Numerical Continuation Research Articles

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.

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.

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

Serpentine structures, composed of straight and circular strips, have garnered significant attention as potential designs for flexible electronics due to their remarkable stretchability. When subjected to stretching, these serpentine strips buckle out of plane, and previous studies have identified two distinct buckling modes whose order of appearance may interchange in serpentine structures with a single cell. In this study, we employ anisotropic rod theory to model serpentine strips as a multi-segment boundary value problem (BVP), with continuity conditions enforced at the interface between the straight and curved strips. We solve the BVP using methods of continuation, and our results reveal that: (1) the exchange of the two buckling modes in a single-cell serpentine strip is induced by a double-eigenvalue and associated secondary bifurcations, which also alter the stability of the two buckling modes; (2) a variety of stable states with reversible symmetry can be manually obtained in tabletop models and are found to be disconnected from the planar branch in numerical continuation. Furthermore, we demonstrate that modulating the strip thickness across different cells leads to the initiation of buckling in the thinnest section, thereby allowing for the tuning of buckling modes in serpentine strips. In structures with two cells, the sequence of the two buckling modes can also be controlled by designing serpentine strips with nonuniform height. This work could enhance the mechanical design of serpentine-interconnect-based flexible structures and could have applications in multistable actuators and mechanical memory devices.

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

This paper deals with mechanical modelling and numerical bifurcation analysis of stick–slip oscillations that plague extremely low feed grinding operations. Based on experimental observations, a novel two degree of freedom mechanical model of a grinding machine feed drive system is formulated, which incorporates Stribeck-type dry friction, position and velocity controller dynamics, and actuator backlash. Loss and re-establishment of contact between the feed drive elements is modelled through both rigid-body impacts and a contact-stiffness model. The resulting piecewise-smooth equations of motion are subjected to detailed stability and bifurcation analysis with the help of shooting and collocation based numerical continuation tools. Major focus is attributed to the influence of the feed velocity and the control-loop parameters as well as the identification of stable, stick–slip free parameter regimes. Finally, a controller enhancement strategy is proposed, based on event-driven integrator reset rules, to help limit the amplitude of arising limit-cycle oscillations.

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.

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.

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

In this work we consider the Allen–Cahn equation, a prototypical model problem in nonlinear dynamics that exhibits bifurcations corresponding to variations of a deterministic bifurcation parameter. Going beyond the state-of-the-art, we introduce a random coefficient in the linear reaction part of the equation, thereby accounting for random, spatially-heterogeneous effects. Importantly, we assume a spatially constant, deterministic mean value of the random coefficient. We show that this mean value is in fact a bifurcation parameter in the Allen–Cahn equation with random coefficients. Moreover, we show that the bifurcation points and bifurcation curves become random objects. We consider two distinct modelling situations: (i) for a spatially homogeneous coefficient we derive analytical expressions for the distribution of the bifurcation points and show that the bifurcation curves are random shifts of a fixed reference curve; (ii) for a spatially heterogeneous coefficient we employ a generalized polynomial chaos expansion to approximate the statistical properties of the random bifurcation points and bifurcation curves. We present numerical examples in 1D physical space, where we combine the popular software package Continuation Core and Toolboxes (COCO) for numerical continuation and the Sparse Grids Matlab Kit for the polynomial chaos expansion. Our exposition addresses both, dynamical systems and uncertainty quantification, highlighting how analytical and numerical tools from both areas can be combined efficiently for the challenging uncertainty quantification analysis of bifurcations in random differential equations.

Determination of electronic resonances by analytic continuation using barycentric formula

Numerical analytical continuation of a function is used to determine its complex roots. The analytical continuation is carried out by means of a barycentric formula. From the knowledge of the complex roots the energy and width of shape resonances as well as of quantum virtual states can be determined. The roots are calculated for several realistic models and the results are compared with other approaches. We also explore and discuss a stability of the predicted resonant roots with respect to changes of the perturbation potential.

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.

