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A loss of ground directional stability can trigger a high-speed Unmanned Aerial Vehicle (UAV) to veer off the runway. In order to investigate the combined effects of the key structural and operational parameters on the UAV ground directional stability from a global perspective, a fully parameterized mathematical high-speed UAV ground nonlinear dynamic model is developed considering several nonlinear factors. The bifurcation analysis procedure of a UAV ground steering system is introduced, following which the simulation efficiency is greatly improved comparing with the time-domain simulation method. Then the numerical continuation method is employed to investigate the influence of the nose wheel steering angle and the global stability region is obtained. The bifurcation parameter plane is divided into several parts with different stability properties by the saddle nodes and the Hopf bifurcation points. We find that the UAV motion states will never cross the bifurcation curve in the nonlinear system. Also, the dual-parameter bifurcation analyses are presented to give a complete description of the possible steering performance. It is also found that BT bifurcation appears when the UAV initial rectilinear velocity and the tire frictional coefficient vary. In addition, results indicate that the influence of tire frictional coefficient has an opposite trend to the influence of initial rectilinear velocity. Overall, using bifurcation analysis method to identify the parameter regions of a UAV nonlinear ground dynamic system helps to improve the development efficiency and quality during UAV designing phase.

Nonlinearities in rotating systems have been seen to cause a wide variety of rich phenomena; however, the understanding of these phenomena has been limited because numerical approaches typically rely on “brute force” time simulation, which is slow due to issues of step size and settling time, cannot locate unstable solution families, and may miss key responses if the correct initial conditions are not used. This work uses numerical continuation to explore the responses of such systems in a more systematic way. A simple isotropic rotor system with a smooth nonlinearity is studied, and the rotating frame is used to obtain periodic solutions. Asynchronous responses with oscillating amplitude are seen to initiate at certain drive speeds due to internal resonance, in a manner similar to that observed for nonsmooth rotor–stator contact systems in the previous literature. These responses are isolated, in the sense that they will only meet the more trivial synchronous responses in the limit of zero damping and out of balance forcing. In addition to increasing our understanding of the responses of these systems, the work establishes the potential of numerical continuation as a tool to systematically explore the responses of nonlinear rotor systems.

We develop the Littlewood-Richardson homotopy algorithm, which uses numerical continuation to compute solutions to Schubert problems on Grassmannians and is based on the geometric Littlewood-Richardson rule. One key ingredient of this algorithm is our new optimal formulation of Schubert problems in local Stiefel coordinates as systems of equations. Our implementation can solve problem instances with tens of thousands of solutions., Comment: 27 pages, many figures

A near rectilinear halo orbit (NRHO) is regarded as a potential orbit for a future deep space station that can effectively support sustainable crewed lunar exploration missions. In this paper, the L2 southern NRHOs are selected as the research object, and a direct transfer trajectory from an NRHO to a low lunar orbit (LLO) is designed and analyzed. First, based on the circular restricted three-body problem, the characteristics of NRHOs are discussed including geometric behaviour, stability and synodic resonance. Second, optimal direct transfer trajectories are obtained by combining a local gradient optimization with a numerical continuation strategy. A reachable domain calculation model is established. Finally, simulations are carried out to verify the feasibility of the trajectory design method. The relationship between the velocity change and the reachable domain is further analysed through simulation calculations.

Dynamic behaviours of an epidemic model of discrete time SIR type, have been discussed. The existence and stability conditions of fixed points and some of the codim-1 bifurcations of this model are investigated in [34], but we make bifurcation analysis more general than their work. It is shown that SIR model undergoes codimension 1 (codim 1) bifurcations such as transcritical, flip (period doubling), Neimark-Sacker, and codimension 2 (codim 2) bifurcations including resonance 1:2, resonance 1:3 and resonance 1:4. For each bifurcation, normal form coefficients along with its scenario are investigated thoroughly. Bifurcation curves of fixed points are drawn with the aid of numerical continuation. Besides, using numerical simulation, in addition to confirming the results of our analysis, more behavior is extracted from the model, such as the bifurcations of higher iterations like the fourth, the eight, etc. It is observed that the discrete epidemic model has richer dynamic behaviours compared to the continuous one.

The vibro-impact capsule system has been studied extensively in the past decade because of its research challenges as a piecewise-smooth dynamical system and broad applications in engineering and healthcare technologies. This paper reports our team’s first attempt to scale down the prototype of the vibro-impact capsule to millimetre size, which is 26 mm in length and 11 mm in diameter, aiming for small-bowel endoscopy. Firstly, an existing mathematical model of the prototype and its mathematical formulation as a piecewise-smooth dynamical system are reviewed in order to carry out numerical optimisation for the prototype by means of path-following techniques. Our numerical analysis shows that the prototype can achieve a high progression speed up to 14.4 mm/s while avoiding the collision between the inner mass and the capsule which could lead to less propulsive force on the capsule so causing less discomfort on the patient. Secondly, the experimental rig and procedure for testing the prototype are introduced, and some preliminary experimental results are presented. Finally, experimental results are compared with the numerical results to validate the optimisation as well as the feasibility of the vibro-impact technique for the potential of a controllable endoscopic procedure.

