Topic
Bifurcation diagram
About: Bifurcation diagram is a research topic. Over the lifetime, 8379 publications have been published within this topic receiving 139664 citations.
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TL;DR: In this article, the response of a damped Duffing oscillator of the softening type to a harmonic excitation is analyzed in a two-parameter space consisting of the frequency and amplitude of the excitation.
Abstract: The response of a damped Duffing oscillator of the softening type to a harmonic excitation is analyzed in a two-parameter space consisting of the frequency and amplitude of the excitation. An approximate procedure is developed for the generation of the bifurcation diagram in the parameter space of interest. It is a combination of second-order perturbation solutions of the system in the neighborhood of its non-linear resonances and Floquet analysis. The results show that the proposed scheme is capable of predicting symmetry-breaking and period-doubling bifurcations as well as Jumps to either bounded or unbounded motions. The results obtained are validated using analogand digital-computer simulations, which show chaos and unbounded motions, among other behaviors.
110 citations
Book•
05 Feb 1997
TL;DR: In this article, the Equivariant Degree of Dold-Ulrich is used for global Hopf Bifurcation theory and Nussbaum-Sadovskii degree for Condensing Fields.
Abstract: Elements of Differential Topology. Degree in Finite--Dimensional Spaces. Leray--Schauder Degree for Compact Fields. Nussbaum--Sadovskii Degree for Condensing Fields. Applications to Bifurcation Theory. S 1 --Equivariant Degree. Global Hopf Bifurcation Theory. Equivariant Degree of Dold--Ulrich. References. Index.
110 citations
TL;DR: In this article, the authors considered a follow-the-leader traffic model, where each car driver chooses his acceleration according to a certain law and the model is represented by a nonlinear system of ODEs.
109 citations
Book•
17 Jul 1998
TL;DR: In this paper, Liapunov-Schmidt reduction and Hopf Bifurcation theory are used to construct Chaotic Regions and Nonlinear Structural Dynamics (NSD) models.
Abstract: 1. Dynamic Systems, Ordinary Differential Equations and Stability of Motion 2. Calculation of Flows 3. Discrete Dynamic Systems 4. Liapunov-Schmidt Reduction 5. Center Manifold Theorem and Normal Form 6. Hopf Bifurcation 7. Averaging Method in Bifurcation Theory 8. Introduction to Chaos 9. Construction of Chaotic Regions 10. Numerical Methods 11. Nonlinear Structural Dynamics.
108 citations
TL;DR: In this paper, the authors investigated the relaxation of a dewetting contact line in the so-called Landau-Levich geometry, in which a vertical solid plate is withdrawn from a bath of partially wetting liquid.
Abstract: The relaxation of a dewetting contact line is investigated theoretically in the so-called ‘Landau–Levich’ geometry in which a vertical solid plate is withdrawn from a bath of partially wetting liquid. The study is performed in the framework of lubrication theory, in which the hydrodynamics is resolved at all length scales (from molecular to macroscopic). We investigate the bifurcation diagram for unperturbed contact lines, which turns out to be more complex than expected from simplified ‘quasi-static’ theories based upon an apparent contact angle. Linear stability analysis reveals that below the critical capillary number of entrainment, Cac, the contact line is linearly stable at all wavenumbers. Away from the critical point, the dispersion relation has an asymptotic behaviour σ∝|q| and compares well to a quasi-static approach. Approaching Cac, however, a different mechanism takes over and the dispersion evolves from ∼|q| to the more common ∼q2. These findings imply that contact lines cannot be described using a universal relation between speed and apparent contact angle, but viscous effects have to be treated explicitly.
108 citations