Stability Of Dynamic Processes Research Articles

Fundamental investigation on the time-variance of process stability

Disturbance factors, such as self-excited tool vibrations, limit the performance of modern machining processes and thus restrict the quality, productivity and sustainability of industrially manufactured components. The dynamic process stability is subjected to significant variances especially at the beginning of the tool life. To precisely quantify these variances, series of milling tests were conducted and analyzed using tools made of high speed steel (HSS) and cemented carbide for the machining of EN AW-7075. In all test series, a critical initial decrease of the stability limit was detected directly at the beginning of the tool life. In the following, a significant increase and subsequent almost constant stability limit was observed. In the case of the HSS cutters the achieved stability limit exceeded the initial level substantially. An increase in the process forces and flank wear was also measured over tool life but the progression cannot directly explain the extend and characteristics regarding the stability limit. The findings described have a considerable impact on the appropriate design of manufacturing processes and experiments to determine the dynamic stability of cutting operations.

Phase transitions and stability of dynamical processes on hypergraphs

Hypergraphs naturally represent higher-order interactions, which persistently appear in social interactions, neural networks, and other natural systems. Although their importance is well recognized, a theoretical framework to describe general dynamical processes on hypergraphs is not available yet. In this paper, we derive expressions for the stability of dynamical systems defined on an arbitrary hypergraph. The framework allows us to reveal that, near the fixed point, the relevant structure is a weighted graph-projection of the hypergraph and that it is possible to identify the role of each structural order for a given process. We analytically solve two dynamics of general interest, namely, social contagion and diffusion processes, and show that the stability conditions can be decoupled in structural and dynamical components. Our results show that in social contagion process, only pairwise interactions play a role in the stability of the absorbing state, while for the diffusion dynamics, the order of the interactions plays a differential role. Our work provides a general framework for further exploration of dynamical processes on hypergraphs.

Eigenvalue Outliers of Non-Hermitian Random Matrices with a Local Tree Structure.

Spectra of sparse non-Hermitian random matrices determine the dynamics of complex processes on graphs. Eigenvalue outliers in the spectrum are of particular interest, since they determine the stationary state and the stability of dynamical processes. We present a general and exact theory for the eigenvalue outliers of random matrices with a local tree structure. For adjacency and Laplacian matrices of oriented random graphs, we derive analytical expressions for the eigenvalue outliers, the first moments of the distribution of eigenvector elements associated with an outlier, the support of the spectral density, and the spectral gap. We show that these spectral observables obey universal expressions, which hold for a broad class of oriented random matrices.

Stability of dynamic processes occurring during landing of descent systems

The dynamics of the landing of descent aeropneumoelastic systems containing a parachute, pneumatic foundation and object is considered. The motion of the pneumoelastic systems on the landing site is studied. The criteria for the stability of motion of various schemes of aeropneumoelastic systems at the final landing phases are derived.

On the Stability of Dynamic Processes in Economic Theory

The notion of stability in the sense of Lyapunov is applied to economic dynamic processes of the Champsaur-Dreze-Henry type. Our purpose in this note is to fill a small gap in the literature concerning dynamic processes in economic theory, of the type presented by Champsaur, Dreze, and Henry [3]. Indeed, these authors do not discuss stability in the sense of Lyapunov [7]. However, a recent result of Maschler and Peleg [9] on this kind of stability (presented in a discrete model) can easily be applied to both continuous and discrete processes used in economics. We shall present this result for a very general class of such processes and conclude with references to a few economic applications. For our purpose a (set valued) dynamic system is simply a pair 〈X,φ〉, where X is a compact subset of R and φ a correspondence from X to its nonempty subsets. Let T be a subset of [0,∞) containing 0 and x0 an element of X. Then a φ-process starting at x0 is a pair of functions: x(·) : T → X, ẋ(·) : T → R, such that: x(0) = x0 and, ∀ t ∈ T , ẋ(t) ∈ φ(x(t)). If T = {0, 1, 2, · · · , } and ẋ(t) = x(t + 1) then the process 〈x(·), ẋ)(·)〉 is called discrete. If T = [0,∞), if x(·) is absolutely continuous on any interval [0, τ ] in T , and ẋ(t) = dx(t)/dt for almost every t in T , then the process 〈x(·), ẋ(·)〉 is called continuous. In the first case the Econometrica, 47(3), 733-737, 1979. As pointed out by Negishi [10], this is the same as Samuelson’s stability of the second kind [11]. The term “stability in the sense of Lyapunov” is used by Arrow and Hahn [1]. Heal [5] and Hori [6] also use this concept of stability. See Champsaur, Dreze, and Henry [3, Section 5].

The Stability of Dynamic Processes

The Stability of Dynamic Processes