Non-linear bifurcation theory and numerical continuation of bifurcations are important methods to predict the oscillation evolution process of a hydro-turbine governing system. The system's oscillation characteristic is directly related to three factors, namely the generator damping, excitation gain and proportion-integration-differentiation controller. Accordingly, three typical bifurcation and continuation scenarios related to these factors are studied, based on a non-linear dynamical model of the governing system in which the excitation system and the power system stabiliser are included. Some important non-linear dynamic phenomena, such as the equilibrium curves stability, bifurcation points location and limit cycle direction, are exhaustively depicted. Moreover, the dynamic behaviour of the system near bifurcation points is also illustrated through both time-domain simulation results and phase trajectory diagrams. The results show that bifurcations of more and more complicated types are found starting from simple objects like equilibria, which is an important route to study the system's dynamic behaviour. An interesting aspect is that the hydro-turbine governing system exhibits multistability, i.e. for some parameter value sets, there is a non-connected set of stable equilibria. Finally, these results provide a predicted reference for the parameter setting to ensure the stability and safety of the hydro-turbine governing system.

This paper proposes tools from bifurcation theory, specifically numerical continuation, as a complementary method for efficiently mapping the state-parameter space of an internal combustion engine model. Numerical continuation allows a steady-state engine response to be traced directly through the state-parameter space, under the simultaneous variation of one or more model parameters. By applying this approach to two nonlinear engine models (a physics-based model and a data-driven model), this work determines how input parameters ‘throttle position’ and ‘desired load torque’ affect the engine’s dynamics. Performing a bifurcation analysis allows the model’s parameter space to be divided into regions of different qualitative types of the dynamic behaviour, with the identified bifurcations shown to correspond to key physical properties of the system in the physics-based model: minimum throttle angles required for steady-state operation of the engine are indicated by fold bifurcations; regions containing self-sustaining oscillations are bounded by supercritical Hopf bifurcations. The bifurcation analysis of a data-driven engine model shows how numerical continuation could be used to evaluate the efficacy of data-driven models.

This paper studies the problems of stick-slip mitigation and speed tuning for a lumped-parameter drill-string system by using a proportional-derivative feedback controller via path-following analysis. In this study, we consider two main control parameters, the weight-on-bit and the desired drill-bit speed. In particular, we determine the combinations of these two parameters for which the proposed control scheme is applicable, which is affected by the non-smooth nature of the system induced by bit-rock interaction. Our analysis using path-following techniques for non-smooth systems reveals the inherent coexistence of stick-slip vibration and constant rotation, and identifies a critical point where the drill-bit speed coincides with the desired angular speed. Furthermore, our analysis proposes a strategy that allows controlling the drill-bit speed to suppress stick-slip, by tuning the controller in a suitable manner.

The aim of this paper is to develop an efficient and robust parallel numerical continuation method for periodic solutions of local nonlinear systems. In this method, a component mode synthesis method is first employed to reduce the nonlinear system with local nonlinearity; the parallel principle is implemented in the correction part of the classic pseudo-arc length continuation, and the step control strategy is dependent on historical and current information from multiple cores. In addition, a numerical continuation (NC) jumping phenomenon may occur alongside the periodic solution tracking process; a geometrical indicator algorithm based on the frequency–amplitude curve is utilized to prevent NC jumping and enhance the robustness of the tracking behavior. By applying the methodology to a nonlinear energy sink and a rod-fastened rolling bearing rotor system, it is shown that numerical continuation can be employed to calculate periodic solutions efficiently and robustly.

In case of a spin-polarized current, the magnetization dynamics in nanowires are governed by the classical Landau-Lifschitz equation with Gilbert damping term, augmented by a typically non-variational Slonczewski term. Taking axial symmetry into account, we study the existence of domain wall type coherent structure solutions, with focus on one space dimension and spin-polarization, but our results also apply to vanishing spin-torque term. Using methods from bifurcation theory for arbitrary constant applied fields, we prove the existence of domain walls with non-trivial azimuthal profile, referred to as inhomogeneous. We present an apparently new type of domain wall, referred to as non-flat, whose approach of the axial magnetization has a certain oscillatory character. Additionally, we present the leading order mechanism for the parameter selection of flat and non-flat inhomogeneous domain walls for an applied field below a threshold, which depends on anisotropy, damping, and spin-transfer. Moreover, numerical continuation results of all these domain wall solutions are presented., Comment: 40 pages, 14 figures

We propose a numerical continuation method for a nonlinear system that consists of scalar and functional equations. At each parameter step, we solve the system by a modified Gauss–Seidel method. An advantage of this method is that the system is divided into two parts and each part is solved by a suitable numerical method with a desired precision. We solve the functional equations self-consistently at each step of the iteration process for the system of scalar equations. We apply the proposed method for calculating temperature dependence of magnetic characteristics of metals in the dynamic spin-fluctuation theory.

In this article, nonlinear forced response of dynamical systems is studied using numerical continuation methods. Several methods are available to calculate nonlinear normal modes. Along with the existing analytical methods, recently, numerical methods, especially the pseudo-arclength continuation method, have attracted many researchers. The pseudo-arclength continuation method is a very powerful method which is capable of handling strongly nonlinear systems. However, as mentioned in recently published article reviews, the computational cost of the method has limited its application. In this research, an updating formula is embedded in the pseudo-arclength continuation algorithm to reduce the computational cost. This modified method is called the efficient path-following method. The assumptions and basis of the efficient path-following method algorithm are same as those presented in other references, but none of them have exploited the updating formula of the efficient path-following method to study the forced response of nonlinear dynamical systems. To investigate the capabilities of the method, forced response of a single-degree-of-freedom Duffing system is computed. It is seen that the efficient path-following method has decreased the computational time significantly up to 70%. The results are in very good conformance with those obtained in other references, which shows the accuracy of this method. To study the ability of the efficient path-following method to handle the multi-degree-of-freedom system, a four-degree-of-freedom nonlinear system is considered, and stable and unstable branches of the solution are computed. It is observed that as the nonlinearity of the system gets stronger, the updating formula becomes more effective.

