Numerical Continuation Research Articles

Bifurcations and model fitting of a discrete epidemic system with incubation period and saturated contact rate

A discrete SIS epidemic system with disease incubation period and saturated contact rate is formulated and studied. The qualitative properties of both the disease-free and endemic equilibria in hyperbolic and non-hyperbolic cases are analysed. All potential local bifurcations of the two equilibria are explored. The direction, stability, and explicit approximate expressions for each type of bifurcation are derived. All possible rational rotation numbers on the family of closed-invariant curves generated through the Neimark–Sacker bifurcation are derived and arranged based on the five-level Farey sequence tree. The 1:5 weak resonance on the invariant curve is analysed. By the study of contact bifurcation, the global dynamic behaviour is discussed. Based on the epidemiological background, the significance of each bifurcation is explained. A large number of numerical simulation and numerical continuation examples support the theoretical analyses. At last, the model is used to fit the number of Hepatitis A infections in Guangdong Province, China, from 2018 to 2024. After analysing the residuals, conducting goodness-of-fit test, Student's t-test and Overfitting test, short-term forecasting for the number of infections is carried out.

Extended Metal-Insulator Crossover with Strong Antiferromagnetic Spin Correlation in Half-Filled 3D Hubbard Model.

The Hubbard model at temperatures above the Néel transition, despite being a paramagnet, can exhibit rich physics due to the interplay of Fermi surface, on-site interaction U and thermal fluctuations. Nevertheless, the understanding of the crossover physics remains only at a qualitative level, because of the intrinsically smooth behavior. Employing an improved variant of the numerically exact auxiliary-field quantum MonteCarlo algorithm equipped with numerical analytic continuation, we obtain a broad variety of thermodynamic and dynamical properties of the three-dimensional Hubbard model at half filling, quantitatively determine the crossover boundaries, and observe that the metal-insulator crossover regime, in which antiferromagnetic spin correlations appear strongest, exists over an extended regime in between the Fermi liquid for small U and the Mott insulator for large U. In particular, the location of the most rapid suppression of double occupancy as U increases, is found to fully reside in the metallic Fermi liquid regime, in contrast to the conventional intuition that it is a representative feature for entering the Mott insulator. Beside providing a reliable methodology for numerical study of crossover physics, our work can also serve as a timely and important guideline for the most recent optical lattice experiments.

Dynamics of the vibro-impact capsule robot with a von Mises truss

AbstractFunctionalised nonlinear structures employing structural instabilities for rapid response shape-shifting are emerging technologies with a wide range of potential applications. The von Mises truss is a widely employed model for such functionalised nonlinear structures; however, few studies have delved into its functionality when integrated with a complex dynamical system. This paper investigates its efficacy on enhancing the progression speed of a vibration-driven robot, known as the vibro-impact capsule robot, which is a piecewise-smooth dynamical system having abundant coexisting attractors. Bifurcation analysis of the capsule robot integrated with a von Mises truss is conducted for this purpose. Our numerical studies focus on the influence of the frequency and amplitude of the robot’s driving force on its progression. Specifically, we compare the periodic responses of both the capsule robots with and without the von Mises truss, utilising the numerical continuation techniques for piecewise-smooth dynamical systems. Our studies confirm the advantage of using the von Mises truss when the driving force of the robot is significantly small. Additionally, we identify an optimal operational regime on the amplitude-frequency control plane, where the maximum robot speed is achieved for a given amount of power consumption. The numerical studies presented in this work provide a promising indication of the advantages offered by the von Mises truss for vibration-driven robots. This research underscores the significant potential of functionalised nonlinear structures to enhance the efficiency of small-scale robots operating under power limitations.

Numerical analysis of binary fluid convection with thermal and solutal lateral gradients.

