611 research outputs found
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
Dissipative Kerr solitons in optical microresonators
This chapter describes the discovery and stable generation of temporal
dissipative Kerr solitons in continuous-wave (CW) laser driven optical
microresonators. The experimental signatures as well as the temporal and
spectral characteristics of this class of bright solitons are discussed.
Moreover, analytical and numerical descriptions are presented that do not only
reproduce qualitative features but can also be used to accurately model and
predict the characteristics of experimental systems. Particular emphasis lies
on temporal dissipative Kerr solitons with regard to optical frequency comb
generation where they are of particular importance. Here, one example is
spectral broadening and self-referencing enabled by the ultra-short pulsed
nature of the solitons. Another example is dissipative Kerr soliton formation
in integrated on-chip microresonators where the emission of a dispersive wave
allows for the direct generation of unprecedentedly broadband and coherent
soliton spectra with smooth spectral envelope.Comment: To appear in "Nonlinear optical cavity dynamics", ed. Ph. Grel
Emergence and combinatorial accumulation of jittering regimes in spiking oscillators with delayed feedback
Interaction via pulses is common in many natural systems, especially
neuronal. In this article we study one of the simplest possible systems with
pulse interaction: a phase oscillator with delayed pulsatile feedback. When the
oscillator reaches a specific state, it emits a pulse, which returns after
propagating through a delay line. The impact of an incoming pulse is described
by the oscillator's phase reset curve (PRC). In such a system we discover an
unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic
regular spiking solution bifurcates with several multipliers crossing the unit
circle at the same parameter value. The number of such critical multipliers
increases linearly with the delay and thus may be arbitrary large. This
bifurcation is accompanied by the emergence of numerous "jittering" regimes
with non-equal interspike intervals (ISIs). Each of these regimes corresponds
to a periodic solution of the system with a period roughly proportional to the
delay. The number of different "jittering" solutions emerging at the
bifurcation point increases exponentially with the delay. We describe the
combinatorial mechanism that underlies the emergence of such a variety of
solutions. In particular, we show how a periodic solution exhibiting several
distinct ISIs can imply the existence of multiple other solutions obtained by
rearranging of these ISIs. We show that the theoretical results for phase
oscillators accurately predict the behavior of an experimentally implemented
electronic oscillator with pulsatile feedback
Phaselocked patterns and amplitude death in a ring of delay coupled limit cycle oscillators
We study the existence and stability of phaselocked patterns and amplitude
death states in a closed chain of delay coupled identical limit cycle
oscillators that are near a supercritical Hopf bifurcation. The coupling is
limited to nearest neighbors and is linear. We analyze a model set of discrete
dynamical equations using the method of plane waves. The resultant dispersion
relation, which is valid for any arbitrary number of oscillators, displays
important differences from similar relations obtained from continuum models. We
discuss the general characteristics of the equilibrium states including their
dependencies on various system parameters. We next carry out a detailed linear
stability investigation of these states in order to delineate their actual
existence regions and to determine their parametric dependence on time delay.
Time delay is found to expand the range of possible phaselocked patterns and to
contribute favorably toward their stability. The amplitude death state is
studied in the parameter space of time delay and coupling strength. It is shown
that death island regions can exist for any number of oscillators N in the
presence of finite time delay. A particularly interesting result is that the
size of an island is independent of N when N is even but is a decreasing
function of N when N is odd.Comment: 23 pages, 12 figures (3 of the figures in PNG format, separately from
TeX); minor additions; typos correcte
A coherent Ising machine for 2000-node optimization problems
The analysis and optimization of complex systems can be reduced to mathematical problems collectively known as combinatorial optimization. Many such problems can be mapped onto ground-state search problems of the Ising model, and various artificial spin systems are now emerging as promising approaches. However, physical Ising machines have suffered from limited numbers of spin-spin couplings because of implementations based on localized spins, resulting in severe scalability problems. We report a 2000-spin network with all-to-all spin-spin couplings. Using a measurement and feedback scheme, we coupled time-multiplexed degenerate optical parametric oscillators to implement maximum cut problems on arbitrary graph topologies with up to 2000 nodes. Our coherent Ising machine outperformed simulated annealing in terms of accuracy and computation time for a 2000-node complete graph
Evaluating spintronics-compatible implementations of Ising machines
The commercial and industrial demand for the solution of hard combinatorial
optimization problems push forward the development of efficient solvers. One of
them is the Ising machine which can solve combinatorial problems mapped to
Ising Hamiltonians. In particular, spintronic hardware implementations of Ising
machines can be very efficient in terms of area and performance, and are
relatively low-cost considering the potential to create hybrid CMOS-spintronic
technology. Here, we perform a comparison of coherent and probabilistic
paradigms of Ising machines on several hard Max-Cut instances, analyzing their
scalability and performance at software level. We show that probabilistic Ising
machines outperform coherent Ising machines in terms of the number of
iterations required to achieve the problem s solution. Nevertheless, high
frequency spintronic oscillators with sub-nanosecond synchronization times
could be very promising as ultrafast Ising machines. In addition, considering
that a coherent Ising machine acts better for Max-Cut problems because of the
absence of the linear term in the Ising Hamiltonian, we introduce a procedure
to encode Max-3SAT to Max-Cut. We foresee potential synergic interplays between
the two paradigms.Comment: 26 pages, 6 Figures, submitted for publication in Phys. Rev. Applied
(it will be presented at intermag 2023 in Japan
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