In this paper we compute the linear stability of similarity solutions of the breakup of viscous liquid threads, in which the viscosity and inertia of the liquid are in balance with the surface tension. The stability of the similarity solution is determined using numerical continuation to find the dominant eigenvalues. Stability of the first two solutions (those with largest minimum radius) is considered. We find that the first similarity solution, which is the one seen in experiments and simulations, is linearly stable with a complex nontrivial eigenvalue, which could explain the phenomenon of break-up producing sequences of small satellite droplets of decreasing radius near a main pinch-off point. The second solution is seen to be linearly unstable. These linear stability results compare favorably to numerical simulations for the stable similarity solution, while a profile starting near the unstable similarity solution is shown to very rapidly leave the linear regime., 14 pages, 8 figures. Accepted version

A variational model describing the nonlinear mode interaction in thin-walled box-section struts under pure axial compression made from a nonlinear material obeying the Ramberg–Osgood law is presented. The formulation combines continuous displacement functions and generalized coordinates, leading to the derivation of a system of differential and integral equations that describe the static equilibrium response of the strut. Solving the system of equations using numerical continuation techniques reveals the strongly unstable post-buckling response arising from combined geometrical and material nonlinearities during the interactive buckling of the global and local buckling modes—the resulting behaviour being more unstable with decreasing material hardening. A finite element (FE) model is also devised and reveals very similar post-buckling behaviour as highlighted in the variational model. The results compare very well in terms of the mechanical destabilization and the post-buckling deformation, which verifies the analytical model.

We herein propose an algorithm for tracking smooth bifurcations of nonlinear systems with interdependent parameters. The approach is based on a complex formulation of the well-known harmonic balance method (HBM). Hill’s method is used to assess the stability of the computed forced response curves, and a minimally extended system is built to allow for the parametric continuation of the detected bifurcation points. The feasibility of coupling HBM-based minimally extended systems and arc-length continuation algorithms is established and demonstrated. The method offers an efficient way of determining the stability regions of the system. The methodology is applied on a spur gear pair model including the backlash nonlinearity and subjected to transmission error and mesh stiffness fluctuation whose harmonic contents depend on several parameters that do not appear explicitly in the equations of motion.

A time-delayed SIR model with general nonlinear incidence rate, pulse vaccination and temporary immunity is developed. The basic reproduction number is derived and it is shown that the disease-free periodic solution generically undergoes a transcritical bifurcation to an endemic periodic solution as the vaccination coverage drops below a critical level. Using numerical continuation and a monodromy operator discretization scheme, we track the bifurcating endemic periodic solution as the vaccination coverage is decreased and a Hopf point is detected. This leads to a bifurcation to an attracting, invariant cylinder. As the vaccination coverage is further decreased, the geometry of the cylinder contracts along its length until it finally collapses to a periodic orbit when the vaccination coverage goes to zero. In the intermediate regime, phase locking on the cylinder is observed.

The generalized lumped-parameter model of the drill-string system is studied in this paper to provide a fundamental understanding of the torsional stick-slip vibrations in downhole drilling. Our investigation focuses on analysing the cause of three coexisting states: bit sticking, stick-slip vibration, and constant rotation. A critical region of multistability is identified based on the lumped-parameter model, and the conditions for switching between these multiple stable states are discussed. Special attention is given to the bifurcation structure of the considered drill-string model, which is obtained via path-following methods for nonsmooth dynamical systems. The bifurcation scenario is compared to the case when a longer drill-string is considered, which amounts to drilling deeper. It is found that the main features of the bifurcation picture persist under variation of the drill-string length, with certain numerical differences regarding for instance the window of multistability.

This paper deals with a modified version of the Circular Restricted Three-Body Problem (CR3BP). In this version, the additional effect of a three-body interaction is taken into account. In particular, we examine numerically the result of this interaction on the evolution of the well-known family of Sitnikov motion of CR3BP as well as that on the families of 3D periodic orbits bifurcating from this family.

Control-based continuation (CBC) is a general and systematic method to probe the dynamics of nonlinear experiments. In this paper, CBC is combined with a novel continuation algorithm that is robust to experimental noise and enables the tracking of geometric features of the response surface such as folds. The method uses Gaussian process regression to create a local model of the response surface on which standard numerical continuation algorithms can be applied. The local model evolves as continuation explores the experimental parameter space, exploiting previously captured data to actively select the next data points to collect such that they maximise the potential information gain about the feature of interest. The method is demonstrated experimentally on a nonlinear structure featuring harmonically-coupled modes. Fold points present in the response surface of the system are followed and reveal the presence of an isola, i.e. a branch of periodic responses detached from the main resonance peak., Comment: 22 pages, 12 figures

To predict the coning motion forms of a rolling projectile with configurational asymmetries, the nonlinear characteristics for the system are investigated in this paper. The nonlinear dynamic model of rolling projectiles in coning motion is built by considering the nonlinear aerodynamics and roll orientation-dependent aerodynamics. The configurational asymmetry is modeled as a periodically parametric excitation in order to study its effect on the periodic response stability of the rolling projectile. Numerical continuation method is resorted to determine the parametric zone for the steady motions, and the possible stable rotational speeds are discussed. The numerical simulations, Lyapunov exponent spectrum analysis and Poincare sections are performed to confirm the existence of chaotic coning motion. The results shown in this study not only contribute to an in-depth understanding for the nonlinear dynamics of rolling projectiles but also provide an important reference for the further study of the control design for the yaw–pitch–roll coupling of asymmetric rolling projectiles with nonlinear aerodynamics.