Travelling pulses on three spatial scales in a Klausmeier-type vegetation-autotoxicity model

Reaction-diffusion models describing interactions between vegetation and water reveal the emergence of several types of patterns and travelling wave solutions corresponding to structures observed in real-life. Increasing their accuracy by also considering the ecological factor known as autotoxicity has lead to more involved models supporting the existence of complex dynamic patterns. In this work, we include an additional carrying capacity for the biomass in a Klausmeier-type vegetation-water-autotoxicity model, which induces the presence of two asymptotically small parameters: ɛ, representing the usual scale separation in vegetation-water models, and δ, directly linked to autotoxicity. We construct three separate types of homoclinic travelling pulse solutions based on two different scaling regimes involving ɛ and δ, with and without a so-called superslow plateau. The relative ordering of the small parameters significantly influences the phase space geometry underlying the construction of the pulse solutions. We complement the analysis by numerical continuation of the constructed pulse solutions, and demonstrate their existence (and stability) by direct numerical simulation of the full partial differential equation model.

Dynamics of particle aggregation in dewetting films of complex liquids

We consider the dynamic wetting and dewetting processes of films and droplets of complex liquids on planar surfaces, focusing on the case of colloidal suspensions, where the particle interactions can be sufficiently attractive to cause agglomeration of the colloids within the film. This leads to an interesting array of dynamic behaviours within the liquid and of the liquid–air interface. Incorporating concepts from thermodynamics and using the thin-film approximation, we construct a model consisting of a pair of coupled partial differential equations that represent the evolution of the liquid film and the effective colloidal height profiles. We determine the relevant phase behaviour of the uniform system, including finding associated binodal and spinodal curves, helping to uncover how the emerging behaviour depends on the particle interactions. Performing a linear stability analysis of our system enables us to identify parameter regimes where agglomerates form, which we independently confirm through numerical simulations and continuation of steady states, to construct bifurcation diagrams. We obtain various dynamics such as uniform colloidal profiles in an unstable situation evolving into agglomerates and thus elucidate the interplay between dewetting and particle aggregation in complex liquids on surfaces.

Primary Resonances in Inclined Cantilever Beam with Tip-Mass: A Parameter-Splitting Multiple-Scales Approach

The transverse static load, resultant from self-weight, significantly impacts both the response and the nonlinear dynamic behavior of a flexible inclined cantilever with a large lumping tip mass under harmonic base motion. However, the transverse static load was not taken into account and the mode shape of a pure cantilever beam was directly adopted during Galerkin’s procedure in the formulations of a cantilever carrying a large lumping tip mass in many previous studies. In this paper, (1) the static load effect caused by the self-weight in an inclined cantilever is considered by using a coordinate transformation, (2) the efficacy of adopting a pure cantilever beam mode shape to analyze a cantilever beam carrying a large lumping mass is studied and (3) a recently proposed semi-analytical method named parameter-splitting multiple-scales method is extended to analyze the cantilever studied to examine its effectiveness. First, the extended Hamilton’s principle is utilized to formulate the equation of motion of the cantilever studied. After that, the mode shape of a pure cantilever and the exact mode shape of a cantilever carrying a lumping mass are separately adopted in Galerkin’s method to transform the partial differential equation into ordinary differential equations. The frequency-response curves of the discretized nonlinear differential equations obtained by the numerical continuation method and the parameter-splitting multiple-scales method are compared to address ① the transverse static load effect on the frequency-response and nonlinear dynamical behavior of the beam, ② the discrepancies between the resultant frequency-response curves obtained by using two different mode shapes and ③ the efficacy of the parameter-splitting multiple-scales method on solving strongly nonlinear practical problem.

