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Location of the Limit Cycle for a Class of Liénard Systems by Means of Dulac-Cherkas Functions
Get PDFDulac-Cherkas functions can be used to estimate the number of limit cycles and to approximate their location. We consider a class of Liénard systems containing the van der Pol system as a special case and present two approaches to construct Dulac-Cherkas functions. By means of two Dulac-Cherkas functions, we improve the Poincaré-Bendixson annulus for the van der Pol system which has been derived in our previous paper [A. A. Grin and K. R. Schneider, Global algebraic Poincaré-Bendixson annulus for the van der Pol syste
The Dynamics of a Collapsing Polyelectrolyte Gel
No full textWe analyse the dynamics of different routes to collapse of a constrained polyelectrolyte gel in contact with an ionic bath. The evolution of the gel is described by a model that incorporates non-linear elasticity, Stefan-Maxwell diffusion and interfacial gradient free energy to account for phase separation of the gel. A bifurcation analysis of the homogeneous equilibrium states reveals three solution branches at low ion concentrations in the bath, giving way to only one above a critical ion concentration. We present numerical solutions that capture both the spatial heterogeneity and the multiple time-scales involved in the process of collapse. These solutions are complemented by two analytical studies. Firstly, a phase-plane analysis that reveals the existence of a depletion front for the transition from the highly swollen to the new collapsed equilibrium state. This depletion front is initiated after the fast ionic diffusion has set the initial condition for this time regime. Secondly, we perform a linear stability analysis about the homogeneous states that show that for a range of ion concentrations in the bath, spinodal decomposition of the swollen state gives rise to localized solvent-rich(poor) and, due to the electro-neutrality condition, ion-poor(rich) phases that coarsen on the route to collapse. This dynamics of a collapsing polyelectrolyte gel has not been described before
Second Order Error Bounds for POD-ROM Methods Based on First Order Divided Differences
No full textThis note proves, for simplicity for the heat equation, that using BDF2 as time stepping scheme in POD-ROM methods with snapshots based on difference quotients gives both the optimal second order error bound in time and pointwise estimates
Balanced-Viscosity solutions to infinite-dimensional multi-rate systems
No full textWe consider generalized gradient systems with rate-independent and rate-dependent dissipation potentials. We provide a general framework for performing a vanishing-viscosity limit leading to the notion of parametrized and true Balanced-Viscosity solutions that include a precise description of the jump behavior developing in this limit. Distinguishing an elastic variable having a viscous damping with relaxation time and an internal variable with relaxation time we obtain different limits for the three cases , and . An application to a delamination problem shows that the theory is general enough to treat nontrivial models in continuum mechanics
Second-order sufficient conditions for sparse optimal control of singular Allen--Cahn systems with dynamic boundary conditions
No full textIn this paper we study the optimal control of a parabolic initial-boundary value problem of Allen--Cahn type with dynamic boundary conditions. Phase field systems of this type govern the evolution of coupled diffusive phase transition processes with nonconserved order parameters that occur in a container and on its surface, respectively. It is assumed that the nonlinear functions driving the physical processes within the bulk and on the surface are double well potentials of logarithmic type whose derivatives become singular at the boundary of their respective domains of definition. For such systems, optimal control problems have been studied in the past. We focus here on the situation when the cost functional of the optimal control problem contains a nondifferentiable term like the L1-norm leading to sparsity of optimal controls. For such cases, we derive second-order sufficient conditions for locally optimal controls
The Emergence of a Giant Component in One-Dimensional Inhomogeneous Networks with Long-Range Effects
No full textWe study the weight-dependent random connection model, a class of sparse graphs featuring many real-world properties such as heavy-tailed degree distributions and clustering. We introduce a coefficient, , measuring the effect of the degree-distribution on the occurrence of long edges. We identify a sharp phase transition in for the existence of a giant component in dimension
Theoretical study of the impact of alloy disorder on carrier transport and recombination processes in deep UV (Al,Ga)N light emitters
Get PDFAluminium gallium nitride ((Al,Ga)N) has gained significant attention in recent years due to its potential for highly efficient light emitters operating in the deep ultra-violet (UV) range (< 280 nm). However, given that current devices exhibit extremely low efficiencies, understanding the fundamental properties of (Al,Ga)N-based systems is of key importance. Here, using a multi-scale simulation framework, we study the impact of alloy disorder on carrier transport, radiative and non-radiative recombination processes in a c-plane Al 0.7 Ga 0.3 N/Al 0.8 Ga 0.2 N quantum well embedded in a p-i-n junction. Our calculations reveal that alloy fluctuations can open "percolative" pathways that promote transport for the electrons and holes into the quantum well region. Such an effect is neglected in conventional, and widely used transport simulations. Moreover, we find also that the resulting increased carrier density and alloy induced carrier localization effects significantly increase non-radiative Auger-Meitner recombination in comparison to the radiative process. Thus, to avoid such non-radiative process and potentially related material degradation, a careful design (wider well, multi quantum wells) of the active region is required to improve the efficiency of deep UV light emitters
Dynamical Gibbs Variational Principles for Irreversible Interacting Particle Systems with Applications to Attractor Properties
No full textWe consider irreversible translation-invariant interacting particle systems on the d-dimensional cubic lattice with finite local state space, which admit at least one Gibbs measure as a time-stationary measure. Under some mild degeneracy conditions on the rates and the specification we prove, that zero relative entropy loss of a translation-invariant measure implies, that the measure is Gibbs w.r.t. the same specification as the time-stationary Gibbs measure. As an application, we obtain the attractor property for irreversible interacting particle systems, which says that any weak limit point of any trajectory of translation-invariant measures is a Gibbs measure w.r.t. the same specification as the time-stationary measure. This extends previously known results to fairly general irreversible interacting particle systems
An introduction to the analysis of gradients systems
Get PDFThe present notes provide an extended version of a small lecture course (of 36 hours) given at the Humboldt-Universität zu Berlin in the Winter Term 2022/23. The material starting in Section 5.4 was added afterwards. The aim of these notes to give an introductory overview on the analytical approaches for gradient-flow equations in Hilbert spaces, Banach spaces, and metric spaces and to show that on the first entry level these theories have a lot in common. The theories and their specific setups are illustrated by suitable examples and counterexamples
Finite sample guarantees
Get PDFThis paper offers a new approach for study the frequentist properties of the penalized MLE for general nonlinear regression models. The idea of the approach is to relax the nonlinear structural equation by introducing an auxiliary parameter for the regression response and replacing the structural equation with a penalty. This leads to a general semiparametric problem which is studied using the SLS approach from \cite{Sp2022}. We state sharp bounds on concentration and on the accuracy of the penalized MLE, Fisher and Wilks expansions, evaluate the risk of estimation over smoothness classes, and a number of further results. All the bounds are given in terms of effective dimension and do not involve the ambient dimension of the parameter space