifurcation of periodic solutions with symmetry /; 1309
    Author: Fiedler, Bernold
    Publisher: Berlin [u.a.] :Springer,
    Year of publication: 1988
    Pages: VIII, 144 S. : , Ill., graph. Darst.
    Series Statement: Lecture notes in mathematics 1309
    ISBN: 3-540-19234-4 , 0-387-19234-4
    Type of Medium: Book
    Language: English
    Parallel Title: obal bifurcation of periodic solutions with symmetry
    URL: http://www.zentralblatt-math.org/zmath/en/search/?an=0644.34038
    URL: http://www.gbv.de/dms/hbz/toc/ht003121723.pdf
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    ZIB Catalog
    Zuse Institute Berlin get availability status
  • 4
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    Unknown
    Homoclinic Bifurcation at Resonant Eigen-values. (1989)
    Details
    Publication Date: 2014-02-26
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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    OPUS
  • 5
    facet.materialart.
    Unknown
    Global Decoupling of Coupled Symmetric Oscillators. (1988)
    Details
    Publication Date: 2014-02-26
    Description: We model a symmetric system of coupled oscillators as a graph with symmetry group $\gamma$. Each vertex of the graph represents an "oscillator" or a "cell" of reactants. The magnitude (concentration) of the reactants in the $ i $ th cell is represented by a vector $ x^i $. The edges represent the coupling of the cells. The cells are assumed to evolve by identical reaction-diffusion equation which depends on the sum of the reactants in the nearest neighbors. Thus the dynamics of the system is described by a nonlinear differential system \begin{flushleft} \[ \mbox {(*) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \dot{x}^i = f (x^i,\sum_{j \in N_i} x^j), \mbox { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \] \end{flushleft} where the sum ranges over the set $ N_i $ of neighbors of cell $ i $ . If $ f $ also has a symmetry (e.g., oddness), there are geometric conditions on the graph such that the nonlinear system $ (*) $ decouples globally into a product flow on certain sums of isotropy subspaces. Thus we may detect higher-dimensional tori of solutions of $ (*) $ which are not amenable to other types of analysis. We present a number of examples, such as bipartite graphs, complete graphs, the square, the octahedron, and a 6-dimensional cube.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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    OPUS
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