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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.

We consider a reaction-advection-diffusion equation that models a population in a bounded habitat with a strong Allee effect and a protection zone. We assume that the population growth function exhibits the strong Allee effect and has a positive integral within the protection zone (indicating population persistence) and a negative integral in the surrounding patches (indicating population decay). We prove that the existence of positive steady-state solutions to this system depends on the length of the protection zone. It is demonstrated that there exists a threshold value H∗ such that, for a protection zone of size H∗, there exists one positive steady-state solution and, for a larger protection zone, there exist multiple positive steady-state solutions. For smaller protection zones, we prove there exists no positive steady-state solution. The dynamics of the equation are further examined via numerical simulations.

Weighted-type finite-difference schemes are a class of widely used nonlinear schemes that can capture strong discontinuities accurately and efficiently. For the Euler equations without source terms, poor convergence of weighted-type schemes is a widely known difficulty in finding steady-state solutions with strong shock waves. The primary reason for this lies in the fact that classical weighted-type schemes produce spurious oscillations near strong discontinuities. Recently, a novel weighted-type scheme has been developed. The nonlinear weights of the new scheme are fourth-order accurate and do not reduce the accuracy at the high-order critical points, which is beneficial for steady-state convergence. In this paper, we compare the convergence performances of classical and new weighted-type schemes in detail. Several benchmark problems containing shock waves, contact discontinuities, and rarefaction waves were used to compare the convergence performance among different weighted-type schemes. The results show that the new weighted-type scheme basically eliminates slight post-shock oscillations, and the residual settles to machine zero. Compared to classical weighted-type schemes, the steady-state convergence performance of the new weighted-type scheme is significantly improved.

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.

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 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.

In this paper, we are concerned about the steady state problem of a degenerate predator-prey model with cross-diffusion. First, some properties of principal eigenvalues are deduced. Next, the stability of the semitrivial steady state solutions are obtained. Finally, existence, uniqueness and stability of positive steady state solutions are derived. The result reveals that the diffusion with spatial degeneracy profoundly influences the structure and stability of semitrivial steady state solutions and the coexistence region, which is a strong contrast to the result with no spatial degeneracy.

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.

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.

Controllability metrics based on system Gramians have been widely adopted to provide quantitative measures of the degree of controllability (DoC) and the disturbance rejection capability (DoDR) of dynamical systems. While steady-state Gramian formulations offer closed-form tractability, they are not applicable when rigid-body modes are present, as the associated poles at the origin cause the conventional Gramians to diverge. This paper presents a novel steady-state DoDR metric for linear time-invariant systems with a rigid-body mode. By block-diagonalizing the dynamics through a similarity transformation and analyzing the asymptotic behavior of the Gramian matrices, we derive an exact closed-form expression for the steady-state DoDR. The resulting formulation is numerically stable and enables systematic evaluation of disturbance-rejection capability even in the presence of a rigid-body mode. The proposed metric is validated using a mass–spring–damper chain model, where its effectiveness is demonstrated in actuator placement problems. The results show that the metric not only remains computationally well-posed but also provides physically meaningful interpretations consistent with modal characteristics. This study establishes a foundation for extending disturbance-rejection metrics to systems with multiple rigid-body modes, thereby broadening the applicability of Gramian-based controllability analysis.