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ABSTRACT A challenge associated with fisheries management is when there are many potential participants, and each participant has virtually zero effect (small agents), but their collective action may be significant. Individual quotas or effort restrictions may not be well suited in such cases. For the abovementioned problem, we explore the mean field games (MFGs) framework, which considers non‐cooperative games of infinitely many agents. The MFGs approach allows considering progressive costs and taxes, which would be difficult or impossible to consider otherwise, and the influence of uncertainties in agents' decisions on their aggregate action. The mean field games are applied to model a fishery with many small fishing firms, which are more or less identical, and to analyze the implications of different management policies on their economic performance and resource sustainability. The main focus is on financial regulations, which make the excessive effort unprofitable.

Increasing the span length of stay cables markedly increases their susceptibility to parametric resonance. This study investigates mitigating parametric resonance in stay cables using a parallel inerter damper and focuses on a methodology to obtain the upper-branch steady-state solution via Runge–Kutta integration. The approach computes both transient and steady-state responses under arbitrary support excitation. Furthermore, we examine the dependence of the Runge–Kutta steady-state solution on initial conditions by analyzing phase portraits of the cable–inerter system. Based on the method of multiple scales, we propose a general procedure to select initial conditions that ensure convergence to the upper-branch steady-state solution. The results show that different initial conditions drive the transient Amplitude along different phase trajectories and may converge to distinct steady-state solutions. The inerter damper modifies the critical phase trajectory of the cable–damper system, thereby shifting the feasible region of initial conditions. Thus, when initial conditions from the undamped case are used in the presence of an inerter damper, the Runge–Kutta solution may converge to the lower-branch stable solution, which may overestimate the damper’s control effectiveness.

The approximation of steady-state vibrations within non-linear dynamical systems is well-established in academics and is becoming increasingly important in industry. However, the complexity and the number of degrees of freedom of application-oriented industrial models demand efficient approximation methods for steady-state solutions. One possible approach to that problem are hybrid approximation schemes, which combine advantages of standard methods from the literature. The common ground of these methods is their description of the steady-state dynamics of a system solely based on the degrees of freedom affected directly by non-linearity—the so-called non-linear degrees of freedom. This contribution proposes a new hybrid method for approximating periodic solutions of systems with localised non-linearities. The motion of the non-linear degrees of freedom is approximated using the Finite Difference  method, whilst the motion of the linear degrees of freedom is treated with the Harmonic Balance  method. An application to a chain of oscillators showing stick-slip oscillations is used to demonstrate the performance of the proposed hybrid framework. A comparison with both pure Finite Difference  and Harmonic Balance  method reveals a noticeable increase in efficiency for larger systems, whilst keeping an excellent approximation quality for the strongly non-linear solution parts.

The paper presents the results of the author's research on effectively determining the initial conditions for the time-stepping model of a high-speed inverter-driven induction machine. The classical time-harmonic and multi-harmonic models based on the multidimensional effective magnetic permeability were used and compared as a preconditioner for the time-stepping model to speed up the steady-state solution. The carried-out simulation experiment proved that using both approaches radically accelerates computations. Furthermore, it has been shown that the multi-harmonic model is much more effective for problems with strong harmonic effects.

This paper is concerned primarily with constructive mathematical analysis of a general system of nonlinear two-point boundary value problem when an empirically constructed candidate for an approximate solution ( quasi-solution ) satisfies verifiable conditions. A local analysis in a neighbour- hood of a quasi-solution assures the existence and uniqueness of solutions and, at the same time, provides error bounds for approximate solutions. Applying this method to a cholera epidemic model, we obtain an analytical approximation of the steady-state solution with rigorous error bounds that also displays dependence on a parameter. In connection with this epidemic model, we also analyse the basic reproduction number, an important threshold quantity in the epidemiology context. Through a complex analytic approach, we determine the principal eigenvalue to be real and positive in a range of parameter values.

Efficient and accurate numerical approximation of the full Boltzmann equation has been a longstanding challenging problem in kinetic theory. This is mainly due to the high dimensionality of the problem and the complicated collision operator. In this work, we propose a highly efficient adaptive low rank method for the Boltzmann equation, concerning in particular the steady state computation. This method employs the fast Fourier spectral method (for the collision operator) and the dynamical low rank method to obtain computational efficiency. An adaptive strategy is introduced to incorporate the boundary information and control the computational rank in an appropriate way. Using a series of benchmark tests in 1D and 2D, we demonstrate the efficiency and accuracy of the proposed method in comparison to the full tensor grid approach.

We construct a new second-order moving-water equilibria preserving central-upwind scheme for the one-dimensional Saint-Venant system of shallow water equations. The idea is based on a reformulation of the source terms as integral in the flux function. Reconstruction of the flux variable yields then a third order equation that can be solved exactly. This procedure does not require any further modification of existing schemes. Several numerical tests are performed to verify the ability of the proposed scheme to accurately capture small perturbations of steady states.

In this paper, we investigate an initial-boundary value problem of a reaction–diffusion equation in a bounded domain with a Robin boundary condition and introduce some particular parameters to consider the non-zero flux on the boundary. This problem arises in the study of mosquito populations under the intervention of the population replacement method, where the boundary condition takes into account the inflow and outflow of individuals through the boundary. Using phase plane analysis, the present paper studies the existence and properties of non-constant steady-state solutions depending on several parameters. Then, we prove some sufficient conditions for their stability. We show that the long-time efficiency of this control method depends strongly on the size of the treated zone and the migration rate. To illustrate these theoretical results, we provide some numerical simulations in the framework of mosquito population control.

This paper is concerned with a delayed reaction–diffusion equation on a unit disk. By means of the singularity theory and Lyapunov–Schmidt reduction, we not only derive universal conclusions about the existence of inhomogeneous steady-state solutions and the equivariant Hopf bifurcation theorems, but also obtain some more extraordinary properties of bifurcating solutions, which are produced by the radial symmetry through abstract methods based on the Lie group representation theory. Meanwhile, we illustrate our results by an application to a population model with a time delay. Furthermore, the methods established in this paper are applicable to specific delayed reaction–diffusion models with other symmetries.

This paper introduces a novel Gramian-based quantitative metric to evaluate the disturbance rejection capabilities of linear unstable systems. The proposed metric addresses key limitations of the previously introduced degree of disturbance rejection (DoDR) metrics, including their dependency on the final time and numerical problems arising from differential equation computations. Specifically, this study defines the steady-state solution of the DoDR metric, which avoids numerical issues by relying only on solving four algebraic equations, even when the Gramian matrices diverge. This study further strengthens its contributions by providing rigorous mathematical proofs supporting the proposed method, ensuring a strong theoretical foundation. The derived results demonstrate that the proposed metric represents the sum of the steady-state input energies required to reject the disturbances in the asymptotically stable and anti-stable subsystems. Numerical examples demonstrated that the proposed metric maintained the physical meaning of the original DoDR while offering practical computational advantages. This study represents a significant step toward the efficient and reliable assessment of disturbance rejection capabilities in unstable systems.