1,176 research outputs found
A Study of the Bogdanov-Takens Bifurcation
A two paraIlleter versal tmfolding for generic nilpotent singular point was studied independently by Takens and Bogdanov and so one now calls it : the Bogdanov-Takens bifurcation. Historically, it was the last codiInension 2 singularity to be treated
Chaos in the Takens-Bogdanov bifurcation with O(2) symmetry
The Takens–Bogdanov bifurcation is a codimension two bifurcation that provides a key to the presence of complex dynamics in many systems of physical interest. When the system is translation invariant in one spatial dimension with no left-right preference the imposition of periodic boundary conditions leads to the Takens–Bogdanov bifurcation with O(2) symmetry. This bifurcation, analyzed by G. Dangelmayr and E. Knobloch, Phil. Trans. R. Soc. London A 322, 243 (1987), describes the interaction between steady states and traveling and standing waves in the nonlinear regime and predicts the presence of modulated traveling waves as well. The analysis reveals the presence of several global bifurcations near which the averaging method (used in the original analysis) fails. We show here, using a combination of numerical continuation and the construction of appropriate return maps, that near the global bifurcation that terminates the branch of modulated traveling waves, the normal form for the Takens–Bogdanov bifurcation admits cascades of period-doubling bifurcations as well as chaotic dynamics of Shil’nikov type. Thus chaos is present arbitrarily close to the codimension two point
Global bifurcations in the Takens-Bogdanov normal form with D_4 symmetry near the O(2) limit
The dynamics of the normal form of the Takens-Bogdanov bifurcation with D_4 symmetry is governed by a one-dimensional map near the gluing bifurcation and near the O(2) integrable limit, rather than the three-dimensional map one would expect. This great simplification allows a quantitative description of the bifurcation sequence through which stability is transfered between invariant subspaces
On the Takens-Bogdanov Bifurcation in the Chua’s Equation
The analysis of the Takens-Bogdanov bifurcation
of the equilibrium at the origin in the Chua’s equation with
a cubic nonlinearity is carried out. The local analysis provides, in
first approximation, different bifurcation sets, where the presence
of several dynamical behaviours (including periodic, homoclinic
and heteroclinic orbits) is predicted. The local results are used
as a guide to apply the adequate numerical methods to obtain
a global understanding of the bifurcation sets. The study of
the normal form of the Takens-Bogdanov bifurcation shows the
presence of a degenerate (codimension-three) situation, which is
analyzed in both homoclinic and heteroclinic cases
A proof of Perko's conjectures for the Bogdanov-Takens system
The Bogdanov-Takens system has at most one limit cycle and, in the parameter
space, it exists between a Hopf and a saddle-loop bifurcation curves. The aim
of this paper is to prove the Perko's conjectures about some analytic
properties of the saddle-loop bifurcation curve. Moreover, we provide sharp
piecewise algebraic upper and lower bounds for this curve
Takens-Bogdanov bifurcation of travelling wave solutions in pipe flow
The appearance of travelling-wave-type solutions in pipe Poiseuille flow that
are disconnected from the basic parabolic profile is numerically studied in
detail. We focus on solutions in the 2-fold azimuthally-periodic subspace
because of their special stability properties, but relate our findings to other
solutions as well. Using time-stepping, an adapted Krylov-Newton method and
Arnoldi iteration for the computation and stability analysis of relative
equilibria, and a robust pseudo-arclength continuation scheme we unfold a
double-zero (Takens-Bogdanov) bifurcating scenario as a function of Reynolds
number (Re) and wavenumber (k). This scenario is extended, by the inclusion of
higher order terms in the normal form, to account for the appearance of
supercritical modulated waves emanating from the upper branch of solutions at a
degenerate Hopf bifurcation. These waves are expected to disappear in
saddle-loop bifurcations upon collision with lower-branch solutions, thereby
leaving stable upper-branch solutions whose subsequent secondary bifurcations
could contribute to the formation of the phase space structures that are
required for turbulent dynamics at higher Re.Comment: 26 pages, 15 figures (pdf and png). Submitted to J. Fluid Mec
Bogdanov-Takens resonance in time-delayed systems
We analyze the oscillatory dynamics of a time-delayed dynamical system
subjected to a periodic external forcing. We show that, for certain values of
the delay, the response can be greatly enhanced by a very small forcing
amplitude. This phenomenon is related to the presence of a Bogdanov- Takens
bifurcation and displays some analogies to other resonance phenomena, but also
substantial differences.Comment: 14 pages, 8 figure
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