35,674 research outputs found
Local Analysis of Dissipative Dynamical Systems
Linear transformation techniques such as singular value decomposition (SVD)
have been used widely to gain insight into the qualitative dynamics of data
generated by dynamical systems. There have been several reports in the past
that had pointed out the susceptibility of linear transformation approaches in
the presence of nonlinear correlations. In this tutorial review, local
dispersion along with the surrogate testing is proposed to discriminate
nonlinear correlations arising in deterministic and non-deterministic settings.Comment: 85 Pages, 13 Figure
Nonintegrability, Chaos, and Complexity
Two-dimensional driven dissipative flows are generally integrable via a
conservation law that is singular at equilibria. Nonintegrable dynamical
systems are confined to n*3 dimensions. Even driven-dissipative deterministic
dynamical systems that are critical, chaotic or complex have n-1 local
time-independent conservation laws that can be used to simplify the geometric
picture of the flow over as many consecutive time intervals as one likes. Those
conserevation laws generally have either branch cuts, phase singularities, or
both. The consequence of the existence of singular conservation laws for
experimental data analysis, and also for the search for scale-invariant
critical states via uncontrolled approximations in deterministic dynamical
systems, is discussed. Finally, the expectation of ubiquity of scaling laws and
universality classes in dynamics is contrasted with the possibility that the
most interesting dynamics in nature may be nonscaling, nonuniversal, and to
some degree computationally complex
Non-stationarity and Dissipative Time Crystals: Spectral Properties and Finite-Size Effects
We discuss the emergence of non-stationarity in open quantum many-body
systems. This leads us to the definition of dissipative time crystals which
display experimentally observable, persistent, time-periodic oscillations
induced by noisy contact with an environment. We use the Loschmidt echo and
local observables to indicate the presence of a finite sized dissipative time
crystal. Starting from the closed Hubbard model we then provide examples of
dissipation mechanisms that yield experimentally observable quantum periodic
dynamics and allow analysis of the emergence of finite sized dissipative time
crystals. For a disordered Hubbard model including two-particle loss and gain
we find a dark Hamiltonian driving oscillations between GHZ states in the
long-time limit. Finally, we discuss how the presented examples could be
experimentally realized.Comment: 31 pages, 5 figures. Submitted to NJP: Focus on Time Crystal
Time Quasilattices in Dissipative Dynamical Systems
We establish the existence of `time quasilattices' as stable trajectories in
dissipative dynamical systems. These tilings of the time axis, with two unit
cells of different durations, can be generated as cuts through a periodic
lattice spanned by two orthogonal directions of time. We show that there are
precisely two admissible time quasilattices, which we term the infinite Pell
and Clapeyron words, reached by a generalization of the period-doubling
cascade. Finite Pell and Clapeyron words of increasing length provide
systematic periodic approximations to time quasilattices which can be verified
experimentally. The results apply to all systems featuring the universal
sequence of periodic windows. We provide examples of discrete-time maps, and
periodically-driven continuous-time dynamical systems. We identify quantum
many-body systems in which time quasilattices develop rigidity via the
interaction of many degrees of freedom, thus constituting dissipative discrete
`time quasicrystals'.Comment: 38 pages, 14 figures. This version incorporates "Pell and Clapeyron
Words as Stable Trajectories in Dynamical Systems", arXiv:1707.09333.
Submission to SciPos
Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects
We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation
in phase space. We demonstrate that it accommodates the phase space
dynamics of low dimensional dissipative systems such as the much studied Lorenz
and R\"{o}ssler Strange attractors, as well as the more recent constructions of
Chen and Leipnik-Newton. The rotational, volume preserving part of the flow
preserves in time a family of two intersecting surfaces, the so called {\em
Nambu Hamiltonians}. They foliate the entire phase space and are, in turn,
deformed in time by Dissipation which represents their irrotational part of the
flow. It is given by the gradient of a scalar function and is responsible for
the emergence of the Strange Attractors.
Based on our recent work on Quantum Nambu Mechanics, we provide an explicit
quantization of the Lorenz attractor through the introduction of
Non-commutative phase space coordinates as Hermitian matrices in
. They satisfy the commutation relations induced by one of the two
Nambu Hamiltonians, the second one generating a unique time evolution.
Dissipation is incorporated quantum mechanically in a self-consistent way
having the correct classical limit without the introduction of external degrees
of freedom. Due to its volume phase space contraction it violates the quantum
commutation relations. We demonstrate that the Heisenberg-Nambu evolution
equations for the Quantum Lorenz system give rise to an attracting ellipsoid in
the dimensional phase space.Comment: 35 pages, 4 figures, LaTe
Mitigation of dynamical instabilities in laser arrays via non-Hermitian coupling
Arrays of coupled semiconductor lasers are systems possessing complex
dynamical behavior that are of major interest in photonics and laser science.
Dynamical instabilities, arising from supermode competition and slow carrier
dynamics, are known to prevent stable phase locking in a wide range of
parameter space, requiring special methods to realize stable laser operation.
Inspired by recent concepts of parity-time () and non-Hermitian
photonics, in this work we consider non-Hermitian coupling engineering in laser
arrays in a ring geometry and show, both analytically and numerically, that
non-Hermitian coupling can help to mitigate the onset of dynamical laser
instabilities. In particular, we consider in details two kinds of
nearest-neighbor non-Hermitian couplings: symmetric but complex mode coupling
(type-I non-Hermitian coupling) and asymmetric mode coupling (type-II
non-Hermitian coupling). Suppression of dynamical instabilities can be realized
in both coupling schemes, resulting in stable phase-locking laser emission with
the lasers emitting in phase (for type-I coupling) or with phase
gradient (for type-II coupling), resulting in a vortex far-field beam. In
type-II non-Hermitian coupling, chirality induced by asymmetric mode coupling
enables laser phase locking even in presence of moderate disorder in the
resonance frequencies of the lasers.Comment: revised version, changed title, added one figure and some reference
Synchronization of coupled stochastic limit cycle oscillators
For a class of coupled limit cycle oscillators, we give a condition on a
linear coupling operator that is necessary and sufficient for exponential
stability of the synchronous solution. We show that with certain modifications
our method of analysis applies to networks with partial, time-dependent, and
nonlinear coupling schemes, as well as to ensembles of local systems with
nonperiodic attractors. We also study robustness of synchrony to noise. To this
end, we analytically estimate the degree of coherence of the network
oscillations in the presence of noise. Our estimate of coherence highlights the
main ingredients of stochastic stability of the synchronous regime. In
particular, it quantifies the contribution of the network topology. The
estimate of coherence for the randomly perturbed network can be used as means
for analytic inference of degree of stability of the synchronous solution of
the unperturbed deterministic network. Furthermore, we show that in large
networks, the effects of noise on the dynamics of each oscillator can be
effectively controlled by varying the strength of coupling, which provides a
powerful mechanism of denoising. This suggests that the organization of
oscillators in a coupled network may play an important role in maintaining
robust oscillations in random environment. The analysis is complemented with
the results of numerical simulations of a neuronal network.
PACS: 05.45.Xt, 05.40.Ca
Keywords: synchronization, coupled oscillators, denoising, robustness to
noise, compartmental modelComment: major revisions; two new section
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