1,700 research outputs found
Time-delayed feedback control of unstable periodic orbits near a subcritical Hopf bifurcation
We show that Pyragas delayed feedback control can stabilize an unstable
periodic orbit (UPO) that arises from a generic subcritical Hopf bifurcation of
a stable equilibrium in an n-dimensional dynamical system. This extends results
of Fiedler et al. [PRL 98, 114101 (2007)], who demonstrated that such feedback
control can stabilize the UPO associated with a two-dimensional subcritical
Hopf normal form. Pyragas feedback requires an appropriate choice of a feedback
gain matrix for stabilization, as well as knowledge of the period of the
targeted UPO. We apply feedback in the directions tangent to the
two-dimensional center manifold. We parameterize the feedback gain by a modulus
and a phase angle, and give explicit formulae for choosing these two parameters
given the period of the UPO in a neighborhood of the bifurcation point. We
show, first heuristically, and then rigorously by a center manifold reduction
for delay differential equations, that the stabilization mechanism involves a
highly degenerate Hopf bifurcation problem that is induced by the time-delayed
feedback. When the feedback gain modulus reaches a threshold for stabilization,
both of the genericity assumptions associated with a two-dimensional Hopf
bifurcation are violated: the eigenvalues of the linearized problem do not
cross the imaginary axis as the bifurcation parameter is varied, and the real
part of the cubic coefficient of the normal form vanishes. Our analysis of this
degenerate bifurcation problem reveals two qualitatively distinct cases when
unfolded in a two-parameter plane. In each case, Pyragas-type feedback
successfully stabilizes the branch of small-amplitude UPOs in a neighborhood of
the original bifurcation point, provided that the phase angle satisfies a
certain restriction.Comment: 35 pages, 19 figure
Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control
For many years it was believed that an unstable periodic orbit with an odd
number of real Floquet multipliers greater than unity cannot be stabilized by
the time-delayed feedback control mechanism of Pyragus. A recent paper by
Fiedler et al uses the normal form of a subcritical Hopf bifurcation to give a
counterexample to this theorem. Using the Lorenz equations as an example, we
demonstrate that the stabilization mechanism identified by Fiedler et al for
the Hopf normal form can also apply to unstable periodic orbits created by
subcritical Hopf bifurcations in higher-dimensional dynamical systems. Our
analysis focuses on a particular codimension-two bifurcation that captures the
stabilization mechanism in the Hopf normal form example, and we show that the
same codimension-two bifurcation is present in the Lorenz equations with
appropriately chosen Pyragus-type time-delayed feedback. This example suggests
a possible strategy for choosing the feedback gain matrix in Pyragus control of
unstable periodic orbits that arise from a subcritical Hopf bifurcation of a
stable equilibrium. In particular, our choice of feedback gain matrix is
informed by the Fiedler et al example, and it works over a broad range of
parameters, despite the fact that a center-manifold reduction of the
higher-dimensional problem does not lead to their model problem.Comment: 21 pages, 8 figures, to appear in PR
Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations
In 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 tool to study the dynamics near
equilibria of such DDEs. The effectiveness is demonstrated on various models
Noise, Bifurcations, and Modeling of Interacting Particle Systems
We consider the stochastic patterns of a system of communicating, or coupled,
self-propelled particles in the presence of noise and communication time delay.
For sufficiently large environmental noise, there exists a transition between a
translating state and a rotating state with stationary center of mass. Time
delayed communication creates a bifurcation pattern dependent on the coupling
amplitude between particles. Using a mean field model in the large number
limit, we show how the complete bifurcation unfolds in the presence of
communication delay and coupling amplitude. Relative to the center of mass, the
patterns can then be described as transitions between translation, rotation
about a stationary point, or a rotating swarm, where the center of mass
undergoes a Hopf bifurcation from steady state to a limit cycle. Examples of
some of the stochastic patterns will be given for large numbers of particles
Multi-stabilities and symmetry-broken one-colour and two-colour states in closely coupled single-mode lasers
We theoretically investigate the dynamics of two mutually coupled identical
single-mode semi-conductor lasers. For small separation and large coupling
between the lasers, symmetry-broken one-colour states are shown to be stable.
In this case the light output of the lasers have significantly different
intensities while at the same time the lasers are locked to a single common
frequency. For intermediate coupling we observe stable symmetry-broken
two-colour states, where both lasers lase simultaneously at two optical
frequencies which are separated by up to 150~GHz. Using a five dimensional
model we identify the bifurcation structure which is responsible for the
appearance of symmetric and symmetry-broken one-colour and two-colour states.
Several of these states give rise to multi-stabilities and therefore allow for
the design of all-optical memory elements on the basis of two coupled
single-mode lasers. The switching performance of selected designs of optical
memory elements is studied numerically.Comment: 12 pages, 15 figure
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