The homotopy methods have long served as powerful tools in solving nonlinear optimal control problems, particularly for which the solutions are highly sensitive to the unknown initial conditions. The principle of homotopy methods is that a homotopic parameter is embedded into the formulations of the optimal control problems, and the original problem can be solved by tracing the optimal solutions of the embedded problems. The existing homotopy methods typically introduce the homotopic parameter into the time variable, the necessary conditions, the performance index or the rightside of the differential equations. In this paper, a new fractional homotopy method is presented, the homotopic parameter of which is embedded into the derivative of the differential equations. By using the proposed method, the optimal solution of the target homotopy problem can be found by solving a series of fractional two-point-boundary-value-problems. Numerical demonstrations in a nonlinear optimal control problem and a three-dimensional minimum-time low-thrust orbital transfer problem are presented to illustrate the applications of the method.

We investigate large amplitude steady water waves in the presence of underlying uniform current with u≠c using a numerical continuation method. The two branches of traveling wave solutions, whose e...

In this paper, we present an algorithm based on the nonlinear time transformation method to approximate homoclinic orbits in planar autonomous nonlinear oscillators. With this approach, a unique perturbation solution up to any desired order can be obtained for them using trigonometric functions. To demonstrate its efficiency, the method is applied to calculate the homoclinic connection, both in the phase space and in the parameter space, of the versal unfolding of the nondegenerate Takens–Bogdanov singularity. Our approach considerably improves the results obtained so far by other methods (Melnikov, Poincare–Lindstedt, regular perturbations, multiple scales, etc.). The approximations achieved to different orders are confirmed by numerical continuation.

A family of planar nilpotent reversible systems with an equilibrium point located at the origin has been studied in the recent paper Algaba et al. (Nonlinear Dyn 87:835–849, 2017). The authors investigate the candidate for an universal unfolding of a codimension-three degenerate case which exhibits a rich bifurcation scenario. However, a codimension-two point is missed in one of the two cases considered. In this paper, we complete the bifurcation set demonstrating the existence of this new organizing center and analyzing the dynamics generated in this case. Moreover, by means of the Melnikov theory, we study analytically four different global connections present in the system under consideration. Numerical continuation of the bifurcation curves illustrates that the first-order analytical approximation is valid in a large region of the parameter space.

We examine doing probabilistic descent over manifolds implicitly defined by a set of polynomials with rational coefficients. The system of polynomials is assumed to be triangularized. An application of Whitney’s embedding theorem allows us to work in a reduced-dimensional embedding space. A numerical continuation method applied to the reduced-dimensional embedded manifold is used to drive the procedure.

In this contribution, heterogeneous connected vehicle systems, that include human-driven as well as connected automated vehicles, are investigated. The reaction time delay of human drivers as well as the communication and actuation delays of connected automated vehicles is incorporated in the model. Saturations due to the speed limit and limited acceleration capabilities of the vehicles are also taken into account. The arising nonlinear delayed system is studied using analytical and numerical bifurcation analysis. Stability analysis is used to identify regions in parameter space where oscillations arise due to loss of linear stability of the equilibrium. Moreover, with the help of numerical continuation, bistability between the equilibrium and oscillations is shown to exist due to the presence of an isola. It is demonstrated that utilizing long-range wireless vehicle-to-vehicle communication the connected automated vehicle is able to eliminate the oscillations and keep the traffic flow smooth.

A procedure named parameter-splitting perturbation method for improving the perturbation solutions to the forced vibrations of strongly nonlinear oscillators is proposed. The idea of the proposed procedure is presented in general first. After that, it is applied to optimize the solutions obtained by the multiple-scales method which is one of well-known perturbation methods. The harmonically forced Duffing oscillator, the harmonically forced oscillator with both nonlinear restoring force and nonlinear inertial force, the harmonically forced purely nonlinear oscillator and harmonically forced two-degree-of-freedom system with cubic nonlinearity are analyzed in various cases to show the advantages of the proposed method. The ratio of nonlinear stiffness coefficient to linear stiffness coefficient is chosen to be larger than one to highlight the validity of the propose method when dealing with strongly nonlinear oscillators. The validity of the proposed procedure is examined by comparing the frequency–response curves obtained by the proposed method, conventional multiple-scales method and numerical continuation method. Moreover, the errors corresponding to the results obtained by multiple-scales method are compared with those obtained by the proposed method to examine the performance of the proposed method. The results show that the proposed method can give much improved solutions in comparison with those obtained by multiple-scales method.

Effects of $$\mathbb {Z}_n$$ -symmetry, ( $$n\ge 2$$ ), on normal form of Neimark–Sacker bifurcation in discrete time dynamical systems are investigated. As an application, we consider three dimensional discrete Hopfield neural network with $$\mathbb {Z}_2$$ -symmetry. We drive analytical conditions for stability and bifurcations of the trivial fixed point of the system and compute analytically the normal form coefficients for the codimension 1 and codimension 2 bifurcation points including pitchfork, period-doubling, Neimark–Sacker, $$\mathbb {Z}_2$$ -symmetric Neimark–Sacker and resonance 1:4. By using numerical continuation in numerical software matcontm, we compute bifurcation curves of trivial fixed point and cycle with period 4 under variation of one and two parameters, and all codimension 1 and codimension 2 bifurcations supported by matcontm, on the corresponding curves are computed.