This paper analyzes the influence of laterally enforced solutal gradients on the steady and bifurcated periodic dynamics in binary fluids contained in horizontally heated slots, taking into account the Soret and Dufour effects. Numerical Newton-Krylov continuation techniques to follow the primary and secondary branches of steady solutions and periodic orbits are applied. The stability of all these branches is also analyzed. A great variety of stable steady and periodic states is found, depending on the ratio of the thermal and solutal gradients. The proximity to parameters that balance the buoyancy forces delays the onset of center-symmetric oscillations to very large values of the thermal Rayleigh number, while large solutal gradients tend to restabilize the steady flows after the onset of center-symmetric oscillations, and to give rise to spatiotemporal symmetric waves of broken center symmetry at larger thermal Rayleigh number. The work done by the thermal buoyancy force is essential for the restabilization of the steady states and the change of the ulterior dynamics. It is also shown that the transition to temporal chaos depends strongly on the absence or intensity of the solutal gradients.

Existence of traveling wave solutions in continuous optimal velocity models

Existence of traveling wave solutions in continuous optimal velocity models

Double-eigenvalue bifurcation and multistability in serpentine strips with tunable buckling behaviors

Double-eigenvalue bifurcation and multistability in serpentine strips with tunable buckling behaviors

Unraveling dengue dynamics with data calibration from Palu and Jakarta: Optimizing active surveillance and fogging interventions

Unraveling dengue dynamics with data calibration from Palu and Jakarta: Optimizing active surveillance and fogging interventions

Method for Constructing a Compact Component Mode Synthesis Model for Analyzing Nonlinear Normal Modes of Structures with Localized Nonlinearities

Abstract Nonlinear dynamics analysis is a crucial topic in mechanical and aerospace engineering. The analysis of nonlinear normal modes (NNMs) provides an effective mathematical tool for interpreting complex nonlinear vibration phenomena. Unlike the invariant normal modes of linear systems, NNMs often exhibit frequency-energy dependence and cannot be computed using the traditional eigen-decomposition method. Calculating NNMs relies on numerical methods that involve expensive iterative computations, especially for systems with numerous degrees-of-freedom. To relieve computational costs, this paper proposes a mode selection method for the component mode synthesis (CMS) technique to enable compact reduced-order modeling of structures with localized nonlinearities. The reduced-order model is then combined with a numerical continuation scheme to establish a low-cost NNM analysis framework. This framework can efficiently predict the NNMs of high-dimensional finite element models. The proposed framework is demonstrated on an I-shaped cantilever beam with localized nonlinear stiffness. The results show that the proposed approach can identify the key modes in the CMS modeling procedure. The NNMs of the nonlinear I-shaped beam structure can then be analyzed with a significant reduction in computational costs.

Stick–slip oscillations in the low feed linear motion of a grinding machine due to dry friction and backlash

Stick–slip oscillations in the low feed linear motion of a grinding machine due to dry friction and backlash

Liquid plugs in narrow tubes: application to airway occlusion

We study liquid plugs in the pulmonary airways based on the two-phase axisymmetric weighted residual integral boundary-layer model of Dietze et al. (J. Fluid Mech., vol. 894, 2020, A17), which was originally developed to study liquid films coating the inner surface of a cylindrical tube in interaction with a core gas flow. The augmented form of this model, which was never applied beyond a proof of concept, allows for the representation of liquid pseudo-plugs. Here, we demonstrate its predictive power vs experiments and direct numerical simulations, in terms of the dynamics of plug formation and the characteristics of developed liquid plugs, such as their shape, flow field, speed and length, as well as the associated wall stresses and their spatial derivatives. In particular, we show that the augmented model allows us to establish a direct continuation path from travelling-wave solutions (TWS) to travelling-plug solutions (TPS). We then apply the model to predict mucus plugs in the conducting zone of the tracheobronchial tree, based on the lung architecture model of Weibel. We proceed by numerical continuation of travelling-state solutions in terms of the airway generation, whereby we impose the wavelength of the linearly most-amplified convective instability (CI) mode or that of the absolute instability (AI) mode. We identify the critical airway generation for liquid-plug formation (TWS/TPS transition), maximum potential for wall-stress-induced epithelial cell damage and CI/AI transition, and investigate how these phenomena are affected by the main control parameters, i.e. airway orientation vs gravity, air flow rate, mucus properties and airway size.

