521 research outputs found
Effect of Cauchy noise on a network of quadratic integrate-and-fire neurons with non-Cauchy heterogeneities
We analyze the dynamics of large networks of pulse-coupled quadratic
integrate-and-fire neurons driven by Cauchy noise and non-Cauchy heterogeneous
inputs. Two types of heterogeneities defined by families of -Gaussian and
flat distributions are considered. Both families are parametrized by an integer
, so that as increases, the first family tends to a normal distribution,
and the second tends to a uniform distribution. For both families, exact
systems of mean-field equations are derived and their bifurcation analysis is
carried out. We show that noise and heterogeneity can have qualitatively
different effects on the collective dynamics of neurons.Comment: 9 pages, 5 figure
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
Properties of generalized synchronization of chaos
A review of recent ideas in the field of generalized synchronization of chaos is presented. This field is concerned with a generalization of the concept of conventional (identical) chaotic synchronization to the case of one-way coupled nonidentical chaotic systems. Generalized synchronization is taken to occur if, ignoring transients, the response system becomes uniquely determined by the current state of the driving system, i. e., all trajectories in the phase space are attracted to a complex synchronization manifold that may have a fractal structure. Different tools for detecting and analyzing the properties of this type of synchronization are discussed
Passage of Abrikosov Vortexes through a Boundary Barrier in Thin Superconducting Film
The model of Abrikosov vortices motion from the edges of thin superconducting film, when the current slightly exceeds the critical current, is considered. It is shown that the additional dynamic barrier is formed in the regions of a weak pinning. It is assumed for simplicity that the group of vortices is derived in the shape of one-dimensional chain. It is demonstrated, that the barrier hinders the penetration of such a chains in the film
An optimal gains matrix for time-delay feedback control
In this paper we propose an optimal time-delayed feedback control (TDFC) for tracking unstable periodic orbits (UPOs). It is shown that TDFC will drive a trajectory onto a periodic orbit while minimising an integral of a cost function of the error in periodicity and the control e®ort. This optimal TDFC relies on the linearisation about the delayed trajectory not the UPO itself and therefore can be implemented without a priori knowledge of a reference orbit. This optimal TDFC is applied to the problem of tracking an unstable periodic orbit in the nonlinear equations describing the circular restricted three-body problem. The results of this investigation demonstrate that TDFC could e±ciently drive a spacecraft onto a periodic orbit in the vicinity of a (UPO) halo orbit
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