International audience; Tuned dynamic absorbers are usually used to counteract vibrations at a given frequency. Presence of non-linearities causes energy-dependent relationship of their resonance and antiresonance frequencies at large amplitude of motion, which consequently leads to adetuning of the absorber from the targeted frequency. This paper presents a procedure to track an extremum point (minimum or maximum) of nonlinear frequency responses, based on a numerical continuation technique coupled to the harmonic balance method to follow periodic solutions in forced steady-state. It thus enable to track a particularantiresonance. The procedure is tested and applied on some application cases to highlight the resonance and antiresonance behavior in presence of geometrically non-linear and/or inertial interactions.

Persistently infecting viruses remain within infected cells for a prolonged period of time without killing the cells and can reproduce via budding virus particles or passing on to daughter cells after division. The ability for populations of infected cells to be long-lived and replicate viral progeny through cell division may be critical for virus survival in examples such as HIV latent reservoirs, tumor oncolytic virotherapy, and non-virulent phages in microbial hosts. We consider a model for persistent viral infection within a replicating cell population with time delay in the eclipse stage prior to infected cell replicative form. We obtain reproduction numbers that provide criteria for the existence and stability of the equilibria of the system and provide bifurcation diagrams illustrating transcritical (backward and forward), saddle-node, and Hopf bifurcations, and provide evidence of homoclinic bifurcations and a Bogdanov-Takens bifurcation. We investigate the possibility of long term survival of the infection (represented by chronically infected cells and free virus) in the cell population by using the mathematical concept of robust uniform persistence. Using numerical continuation software with parameter values estimated from phage-microbe systems, we obtain two parameter bifurcation diagrams that divide parameter space into regions with different dynamical outcomes. We thus investigate how varying different parameters, including how the time spent in the eclipse phase, can influence whether or not the virus survives.

We propose a new two-parameter family of hybrid traveling-standing (TS) water waves in infinite depth that evolve to a spatial translation of their initial condition at a later time. We use the square root of the energy as an amplitude parameter and introduce a traveling parameter that naturally interpolates between pure traveling waves moving in either direction and pure standing waves in one of four natural phase configurations. The problem is formulated as a two-point boundary value problem and a quasi-periodic torus representation is presented that exhibits TS-waves as nonlinear superpositions of counter-propagating traveling waves. We use an overdetermined shooting method to compute nearly 50,000 TS-wave solutions and explore their properties. Examples of waves that periodically form sharp crests with high curvature or dimpled crests with negative curvature are presented. We find that pure traveling waves maximize the magnitude of the horizontal momentum among TS-waves of a given energy. Numerical evidence suggests that the two-parameter family of TS-waves contains many gaps and disconnections where solutions with the given parameters do not exist. Some of these gaps are shown to persist to zero-amplitude in a fourth-order perturbation expansion of the solutions in powers of the amplitude parameter. Analytic formulas for the coefficients of this perturbation expansion are identified using Chebyshev interpolation of solutions computed in quadruple-precision.

Optomechanical systems are known to exhibit a rich set of complex dynamical features including various types of chaotic behavior and multi-stability. Although this exotic behavior has attracted an intense research interest, the utilization of optomechanical systems in technological applications, in most cases necessitates a complex, yet predictable and controllable, oscillatory response. In fact, the various types of robust oscillations supported by optomechanical systems are nested in either the same or neighboring regions of the parameter space, where chaos exists. In this work we systematically dissect the parameter space of the fundamental optomechanical oscillator in order to identify regions where stable self-sustained and self-modulated oscillations exist, by utilizing bifurcation analysis and advanced numerical continuation techniques. Moreover,in cases where bistability occurs, we study the accessibility of these oscillatory states in terms of initial conditions and their location with respect to well-defined basins of attraction. The results provide specific knowledge for the parameter sets enabling the appropriate oscillatory response for different types of applications., 21 pages, 6 figures

The present work studies the continuation class of the regular n-gon solution of the n-body problem. For odd numbers of bodies between $$n = 3$$ and $$n = 15$$ , we apply one parameter numerical continuation algorithms to the energy/frequency variable and find that the figure eight choreography can be reached starting from the regular n-gon. The continuation leaves the plane of the n-gon and passes through families of spatial choreographies with the topology of torus knots. Numerical continuation out of the n-gon solution is complicated by the fact that the kernel of the linearization there is high dimensional. Our work exploits a symmetrized version of the problem which admits dense sets of choreography solutions and which can be written as a delay differential equation in terms of one of the bodies. This symmetrized setup simplifies the problem in several ways. On the one hand, the direction of the kernel is determined automatically by the symmetry. On the other hand, the set of possible bifurcations is reduced and the n-gon continues to the eight after a single symmetry breaking bifurcation. Based on the calculations presented here, we conjecture that the n-gon and the eight are in the same continuation class for all odd numbers of bodies.