Exploring the dynamics of a four-dimensional Chua-like system: Numerical simulations and hardware investigations

In this work, we employ simulations to investigate deterministic and stochastic dynamics of a system governed by four coupled nonlinear ordinary differential equations (ODEs) derived from the canonical three-dimensional Chua model, replacing the piecewise nonlinearity with a cubic function. We employed two simulation approaches: (i) using software — where we performed numerical simulations (digital computation) in the ODEs model, and (ii) using hardware — where we performed experimental studies (analog computation) in the ODEs model. In the initial software-based approach for simulations, we performed stability and bifurcation analyses of the trivial equilibrium point, which were analytically conducted, along with the numerical continuation method. We solved the model using digital computation and explored its intricate dynamical behavior through nonlinear dynamics techniques applied to the parameter planes. In the second approach, employing hardware for simulations through analog computation, we designed an electronic circuit using analog electronics to solve the model in a continuous-time integration. The experimental setup mirrored the numerical simulations, yielding comparable results in terms of dynamical behaviors. This study underscores significant agreement between numerical and experimental findings, highlighting the robustness of the model across digital and analog contexts and offering practical insights into the various nonlinear dynamics presented in this system.

Numerical Continuation and Bifurcation for Differential Geometric PDEs

Numerical Continuation and Bifurcation for Differential Geometric PDEs

Effect of the dynamic contact angle on electromagnetically driven flows in free liquid films

We study the effect of a dynamic contact angle on an electromagnetically driven flow in a horizontal free electrolyte film stretched between two coaxial cylindrical electrodes and placed in a uniform magnetic field. The flow dynamics and film deformation are described using the reduced hydrodynamic model derived in the lubrication approximation in [A. Pototsky and S. A. Suslov, J. Fluid Mech., 984, A75 (2024)]. The linearized molecular kinetic model is used to relate the dynamic and static contact angles to the wetting-line friction coefficient. Steady azimuthal flow is found for arbitrary static contact angles. Linear stability of the base azimuthal flow with respect to azimuthally invariant perturbations is studied using the numerical continuation method. We find that the flow stability is highly sensitive to variations of the wetting-line friction coefficient and the static contact angle. The least stable azimuthal flow is found for the frictionless contact line corresponding to a free film that remains perpendicular to the surface of the electrodes. The azimuthal flows are found to experience a supercritical Hopf bifurcation upon which they transition to a stable oscillatory dynamic regime characterized by alternating squeezing and swelling of the film near the inner and outer electrodes.

Spin cones in random-field XY models.

We determine the arrangement of spins in the ground state of the XY model with quenched, random fields, on a fully connected graph. Two types of disordered fields are considered, namely, randomly oriented magnetic fields and randomly oriented crystal fields. Orientations are chosen from a uniformly isotropic distribution, but disorder fluctuations in each realization of a finite system lead to a breaking of rotational symmetry. The result is an interesting pattern of spin orientations found by solving a system of coupled, nonlinear equationswithin perturbation theory and also by exact numerical continuation. All spins lie within a cone for small enough ratio of field to coupling strength, with an interesting distribution of spin orientations, with peaks at the cone edges. The orientation of the cone depends strongly on the realization of disorder, but the opening angle does not. In the case of random magnetic fields, the cone angle widens as the ratio increases till a critical value at which there is a first-order phase transition and the cone disappears. With random crystal fields, there is no phase transition and the cone angle approaches 180^{∘} for large values of the ratio. At finite low temperatures, Monte Carlo simulations show that the formation of a cone and its subsequent alignment along the equilibrium direction occur on two different timescales.

Backbone curve tailoring via Lyapunov subcenter manifold optimization

We present a technique for the direct optimization of conservative backbone curves in nonlinear mechanical systems. The periodic orbits on the conservative backbone are computed analytically using the reduced dynamics of the corresponding Lyapunov subcenter manifold (LSM). In this manner, we avoid expensive full-system simulations and numerical continuation to approximate the nonlinear response. Our method aims at tailoring the shape of the backbone curve using a gradient-based optimization with respect to the system’s parameters. To this end, we formulate the optimization problem by imposing constraints on the frequency-amplitude relation. Sensitivities are computed analytically by differentiating the backbone expression and the corresponding LSM. At each iteration, only the reduced-order model construction and sensitivity computation are performed, making our approach robust and efficient.