3,601 research outputs found
Realizability of the normal form for the triple-zero nilpotency in a class of delayed nonlinear oscillators
The effects of delayed feedback terms on nonlinear oscillators has been
extensively studied, and have important applications in many areas of science
and engineering. We study a particular class of second-order delay-differential
equations near a point of triple-zero nilpotent bifurcation. Using center
manifold and normal form reduction, we show that the three-dimensional
nonlinear normal form for the triple-zero bifurcation can be fully realized at
any given order for appropriate choices of nonlinearities in the original
delay-differential equation.Comment: arXiv admin note: text overlap with arXiv:math/050539
Delay-induced multistability near a global bifurcation
We study the effect of a time-delayed feedback within a generic model for a
saddle-node bifurcation on a limit cycle. Without delay the only attractor
below this global bifurcation is a stable node. Delay renders the phase space
infinite-dimensional and creates multistability of periodic orbits and the
fixed point. Homoclinic bifurcations, period-doubling and saddle-node
bifurcations of limit cycles are found in accordance with Shilnikov's theorems.Comment: Int. J. Bif. Chaos (2007), in prin
Zero-Hopf bifurcation in the Van der Pol oscillator with delayed position and velocity feedback
In this paper, we consider the traditional Van der Pol Oscillator with a
forcing dependent on a delay in feedback. The delay is taken to be a nonlinear
function of both position and velocity which gives rise to many different types
of bifurcations. In particular, we study the Zero-Hopf bifurcation that takes
place at certain parameter values using methods of centre manifold reduction of
DDEs and normal form theory. We present numerical simulations that have been
accurately predicted by the phase portraits in the Zero-Hopf bifurcation to
confirm our numerical results and provide a physical understanding of the
oscillator with the delay in feedback
Qualitative analysis of the dynamics of the time delayed Chua's circuit
IEEE TRANS. CIRCUITS SYST.
Asymptotic methods for delay equations.
Asymptotic methods for singularly perturbed delay differential equations are in many ways more challenging to implement than for ordinary differential equations. In this paper, four examples of delayed systems which occur in practical models are considered: the delayed recruitment equation, relaxation oscillations in stem cell control, the delayed logistic equation, and density wave oscillations in boilers, the last of these being a problem of concern in engineering two-phase flows. The ways in which asymptotic methods can be used vary from the straightforward to the perverse, and illustrate the general technical difficulties that delay equations provide for the central technique of the applied mathematician. © Springer 2006
Complex oscillations in the delayed Fitzhugh-Nagumo equation
Motivated by the dynamics of neuronal responses, we analyze the dynamics of
the Fitzhugh-Nagumo slow-fast system with delayed self-coupling. This system
provides a canonical example of a canard explosion for sufficiently small
delays. Beyond this regime, delays significantly enrich the dynamics, leading
to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a
delay-induced subcritical Bogdanov-Takens instability arising at the fold
points of the S-shaped critical manifold. Underlying the transition from
canard-induced to delay-induced dynamics is an abrupt switch in the nature of
the Hopf bifurcation
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