International audience; This article presents a Harmonic Balance Method-based numerical strategy to provide a qualitative numerical characterization of a compressor blade's dynamics when structural contacts occur with the surrounding casing. The mitigation of the Gibbs phenomenon follows a two-pronged approach: (1) a regularization of the unilateral contact law and (2) the adjustment of the Fourier coefficients by means of a Lanczos filtering technique. In order to validate the proposed approach, it is first applied to an academic cantilever rod undergoing unilateral contact constraints. An in-depth comparative analysis of the obtained results-with an emphasis on displacements, contact forces and velocities-with respect to a time integration-based reference numerical strategy underlines the relevance and accuracy of the proposed methodology. The latter is then applied to the vibration analysis of an industrial compressor blade: NASA rotor 37. For a given contact configuration, obtained results are thoroughly compared to those obtained with a previously published time integration-based numerical strategy with a distinct contact treatment algorithm. Particular attention is paid to demonstrate the accuracy of the methodology for the prediction of displacements, contact forces, velocities as well as stress fields within the blade. Notably, it is evidenced that the proposed methodology, contrary to the reference time integration-based numerical strategy, is able to capture the exact location of the blade's nonlinear resonance.; Cet article présente une stratégie numérique basée sur la méthode de l'équilibrage harmonique visant à fournir une caractérisation numérique qualitative de la dynamique d'une aube de compresseur lorsque des contacts structurels se produisent avec le carter environnant. L'atténuation du phénomène de Gibbs repose sur une double approche~: (1) une régularisation de la loi de contact unilatéral et (2) l'ajustement des coefficients de Fourier au moyen d'une technique de filtrage de Lanczos. Afin de valider l'approche proposée, celle-ci est d'abord appliquée à un modèle académique de barre encastrée/libre soumise à des contraintes de contact unilatéral. Une analyse comparative approfondie des résultats obtenus--mettant l'accent sur les déplacements, les efforts de contact et les vitesses--avec une stratégie numérique de référence basée sur l'intégration temporelle souligne la pertinence et la précision de la méthodologie proposée. Cette dernière est ensuite appliquée à l'analyse des vibrations d'une aube de compresseur industriel~: le rotor 37 de la NASA. Pour une configuration de contact donnée, les résultats obtenus sont minutieusement comparés à ceux obtenus à l'aide d'une stratégie numérique de référence basée sur l'intégration temporelle publiée antérieurement, utilisant notamment un algorithme de gestion du contact distinct. Une attention particulière est accordée à la démonstration de la précision de la méthodologie en termes de prédiction des déplacements, des efforts de contact, des vitesses ainsi que des champs de contrainte à l'intérieur de l'aube. Il est notamment démontré que la méthodologie proposée, à la différence de la stratégie numérique de référence, est capable de localiser exactement la résonance non-linéaire de l'aube.

In this study, we explore the mechanics of a bigon and a bigon ring from a combination of experiments and numerical simulations. A bigon is a simple elastic network consisting of two initially straight strips that are deformed to intersect with each other through a fixed intersection angle at each end. A bigon ring is a novel multistable structure composed of a series of bigons arranged to form a loop. We find that a bigon ring usually contains several families of stable states and one of them is a multiply-covered loop, which is similar to the folding behavior of a bandsaw blade. To model bigons and bigon rings, we propose a numerical framework combining several existing techniques to study mechanics of elastic networks consisting of thin strips. Each strip is modeled as a Kirchhoff rod, and the entire strip network is formulated as a two-point boundary value problem (BVP) that can be solved by a general-purpose BVP solver. Together with numerical continuation, we apply the numerical framework to study static equilibria and bifurcations of the bigons and bigon rings. Both numerical and experimental results show that the intersection angle and the aspect ratio of the strip’s cross section contribute to the bistability of a bigon and the multistability of a bigon ring; the latter also depends on the number of bigon cells in the ring. The numerical results further reveal interesting connections among various stable states in a bigon ring. Our numerical framework can be applied to general elastic rod networks that may contain flexible joints, naturally curved strips of different lengths, etc. The folding and multistable behaviors of a bigon ring may inspire the design of novel deployable and morphable structures.

The lowest fingerings of the saxophone can lead to several different regimes, depending on the musician’s control and the characteristics of the instrument. This is explored in this paper through a physical model of saxophone. The harmonic balance method shows that for many combinations of musician control parameters, several regimes are stable. Time-domain synthesis is used to show how different regimes can be selected through initial conditions and the initial evolution (rising time) of the blowing pressure, which is explained by studying the attraction basin of each stable regime. These considerations are then applied to study how the produced regimes are affected by properties of the resonator. The inharmonicity between the first two resonances is varied in order to find the value leading to the best suppression of unwanted overblowing. Overlooking multistability in this description can lead to biased conclusions. Results for all the lowest fingerings show that a slightly positive inharmonicity, close to that measured on a saxophone, leads to first register oscillations for the greatest range of control parameters. A perfect harmonicity (integer ratio between the first two resonances) decreases first register production, which adds nuance to one of Benade’s guidelines for understanding sound production. Thus, this study provides some a posteriori insight into empirical design choices relative to the saxophone.

