1,630 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
Galerkin approximations for the optimal control of nonlinear delay differential equations
Optimal control problems of nonlinear delay differential equations (DDEs) are
considered for which we propose a general Galerkin approximation scheme built
from Koornwinder polynomials. Error estimates for the resulting
Galerkin-Koornwinder approximations to the optimal control and the value
function, are derived for a broad class of cost functionals and nonlinear DDEs.
The approach is illustrated on a delayed logistic equation set not far away
from its Hopf bifurcation point in the parameter space. In this case, we show
that low-dimensional controls for a standard quadratic cost functional can be
efficiently computed from Galerkin-Koornwinder approximations to reduce at a
nearly optimal cost the oscillation amplitude displayed by the DDE's solution.
Optimal controls computed from the Pontryagin's maximum principle (PMP) and the
Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE
systems, are shown to provide numerical solutions in good agreement. It is
finally argued that the value function computed from the corresponding reduced
HJB equation provides a good approximation of that obtained from the full HJB
equation.Comment: 29 pages. This is a sequel of the arXiv preprint arXiv:1704.0042
Self-tuning to the Hopf bifurcation in fluctuating systems
The problem of self-tuning a system to the Hopf bifurcation in the presence
of noise and periodic external forcing is discussed. We find that the response
of the system has a non-monotonic dependence on the noise-strength, and
displays an amplified response which is more pronounced for weaker signals. The
observed effect is to be distinguished from stochastic resonance. For the
feedback we have studied, the unforced self-tuned Hopf oscillator in the
presence of fluctuations exhibits sharp peaks in its spectrum. The implications
of our general results are briefly discussed in the context of sound detection
by the inner ear.Comment: 37 pages, 7 figures (8 figure files
Control of Dynamic Hopf Bifurcations
The slow passage through a Hopf bifurcation leads to the delayed appearance
of large amplitude oscillations. We construct a smooth scalar feedback control
which suppresses the delay and causes the system to follow a stable equilibrium
branch. This feature can be used to detect in time the loss of stability of an
ageing device. As a by-product, we obtain results on the slow passage through a
bifurcation with double zero eigenvalue, described by a singularly perturbed
cubic Lienard equation.Comment: 25 pages, 4 figure
Suppression of Limit Cycle Oscillations using the Nonlinear Tuned Vibration Absorber
The objective of the present study is to mitigate, or even completely
eliminate, the limit cycle oscillations in mechanical systems using a passive
nonlinear absorber, termed the nonlinear tuned vibration absorber (NLTVA). An
unconventional aspect of the NLTVA is that the mathematical form of its
restoring force is not imposed a priori, as it is the case for most existing
nonlinear absorbers. The NLTVA parameters are determined analytically using
stability and bifurcation analyses, and the resulting design is validated using
numerical continuation. The proposed developments are illustrated using a Van
der Pol-Duffing primary system
Identifying dynamical systems with bifurcations from noisy partial observation
Dynamical systems are used to model a variety of phenomena in which the
bifurcation structure is a fundamental characteristic. Here we propose a
statistical machine-learning approach to derive lowdimensional models that
automatically integrate information in noisy time-series data from partial
observations. The method is tested using artificial data generated from two
cell-cycle control system models that exhibit different bifurcations, and the
learned systems are shown to robustly inherit the bifurcation structure.Comment: 16 pages, 6 figure
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