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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Zeitschrift für angewandte Mathematik und Physik 43 (1992), S. 292-318 
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We show that in conservative systems each non-degenerate homoclinic orbit asymptotic to a hyperbolic equilibrium possesses an associated family of periodic orbits. The family is parametrized by the period, and the periodic orbits accumulate on the homoclinic orbit as the period tends to infinity. A similar result holds for symmetric homoclinic orbits in reversible systems. Our results extend earlier work by Devaney and Henrard, and provide a positive answer to a conjecture of Strömgren. We present a unified approach to both the conservative and the reversible case, based on a technique introduced recently by X.-B. Lin.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Archive for rational mechanics and analysis 107 (1989), S. 325-345 
    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
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Archive for rational mechanics and analysis 145 (1998), S. 129-159 
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. The equivariant dynamics near relative equilibria to actions of noncompact, finite‐dimensional Lie groups G can be described by a skew‐product flow on a center manifold: $\dot{g} = g{\textbf a}(v), \dot{v} = \varphi (v)$ with $g\in G$ , with v in a slice transverse to the group action, and a(v) in the Lie algebra of G. We present a normal form theory near relative equilibria $\varphi(v$=$0)=0,$ in this general case. For the specific case of the Euclidean groups $SE(N),$ the skew product takes the form $$\dot{R} = R {\textbf r}(v),\qquad \dot{S} = R {\textbf s}(v),\qquad \dot{v} = \varphi (v)$$ with ${\textbf r}(v)\in SO(N),\;{\textbf s}(v)\in\mathr^N$ . We give a precise meaning to the intuitive idea of tip motion of a meandering spiral: it corresponds to the dynamics of $S(t)$ . This clarifies the notion of meander radii and drift resonance in the plane $N=2$ . For illustration, we discuss the unbounded tip motions associated with a weak focus in v, on the verge of Hopf bifurcation, in the case of resonant Hopf and rotation frequencies of the spiral, and study resonant relative Hopf bifurcation. We also encounter random Brownian tip motions for trajectories $v(t)\rightarrow \Gamma,$ which become homoclinic for $t\rightarrow +\infty$ . We conclude with some comments on the homoclinic tip shifts and drift resonance velocities in the Bogdanov‐Takens bifurcation, which turn out to be small beyond any finite order.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Archive for rational mechanics and analysis 119 (1992), S. 145-196 
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Archive for rational mechanics and analysis 94 (1986), S. 59-81 
    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
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Journal of dynamics and differential equations 2 (1990), S. 177-244 
    ISSN: 1572-9222
    Keywords: homoclinic orbit ; period doubling ; pathfollowing ; global bifurcation ; resonance ; 34C15 ; 34C35 ; 58F14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider a bifurcation of homoclinic orbits, which is an analogue of period doubling in the limit of infinite period. This bifurcation can occur in generic two parameter vector fields when a homoclinic orbit is attached to a stationary point with resonant eigenvalues. The resonance condition requires the eigenvalues with positive/negative real part closest to zero to be real, simple, and equidistant to zero. Under an additional global twist condition, an exponentially flat bifurcation of double homoclinic orbits from the primary homoclinic branch is established rigorously. Moreover, associated period doublings of periodic orbits with almost infinite period are detected. If the global twist condition is violated, a resonant side switching occurs. This corresponds to an exponentially flat bifurcation of periodic saddle-node orbits from the homoclinic branch.
    Type of Medium: Electronic Resource
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  • 7
    ISSN: 1573-269X
    Keywords: homoclinic bifurcations criteria ; elliptic Lindstedt–Poincaré method ; Melnikov function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A criterion to predict bifurcation of homoclinic orbits instrongly nonlinear self-excited one-degree-of-freedom oscillator $$\ddot x + c_1 x + c_2 f(x) = \varepsilon g(\mu ,x,\dot x),$$ is presented. TheLindstedt–Poincaré perturbation method is combined formally withthe Jacobian elliptic functions to determine an approximation of thelimit cycles near homoclinicity. We then apply a criterion forpredicting homoclinic orbits, based on the collision of the bifurcatinglimit cycle with the saddle equilibrium. In particular we show that thiscriterion leads to the same results, formally and to leading order, asthe standard Melnikov technique. Explicit applications of this criterionto quadratic or cubic nonlinearities f(x) are included.
    Type of Medium: Electronic Resource
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  • 8
    Book
    Book
    Berlin :Konrad-Zuse-Zentrum für Informationstechnik,
    Title: Global Decoupling of Coupled Symmetric Oscillators. /; Preprint SC 88-03
    Author: Alexander, J. C.
    Contributer: Fiedler, Bernold
    Publisher: Berlin :Konrad-Zuse-Zentrum für Informationstechnik,
    Year of publication: 1988
    Series Statement: Preprint / Konrad-Zuse-Zentrum für Informationstechnik Berlin Preprint SC 88-03
    ISSN: 0933-7911
    Type of Medium: Book
    Language: English
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  • 9
    Book
    Book
    Berlin :Konrad-Zuse-Zentrum für Informationstechnik,
    Title: Discretization of Homoclinic Orbits, Rapid Forcing and "Invisible" Chaos. /; Preprint SC 91-05
    Author: Fiedler, Bernold
    Contributer: Scheurle, J.
    Publisher: Berlin :Konrad-Zuse-Zentrum für Informationstechnik,
    Year of publication: 1991
    Pages: 64 S.
    Series Statement: Preprint / Konrad-Zuse-Zentrum für Informationstechnik Berlin Preprint SC 91-05
    ISSN: 0933-7911
    Type of Medium: Book
    Language: English
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  • 10
    Title: Ergodic theory, analysis and efficient simulation of dynamical systems /
    Contributer: Fiedler, Bernold
    Publisher: Berlin [u.a.] :Springer,
    Year of publication: 2001
    Pages: XI, 820 S. : , Ill., graph. Darst. ; , 25 cm
    ISBN: 3-540-41290-5
    Type of Medium: Book
    Language: English
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