scholarly journals Stability of large complex systems with heterogeneous relaxation dynamics

2021 ◽  
Vol 2021 (12) ◽  
pp. 123301
Author(s):  
Pierre Mergny ◽  
Satya N Majumdar
Keyword(s):  
Large Deviation ◽  
Interaction Strength ◽  
Coulomb Gas ◽  
Initial Position ◽  
Decay Rates ◽  
Stable Phase ◽  
Relaxation Dynamics ◽  
Large Complex ◽  
Unstable Phase ◽  
Large Complex System

Abstract We study the probability of stability of a large complex system of size N within the framework of a generalized May model, which assumes a linear dynamics of each population size n i (with respect to its equilibrium value): d n i d t = − a i n i − T ∑ j J i j n j . The a i > 0’s are the intrinsic decay rates, J ij is a real symmetric (N × N) Gaussian random matrix and T measures the strength of pairwise interaction between different species. Our goal is to study how inhomogeneities in the intrinsic damping rates a i affect the stability of this dynamical system. As the interaction strength T increases, the system undergoes a phase transition from a stable phase to an unstable phase at a critical value T = T c. We reinterpret the probability of stability in terms of the hitting time of the level b = 0 of an associated Dyson Brownian motion (DBM), starting at the initial position a i and evolving in ‘time’ T. In the large N → ∞ limit, using this DBM picture, we are able to completely characterize T c for arbitrary density μ(a) of the a i ’s. For a specific flat configuration a i = 1 + σ i − 1 N , we obtain an explicit parametric solution for the limiting (as N → ∞) spectral density for arbitrary T and σ. For finite but large N, we also compute the large deviation properties of the probability of stability on the stable side T < T c using a Coulomb gas representation.

Healing of Carotid Stents: A Prospective Duplex Ultrasound Study

2003 ◽  
Vol 10 (3) ◽  
pp. 636-642 ◽  
Author(s):  
Andrea Willfort-Ehringer ◽  
Ramazanali Ahmadi ◽  
Michael E. Gschwandtner ◽  
Angelika Haumer ◽  
Gottfried Heinz ◽  
...  
Keyword(s):  
Carotid Arteries ◽  
Duplex Ultrasound ◽  
Stable Phase ◽  
Carotid Stent ◽  
Time Intervals ◽  
Unstable Phase ◽  
Ultrasound Study ◽  
Maximal Thickness ◽  
Second Year

Purpose: To study the dynamics of carotid stent healing over a 2-year period using duplex ultrasound imaging. Methods: One hundred twelve patients with 121 successfully stented carotid arteries were examined with color-coded duplex ultrasound the day after the stent procedure and at 1, 3, 6, 12, and 24 months in follow-up. The maximal thickness and echogenicity of the layer between the stent and the perfused lumen (SPL) were evaluated. Echogenicity was classified as echogenic if the SPL layer was clearly detected in B mode and echolucent if the SPL layer was barely visible in B mode, its border defined by assistance of color-coded flow. Results: At day 1, an echolucent SPL layer with a median thickness of 0.7 mm was interpreted as a thrombotic layer, which decreased at 1 month to practically zero (i.e., not detectable). In follow-up, increases in thickness (mainly up to 6 months) and echogenicity (up to 12 months) of the SPL layer were interpreted as neointimal ingrowth. At 3, 6, and 12 months, the median maximal thickness of the SPL layer was 0.5 mm, 0.9 mm, and 1.0 mm, respectively, whereas the percentage of patients with an echogenic SPL layer was 27% (32/119), 56% (66/117), and 95% (105/110), respectively, at the same time intervals. No further change was observed at the 24-month examination. Conclusions: Three phases of carotid stent incorporation are defined: (1) an early unstable period soon after stent placement with an echolucent (thrombotic) SPL layer, (2) a moderately unstable phase with ingrowing neointima (1–12 months), and (3) a stable phase from the second year on. These data may indicate the need for different intensities of therapy and surveillance intervals.