Improved Detuned Multiple-scales Method for the Forced Vibration of Strongly Nonlinear Oscillators

Abstract A novel semi-analytical technique is introduced for examining the forced oscillations of systems with strong nonlinearity, by integrating the parameter-splitting technique with the detuned multiple-scales approach. This method leverages the benefits of parameter splitting, where system parameters are initially split using the parameter-splitting technique. Subsequently, the system with these splitting parameters is tackled using the detuned multiple-scales method. The analytical solution derived from the detuned multiple-scales method is then integrated into the equation of motion, with the aim of minimizing the cumulative error in the equation to ascertain the unknown parameters resulting from the splitting procedure. The efficacy of this proposed approach is demonstrated through the analysis of the forced vibrations of a Helmholtz-Duffing oscillator and a Duffing oscillator. The steady-state response is evaluated by comparing the frequency-response curves generated by the proposed method against those produced by numerical continuation and the traditional detuned multiple-scales method. Ultimately, through convergence checks, it is established that corrections are essential for erroneous solutions that are directly derived from the classical detuned multiple-scales method.

Cross validation in stochastic analytic continuation.

Stochastic analytic continuation (SAC) of quantum Monte Carlo (QMC) imaginary-time correlation function data is a valuable tool in connecting many-body models to experimentally measurable dynamic response functions. Recent developments of the SAC method have allowed for spectral functions with sharp features, e.g., narrow peaks and divergent edges, to be resolved with unprecedented fidelity. Often it is not known what exact sharp features, if any, are present a priori, and, due to the ill-posed nature of the analytic continuation problem, multiple spectral representations may be acceptable. In this work we borrow from the machine learning and statistics literature and implement a cross validation technique to provide an unbiased method to identify the most likely spectrum among a set obtained with different spectral parametrizations and imposed constraints. We demonstrate the power of this method with examples using imaginary-time data generated by QMC simulations and synthetic data generated from artificial spectra. Our procedure, which can be considered a form of model selection, can be applied to a variety of numerical analytic continuation methods, beyond just SAC.

DYNAMICAL BEHAVIOR IN THE COMPETITIVE MODEL INCORPORATING THE FEAR EFFECT OF PREY DUE TO ALLELOPATHY WITH SHARED BIOTIC RESOURCES

This research develops a mathematical model of a natural phenomenon, namely sea snails that can release toxins (allelopathy) so that non-toxic sea snails become afraid. In addition, toxic and non-toxic sea snails share biotic resources. Based on the existing phenomenon, the model of fear effect caused by allelopathy in the competitive interaction model with shared biotic resources will be studied. In this system, three equilibrium points are obtained: extinction point of prey, extinction point of predator, and coexisting point under certain conditions. Analysis of local stability at equilibrium points by linearization shows that all equilibrium points are asymptotically stable with certain conditions. Numerical simulations at the equilibrium point show the same results as the analysis results. Then, numerical continuity was carried out by selecting variation of the fear effect parameter for Numerical continuity results show that changes in these parameters affect the population of toxic and non-toxic species, marked by the emergence of Transcritical bifurcations, Bifurcation occurs at the first Saddle-Node bifurcation at and the second Saddle-Node bifurcation at

Computing the dynamic response of a periodic structure coupled with a nonlinear junction using the harmonic balance method and Floquet-Bloch modelling

Structures constituted of assembled sub-components are often subject to nonlinear effects due to the presence of joints or contacts, which can induce higher-order harmonics of the source excitation. In the context of periodic structures, these nonlinearities cannot be tackled using the Floquet-Bloch theory in its usual harmonic formulation. This study hence presents an adaptation of the classical Floquet-Bloch theory to address multi-harmonic systems arising from a finite number of localized nonlinearities in periodic waveguides. We adopt the Harmonic Balance Method to recast the governing equations into a nonlinear algebraic system, which can then be solved using numerical continuation algorithms. To demonstrate the validity and benefit of this approach, the predictions derived from the proposed methodology are compared to results obtained through conventional Finite Element analysis. Excellent agreement and a notable reduction in computational cost are observed. This efficient procedure for analysing the nonlinear dynamic response of periodic waveguides opens up new possibilities in the study of damaged composite structures.

