1,721,063 research outputs found
Periodic Center Manifolds for Nonhyperbolic Limit Cycles in ODEs
Get PDFIn this paper, we deal with a classical object, namely, a nonhyperbolic limit
cycle in a system of smooth autonomous ordinary differential equations. While
the existence of a center manifold near such a cycle was assumed in several
studies on cycle bifurcations based on periodic normal forms, no proofs were
available in the literature until recently. The main goal of this paper is to
give an elementary proof of the existence of a periodic smooth locally
invariant center manifold near a nonhyperbolic cycle in finite-dimensional
ordinary differential equations by using the Lyapunov-Perron method. In
addition, we provide several explicit examples of analytic vector fields
admitting (non)-unique, (non)--smooth and (non)-analytic periodic
center manifolds.Comment: 35 pages, 4 figure
Interplay between Normal Forms and Center Manifold Reduction for Homoclinic Predictors near Bogdanov-Takens Bifurcation
No full textThis paper provides for the first time correct third-order homoclinic predictors in n-dimensional ODEs near a generic Bogdanov-Takens bifurcation point, which can be used to start the numerical continuation of the appearing homoclinic orbits. To achieve this, higher-order time approximations to the nonlinear time transformation in the Lindstedt-Poincar\'e method are essential. Moreover, a correct transform between approximations to solutions in the normal form and approximations to solutions on the parameter-dependent center manifold is derived rigorously. A detailed comparison is done between applying different normal forms (smooth and orbital), different phase conditions, and different perturbation methods (regular and Lindstedt-Poincar\'e) to approximate the homoclinic solution near Bogdanov-Takens points. Examples demonstrating the correctness of the predictors are given. The new homoclinic predictors are implemented in the open-source MATLAB/GNU Octave continuation package MatCont.</p
Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations
No full textIn this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual
semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative.
The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more
generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful toolto study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models
Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations
No full textIn this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual
semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative.
The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more
generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful toolto study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Periodic Normal Forms for Bifurcations of Limit Cycles in DDEs
Get PDFA recent work by the authors on the existence of a periodic smooth
finite-dimensional center manifold near a nonhyperbolic cycle in delay
differential equations motivates the derivation of periodic normal forms. In
this paper, we prove the existence of a special coordinate system on the center
manifold that will allow us to describe the local dynamics on the center
manifold near the cycle in terms of these periodic normal forms. To construct
the linear part of this coordinate system, we prove the existence of time
periodic smooth Jordan chains for the original and adjoint system. Moreover, we
establish duality and spectral relations between both systems by using tools
from the theory of delay equations and Volterra integral equations, dual
perturbation theory, duality theory and evolution semigroups.Comment: 53 pages, 1 figure. arXiv admin note: text overlap with
arXiv:2207.0248
Periodic Center Manifolds for DDEs in the Light of Suns and Stars
No full textIn this paper, we prove the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic cycle in classical delay differential equations by using the Lyapunov-Perron method. The results are based on the rigorous functional analytic perturbation framework for dual semigroups (sun-star calculus). The generality of the dual perturbation framework ensures that the results extend to a much broader class of evolution equations
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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