1,545 research outputs found
Dynamics of a Limit Cycle Oscillator under Time Delayed Linear and Nonlinear Feedbacks
We study the effects of time delayed linear and nonlinear feedbacks on the
dynamics of a single Hopf bifurcation oscillator. Our numerical and analytic
investigations reveal a host of complex temporal phenomena such as phase slips,
frequency suppression, multiple periodic states and chaos. Such phenomena are
frequently observed in the collective behavior of a large number of coupled
limit cycle oscillators. Our time delayed feedback model offers a simple
paradigm for obtaining and investigating these temporal states in a single
oscillator.We construct a detailed bifurcation diagram of the oscillator as a
function of the time delay parameter and the driving strengths of the feedback
terms. We find some new states in the presence of the quadratic nonlinear
feedback term with interesting characteristics like birhythmicity, phase
reversals, radial trapping, phase jumps and spiraling patterns in the amplitude
space. Our results may find useful applications in physical, chemical or
biological systems.Comment: VERSION 4: Fig. 10(d) added, an uncited reference removed; (To appear
in Physica D) (17 pages, 21 figures, two column, aps RevTeX); VERSION 3:
Revised. In Section 2, small tau approximation added; Section 3 is divided
into subsections; periodic solution discussed in detail; Figs. 7 and 11
discarded; Figs. 12 and 14 altered; three new figures (now Figs. 10, 11 and
21) added. VERSION 2: Figs. 1 and 2 replace
Dynamics of neural systems with discrete and distributed time delays
In real-world systems, interactions between elements do not happen instantaneously, due to the time
required for a signal to propagate, reaction times of individual elements, and so forth. Moreover,
time delays are normally nonconstant and may vary with time. This means that it is vital to introduce
time delays in any realistic model of neural networks. In order to analyze the fundamental
properties of neural networks with time-delayed connections, we consider a system of two coupled
two-dimensional nonlinear delay differential equations. This model represents a neural network,
where one subsystem receives a delayed input from another subsystem. An exciting feature of the
model under consideration is the combination of both discrete and distributed delays, where distributed
time delays represent the neural feedback between the two subsystems, and the discrete
delays describe the neural interaction within each of the two subsystems. Stability properties are
investigated for different commonly used distribution kernels, and the results are compared to the
corresponding results on stability for networks with no distributed delays. It is shown how approximations
of the boundary of the stability region of a trivial equilibrium can be obtained analytically
for the cases of delta, uniform, and weak gamma delay distributions. Numerical techniques are used
to investigate stability properties of the fully nonlinear system, and they fully confirm all analytical
findings
Time Delay Effects on Coupled Limit Cycle Oscillators at Hopf Bifurcation
We present a detailed study of the effect of time delay on the collective
dynamics of coupled limit cycle oscillators at Hopf bifurcation. For a simple
model consisting of just two oscillators with a time delayed coupling, the
bifurcation diagram obtained by numerical and analytical solutions shows
significant changes in the stability boundaries of the amplitude death, phase
locked and incoherent regions. A novel result is the occurrence of amplitude
death even in the absence of a frequency mismatch between the two oscillators.
Similar results are obtained for an array of N oscillators with a delayed mean
field coupling and the regions of such amplitude death in the parameter space
of the coupling strength and time delay are quantified. Some general analytic
results for the N tending to infinity (thermodynamic) limit are also obtained
and the implications of the time delay effects for physical applications are
discussed.Comment: 20 aps formatted revtex pages (including 13 PS figures); Minor
changes over the previous version; To be published in Physica
Time-delayed models of gene regulatory networks
We discuss different mathematical models of gene regulatory networks as relevant to the onset and development of cancer. After discussion of alternativemodelling approaches, we use a paradigmatic two-gene network to focus on the role played by time delays in the dynamics of gene regulatory networks. We contrast the dynamics of the reduced model arising in the limit of fast mRNA dynamics with that of the full model. The review concludes with the discussion of some open problems
Mammalian Brain As a Network of Networks
Acknowledgements AZ, SG and AL acknowledge support from the Russian Science Foundation (16-12-00077). Authors thank T. Kuznetsova for Fig. 6.Peer reviewedPublisher PD
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