The active phase-field-crystal (active PFC) model provides a simple microscopic mean field description of crystallization in active systems. It combines the PFC model (or conserved Swift-Hohenberg equation) of colloidal crystallization and aspects of the Toner-Tu theory for self-propelled particles. We employ the active PFC model to study the occurrence of localized and periodic active crystals in two spatial dimensions. Due to the activity, crystalline states can undergo a drift instability and start to travel while keeping their spatial structure. Based on linear stability analyses, time simulations and numerical continuation of the fully nonlinear states, we present a detailed analysis of the bifurcation structure of resting and traveling states. We explore, for instance, how the slanted homoclinic snaking of steady localized states found for the passive PFC model is modified by activity. The analysis is carried out for the model in two spatial dimensions. Morphological phase diagrams showing the regions of existence of various solution types are presented merging the results from all the analysis tools employed. We also study how activity influences the crystal structure with transitions from hexagons to rhombic and stripe patterns. This in-depth analysis of a simple PFC model for active crystals and swarm formation provides a clear general understanding of the observed multistability and associated hysteresis effects, and identifies thresholds for qualitative changes in behavior.

This paper presents different families of transfers from Low Lunar Orbit to Distant Retrograde Orbits in the Earth-Moon system and analyzes their characteristics from the perspective of velocity increment requirement, transfer time, and sensitivity to small perturbations. Initially, families of transfer trajectories in the Circular Restricted Three-Body Problem are found by the predictor-corrector method and numerical continuation. Each transfer uses two impulsive maneuvers. These families are computed for different parking orbit altitudes, constants of motion, and insertion points. Then, to investigate the sensitivity characteristics of each family, a metric based on Local Lyapunov Exponent is introduced and computed along each trajectory. This work provides various choices for future cargo and human-crewed missions in the cislunar space.

We show that one can employ well-established numerical continuation methods to efficiently calculate the phase diagram for thermodynamic systems described by a suitable free energy functional. In particular, this involves the determination of lines of phase coexistence related to first order phase transitions and the continuation of triple points. To illustrate the method we apply it to a binary phase-field-crystal model for the crystallisation of a mixture of two types of particles. The resulting phase diagram is determined for one- and two-dimensional domains. In the former case it is compared to the diagram obtained from a one-mode approximation. The various observed liquid and crystalline phases and their stable and metastable coexistence are discussed as well as the temperature-dependence of the phase diagrams. This includes the (dis)appearance of critical points and triple points. We also relate bifurcation diagrams for finite-size systems to the thermodynamics of phase transitions in the infinite-size limit.

Hysteresis dynamics has been described in a vast number of biological experimental studies. Many such studies are phenomenological and a mathematical appreciation has not attracted enough attention. In the paper, we explore the nature of hysteresis and study it from the dynamical system point of view by using the bifurcation and perturbation theories. We firstly make a classification of hysteresis according to the system behaviours transiting between different types of attractors. Then, we focus on a mathematically amenable situation where hysteretic movements between the equilibrium point and the limit cycle are initiated by a subcritical Hopf bifurcation and a saddle-node bifurcation of limit cycles. We present a analytical framework by using the method of multiple scales to obtain the normal form up to the fifth order. Theoretical results are compared with time domain simulations and numerical continuation, showing good agreement. Although we consider the time-delayed FitzHugh-Nagumo neural system in the paper, the generalization should be clear to other systems or parameters. The general framework we present in the paper can be naturally extended to the notion of bursting activity in neuroscience where hysteresis is a dominant mechanism to generate bursting oscillations.

Many dynamic processes involve time delays, thus their dynamics are governed by delay differential equations (DDEs). Studying the stability of dynamic systems is critical, but analyzing the stability of time-delay systems is challenging because DDEs are infinite-dimensional. We propose a new approach to quickly generate stability charts for DDEs using continuation of characteristic roots (CCR). In our CCR method, the roots of the characteristic equation of a DDE are written as implicit functions of the parameters of interest, and the continuation equations are derived in the form of ordinary differential equations (ODEs). Numerical continuation is then employed to determine the characteristic roots at all points in a parametric space; the stability of the original DDE can then be easily determined. A key advantage of the proposed method is that a system of linearly independent ODEs is solved rather than the typical strategy of solving a large eigenvalue problem at each grid point in the domain. Thus, the CCR method can significantly reduce the computational effort required to determine the stability of DDEs. As we demonstrate with several examples, the CCR method generates highly accurate stability charts, and does so up to 10 times faster than the Galerkin approximation method.

International audience; The system formed by a trumpet player and his/her instrument can be seen as a non-linear dynamical system, and modeled by physical equations. Numerical tools can then be used to study these models and clarify the influence of the model parameters. The acoustic input impedance, for instance, is strongly dependent on the geometry of the air column and is therefore of primary interest for a musical instrument maker. In this study, a method of continuation of periodic solutions based on the combination of the Harmonic Balance Method (HBM) and the Asymptotic Numerical Method (ANM), is applied to a physical model of brass instruments. It allows the study of the evolution of the system where one parameter of the model (static mouth pressure) varies. This method is used to compare different B trumpets on the basis of two descriptors (hysteresis behavior and dynamic range) computed from the continuation outputs. Results show that this methodology enables to differentiate instruments in the space of the calculated descriptors. Calculations for different values of the lip parameters are also performed to confirm that the obtained categorization is independent of variations of lip parameters.