The Problem of Size in Robust Design

1997 ◽  
Author(s):  
Patrick N. Koch ◽  
Janet K. Allen ◽  
Farrokh Mistree ◽  
Dimitri Mavris
Keyword(s):  
Robust Design ◽  
High Speed ◽  
Model Building ◽  
Parametric Design ◽  
Design Approach ◽  
Preliminary Design ◽  
Fast Analysis ◽  
Large Complex ◽  
Complex Design ◽  
Large Complex System

Abstract To facilitate the effective solution of multidisciplinary, multiobjective complex design problems, a departure from the traditional parametric design analysis and single objective optimization approaches is necessary in the preliminary stages of design. A necessary tradeoff becomes one of efficiency vs. accuracy as approximate models are sought to allow fast analysis and effective exploration of a preliminary design space. In this paper we apply a general robust design approach for efficient and comprehensive preliminary design to a large complex system: a high speed civil transport (HSCT) aircraft. Specifically, we investigate the HSCT wing configuration design, incorporating life cycle economic uncertainties to identify economically robust solutions. The approach is built on the foundation of statistical experimentation and modeling techniques and robust design principles, and is specialized through incorporation of the compromise Decision Support Problem for multiobjective design. For large problems however, as in the HSCT example, this robust design approach developed for efficient and comprehensive design breaks down with the problem of size — combinatorial explosion in experimentation and model building with number of variables — and both efficiency and accuracy are sacrificed. Our focus in this paper is on identifying and discussing the implications and open issues associated with the problem of size for the preliminary design of large complex systems.

scholarly journals The Yamada Model for a Self-Pulsing Laser: Bifurcation Structure for Nonidentical Decay Times of Gain and Absorber

2020 ◽  
Vol 30 (14) ◽  
pp. 2030039
Author(s):  
Robert Otupiri ◽  
Bernd Krauskopf ◽  
Neil G. R. Broderick
Keyword(s):  
Three Dimensional ◽  
Equation Model ◽  
Decay Rates ◽  
Phase Portraits ◽  
Stable Phase ◽  
Bifurcation Structure ◽  
Decay Times ◽  
Comprehensive Picture ◽  
Two Parameter ◽  
Parameter Bifurcation

We consider self-pulsing in lasers with a gain section and an absorber section via a mechanism known as [Formula: see text]-switching, as described mathematically by the Yamada ordinary differential equation model for the gain, the absorber and the laser intensity. More specifically, we are interested in the case that gain and absorber decay on different time-scales. We present an overall bifurcation structure by showing how the two-parameter bifurcation diagram in the plane of pump strength versus decay rate of the gain changes with the ratio between the two decay rates. In total, there are ten cases BI to BX of qualitatively different two-parameter bifurcation diagrams, which we present with an explanation of the transitions between them. Moroever, we show for each of the associated eleven cases of structurally stable phase portraits (in open regions of the parameter space) a three-dimensional representation of the organization of phase space by the two-dimensional manifolds of saddle equilibria and saddle periodic orbits. The overall bifurcation structure provides a comprehensive picture of the observable dynamics, including multistability and excitability, which we expect to be of relevance for experimental work on [Formula: see text]-switching lasers with different kinds of saturable absorbers.

When will a large complex system be stable?

1985 ◽  
Vol 113 (1) ◽  
pp. 153-156 ◽  
Author(s):  
Joel E. Cohen ◽  
Charles M. Newman
Keyword(s):  
Complex System ◽  
Large Complex ◽  
Large Complex System

scholarly journals LARGE DEVIATIONS FOR A NON-CENTERED WISHART MATRIX

2013 ◽  
Vol 02 (01) ◽  
pp. 1250016 ◽  
Author(s):  
ADRIEN HARDY ◽  
ARNO B. J. KUIJLAARS
Keyword(s):  
Large Deviation ◽  
Variational Problems ◽  
Logarithmic Potential ◽  
Rate Function ◽  
Coulomb Gas ◽  
Logarithmic Potential Theory ◽  
Additive Perturbation ◽  
Two Measures ◽  
Multiple Orthogonal Polynomials ◽  
Vector Equilibrium

We investigate an additive perturbation of a complex Wishart random matrix and prove that a large deviation principle holds for the spectral measures. The rate function is associated to a vector equilibrium problem coming from logarithmic potential theory, which in our case is a quadratic map involving the logarithmic energies, or Voiculescu's entropies, of two measures in the presence of an external field and an upper constraint. The proof is based on a two type particles Coulomb gas representation for the eigenvalue distribution, which gives a new insight on why such variational problems should describe the limiting spectral distribution. This representation is available because of a Nikishin structure satisfied by the weights of the multiple orthogonal polynomials hidden in the background.

Sign in / Sign up

Export Citation Format

Share Document