Uncertainty quantification analysis of bifurcations of the Allen–Cahn equation with random coefficients

Uncertainty quantification analysis of bifurcations of the Allen–Cahn equation with random coefficients

A Well-Tested Pump Card Classification and Identification Algorithm Based on Time Series Analysis Techniques

Abstract To accurately identify the operational conditions of oil pumping wells, promptly monitor the working status of oil wells, expedite responses to sudden pumping well accidents, and ensure orderly production at the oil extraction site, this paper introduces a time segment-based oil pumping well condition identification algorithm. This method no longer relies on drawn dynamometer cards for condition identification. Instead, it solely utilizes sensor-provided polished rod displacement and load time series, aligning more closely with the data’s intrinsic nature. The algorithm is sufficiently lightweight, highly responsive, and does not require GPU resources, making it suitable for edge deployment. Furthermore, this approach provides evidence of the time continuity and numerical continuity of time series data. Compared to traditional identification methods, this approach maintains a high level of accuracy, offers a streamlined model, and is interpretable, presenting a novel perspective for oil pumping well condition identification.

Determination of electronic resonances by analytic continuation using barycentric formula

Determination of electronic resonances by analytic continuation using barycentric formula

The role of dynamic friction in the appearance of periodic oscillations in mechanical systems

This article investigates the appearance of periodic mechanical oscillations associated with the transition between static and dynamic friction regimes. The study employs a mechanical system with one degree of freedom and a friction model recently proposed by Brown and McPhee, whose continuity and differentiability properties make it particularly appropriate for an analytical treatment of the equations. A bifurcation study of the system, including stability analysis, transformation to normal form and numerical continuation techniques, reveals that stable periodic orbits can be created either by a supercritical Hopf bifurcation or by a saddle-node bifurcation of limit cycles. The influence of all system parameters on the appearance of periodic oscillations is investigated in detail. In particular, the effect of the friction model parameters (static-to-dynamic friction ratio and transition speed between the static and dynamic regimes) on the bifurcation behavior of the system is addressed.

Family of 2:1 resonant quasi-periodic distant retrograde orbits in cislunar space

Given the current enthusiasm for lunar exploration, the 2:1 resonant distant retrograde orbit (DRO) in Earth-Moon space is of interest. To gain an in-depth understanding of the complex dynamic environment in cislunar space and provide more options for parking orbits, this paper investigates the existence of quasi-periodic orbits near the 2:1 resonant DRO in the circular restricted three-body problem (CR3BP). Firstly, the numerical computation approach, continuation strategy, and stability analysis method of quasi-periodic orbits are introduced. Then, addressing the primary challenges in the continuation progress, we have developed an adaptive continuation algorithm with automatic adjustment of the step size and the number of discrete points that characterize the invariant torus. Finally, two types of 2D quasi-DROs and their linear stability properties are explored. Using Poincaré sections, we investigated the boundaries of the maximum extent attainable by both 2D quasi-DRO families in the CR3BP at a specific Jacobi energy, confirming that both types of quasi-periodic families have reached their respective boundaries. The algorithm described in this paper is beneficial for facilitating the computation of quasi-periodic families and aids in discovering additional potential dynamical structures.

Large basins of attraction for control-based continuation of unstable periodic states

Numerical continuation tools are nowadays standard to analyse nonlinear dynamical systems by numerical means. These powerful methods are unfortunately not available in real experiments without having access to an accurate mathematical model. Implementing such a concept in real world experiments using control and data processing to track unstable states and their bifurcations, requires robust control techniques with large basins and good global properties. Here we propose design principles for control techniques for periodic states which lead to large basins and which are robust, without the need to have access to a detailed mathematical model. Our analytic considerations for the control design will be based on weakly nonlinear analysis of periodically driven oscillator systems. We then demonstrate by numerical means that in strong nonlinear regimes successful control with large basins of attraction can be achieved when only plain time series data are available.