This article proposes a framework which allows the study of stability robustness of equilibria of a nonlinear system in the face of parametric uncertainties from the point of view of bifurcation theory. In this context, a branch of equilibria is stable if bifurcations (i.e., qualitative changes of the steady-state solutions) do not occur as one or more bifurcation parameters are varied. The work focuses specifically on Hopf bifurcations, where a stable branch of equilibria meets a branch of periodic solutions. It is of practical interest to evaluate how the presence of uncertain parameters in the system alters the result of analyses performed with respect to a nominal vector field. Note that in this article bifurcation parameters have a different meaning than uncertain parameters. To answer the question, the concept of robust bifurcation margins is proposed based on the idea of describing the uncertain system in a Linear Fractional Transformation fashion. The robust bifurcation margins can be interpreted as nonlinear analogues of the structural singular value, or $\mu$, which provides robust stability margins for linear time invariant systems. Their computation is formulated as a nonlinear program aided by a continuation-based multistart strategy to mitigate the issue of local minima. Application of the framework is demonstrated on two case studies from the power system and aerospace literature., SIAM Journal on Applied Dynamical Systems, 19 (3), ISSN:1536-0040

We derive bifurcation test equations for A-series singularities of nonlinear functionals and, based on these equations, we propose a numerical method for detecting high codimensional bifurcations in parameter-dependent PDEs such as parameter-dependent semilinear Poisson equations. As an example, we consider a Bratu-type problem and show how high codimensional bifurcations such as the swallowtail bifurcation can be found numerically. In particular, our original contributions are (1) the use of the Infinite-Dimensional Splitting Lemma, (2) the unified and simplified treatment of all A-series bifurcations, (3) the presentation in Banach spaces, i.e. our results apply both to the PDE and its (variational) discretization, (4) further simplifications for parameter-dependent semilinear Poisson equations (both continuous and discrete), and (5) the unified treatment of the continuous problem and its discretisation.

We consider the Yamada model for an excitable or self-pulsating laser with saturable absorber, and study the effects of delayed optical self-feedback in the excitable case. More specifically, we are concerned with the generation of stable periodic pulse trains via repeated self-excitation after passage through the delayed feedback loop, as well as their bifurcations. We show that onset and termination of such pulse trains correspond to the simultaneous bifurcation of countably many fold periodic orbits with infinite period in this delay differential equation. We employ numerical continuation and the concept of reappearance of periodic solutions to show that these bifurcations coincide with codimension-two points along families of connecting orbits and fold periodic orbits in a related advanced differential equation. These points include heteroclinic connections between steady states, as well as homoclinic bifurcations with non-hyperbolic equilibria. Tracking these codimension-two points in parameter space reveals the critical parameter values for the existence of periodic pulse trains. We use the recently developed theory of temporal dissipative solitons to infer necessary conditions for the stability of such pulse trains., 31 pages, 7 figures

The widely cited Haken–Kelso–Bunz (HKB) model of motor coordination is used in an enormous range of applications. In this paper, we show analytically that the weakly damped, weakly coupled HKB model of two oscillators depends on only two dimensionless parameters; the ratio of the linear damping coefficient and the linear coupling coefficient and the ratio of the combined nonlinear damping coefficients and the combined nonlinear coupling coefficients. We illustrate our results with a mechanical analogue. We use our analytic results to predict behaviours in arbitrary parameter regimes and show how this led us to explain and extend recent numerical continuation results of the full HKB model. The key finding is that the HKB model contains a significant amount of behaviour in biologically relevant parameter regimes not yet observed in experiments or numerical simulations. This observation has implications for the development of virtual partner interaction and the human dynamic clamp, and potentially for the HKB model itself.

The passive conserved Swift-Hohenberg equation (or phase-field-crystal [PFC] model) corresponds to a gradient dynamics for a single order parameter field related to density. It provides a simple microscopic description of the thermodynamic transition between liquid and crystalline states. In addition to spatially extended periodic structures, the model describes a large variety of steady spatially localized structures. In appropriate bifurcation diagrams the corresponding solution branches exhibit characteristic slanted homoclinic snaking. In an active PFC model, encoding for instance the active motion of self-propelled colloidal particles, the gradient dynamics structure is broken by a coupling between density and an additional polarization field. Then, resting and traveling localized states are found with transitions characterized by parity-breaking drift bifurcations. Here, we first briefly review the snaking behavior of localized states in passive and active PFC models before discussing the bifurcation behavior of localized states in systems of (i) two coupled passive PFC equations described by common gradient dynamics, (ii) two coupled passive PFC where the coupling breaks the gradient dynamics structure, and (iii) a passive PFC coupled to an active PFC., Comment: submitted to the IMA Journal of Applied Mathematics' Special Issue on Homoclinic Snaking at 21, in memory of Patrick Woods

Motivated by the inadequacy of conducting atomistic simulations of crack propagation using static boundary conditions that do not reflect the movement of the crack tip, we extend Sinclair's flexible boundary condition algorithm [Philos. Mag. 31, 647-671 (1975)] and propose a numerical-continuation-enhanced flexible boundary (NCFlex) scheme, enabling full solution paths for cracks to be computed with pseudo-arclength continuation, and present a method for incorporating more detailed far-field information into the model for next to no additional computational cost. The new algorithms are ideally suited to study details of lattice trapping barriers to brittle fracture and can be incorporated into density functional theory and multiscale quantum/classical QM/MM calculations. We demonstrate our approach for Mode III fracture with a 2D toy model and mploy it to conduct a 3D study of Mode I fracture of silicon using realistic interatomic potentials, highlighting the superiority of the new approach over employing a corresponding static boundary condition. In particular, the inclusion of numerical continuation enables converged results to be obtained with realistic model systems containing a few thousand atoms, with very few iterations required to compute each new solution. We also introduce a method to estimate the lattice trapping range of admissible stress intensity factors $K_- < K < K_+$ very cheaply and demonstrate its utility on both the toy and realistic model systems., 16 pages, 13 figures