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Summary: We present RCO (regularized Cholesky optimization): a numerical algorithm for finding a symmetric positive definite (PD) n×n$$ n\times n $$ matrix with a bounded condition number that minimizes an objective function. This task arises when estimating a covariance matrix from noisy data or due to model constraints, which can cause spurious small negative eigenvalues. A special case is the problem of finding the nearest well‐conditioned PD matrix to a given matrix. RCO explicitly optimizes the entries of the Cholesky factor. This requires solving a regularized non‐linear, non‐convex optimization problem, for which we apply Newton‐CG and exploit the Hessian's sparsity. The regularization parameter is determined via numerical continuation with an accuracy‐conditioning trade‐off criterion. We apply RCO to our motivating educational measurement application of estimating the covariance matrix of an empirical best linear prediction (EBLP) of school growth scores. We present numerical results for two empirical datasets, state and urban. RCO outperforms general‐purpose near‐PD algorithms, obtaining 10×$$ 10\times $$‐smaller matrix reconstruction bias and smaller EBLP estimator mean‐squared error. It is in fact the only algorithm that solves the right minimization problem, which strikes a balance between the objective function and the condition number. RCO can be similarly applied to the stable estimation of other posterior means. For the task of finding the nearest PD matrix, RCO yields similar condition numbers to near‐PD methods, but provides a better overall near‐null space. [ABSTRACT FROM AUTHOR]

The paper utilizes a novel approach for the dynamic design of automotive turbocharger rotors by employing nonlinear dynamics of time periodic systems, emphasizing the influence of bearing design variables to prevent sub-synchronous components in system's response. The investigation focuses on the response of an unbalanced turbocharger rotor on floating ring bearings using a collocation-type method which has been developed for the needs of the work, being then integrated with pseudo arc length continuation for the calculation of unstable solution branches of the system, in several design sets, and with poor initial values. The analytical model includes a rigid rotor model and short bearing approximations for the two floating ring bearings, which introduce strong nonlinear forces in series (inner film and outer film at each bearing). Floquet theory is employed to analyse the non-autonomous dynamic system, and stability characteristics of the response limit cycles are assessed through Floquet multipliers, whose magnitude serves as a stability index in the algorithm. A genetic algorithm based multi-objective optimization is combined to the robust collocation-type method to achieve reduced values for Floquet multipliers, ensuring that response limit cycles maintain stability and periodicity, thereby preventing the occurrence of bifurcations which normally lead to sub-synchronous response components. Twelve design variables are computed to satisfy low rotor eccentricity and power loss. Acceptable design sets are verified for efficacy by assessing system response through time integration, akin to a virtual experiment. This approach significantly reduces computational time and resource requirements compared to traditional Design of Experiment (DoE) procedures and is not constrained by complex models of the rotor, bearings, or other components. A feature of the method is that it offers an insight on the stability and its quality, rather than simply assessing a threshold speed of instability. [ABSTRACT FROM AUTHOR]

This paper examines a three-dimensional delayed discrete neural network model analytically and numerically to determine the existence of different types of bifurcations of the involved fixed points. The model exhibits different bifurcations such as pitchforks, flips, Neimark–Sackers, and flip-Neimark–Sackers. The critical coefficients are used to determine the structure of each bifurcation. The curves are calculated and plotted for each bifurcation when the parameters are changed. Further, these bifurcations are theoretically analyzed and numerically verified. From the obtained results, we observed that by drawing the curves associated with each bifurcation, the numerical simulations are consistent with the analytical results. [ABSTRACT FROM AUTHOR]

Flexoelectricity is present in nonuniformly deformed dielectric materials and has size-dependent properties, making it useful for microelectromechanical systems. Flexoelectricity is small compared to piezoelectricity; therefore, producing a large-scale flexoelectric effect is of great interest. In this paper, we explore a way to enhance the flexoelectric effect by utilizing the snap-through instability and a stiffness gradient present along the length of a curved dielectric plate. To analyze the effect of stiffness profiles on the plate, we employ numerical parameter continuation. Our analysis reveals a nonlinear relationship between the effective electromechanical coupling coefficient and the gradient of Young's modulus. Moreover, we demonstrate that the quadratic profile is more advantageous than the linear profile. For a dielectric plate with a quadratic profile and a modulus gradient of − 0.9, the effective coefficient can reach as high as 15.74 pC/N, which is over three times the conventional coupling coefficient of piezoelectric material. This paper contributes to our understanding of the amplification of flexoelectric effects by harnessing snapping surfaces and stiffness gradient design. [ABSTRACT FROM AUTHOR]

This work provides a systematic recipe for computing accurate high order Fourier expansions of quasiperiodic invariant circles (and systems of such circles) in area preserving maps. The recipe requires only a finite data set sampled from the quasiperiodic circle. Our approach, being based on the parameterization method of [A. Haro and R. de la Llave, SIAM J. Appl. Dyn. Syst., 6 (2007), pp. 142--207; A. Haro and R. de la Llave, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), pp. 1261--1300; A. Haro and R. de la Llave, J. Differential Equations, 228 (2006), pp. 530--579], uses a Newton scheme to iteratively solve a conjugacy equation describing the invariant circle (or systems of circles). A critical step in properly formulating the conjugacy equation is to determine the rotation number of the quasiperiodic subsystem. For this we exploit the weighted Birkhoff averaging method of [S. Das et al., Nonlinearity, 30 (2017), pp. 4111--4140; S. Das et al., The Foundations of Chaos Revisited, Springer, Cham, 2016, pp. 103--118; S. Das and J. A. Yorke, Nonlinearity, 31 (2018), pp. 491--501]. This approach facilities accurate computation of the rotation number given nothing but the already mentioned orbit data. The weighted Birkhoff averages also facilitate the computation of other integral observables like Fourier coefficients of the parameterization of the invariant circle. Since, the parameterization method is based on a Newton scheme, we only need to approximate a small number of Fourier coefficients with low accuracy (say, a few correct digits) to find a good enough initial approximation so that Newton converges. Moreover, the Fourier coefficients may be computed independently, so we can sample the higher modes to guess the decay rate of the Fourier coefficients. This allows us to choose, a priori, an appropriate number of modes in the truncation. We illustrate the utility of the approach for explicit example systems including the area preserving H\'enon map and the standard map (polynomial and trigonometric nonlinearity respectively). We present example computations for invariant circles and for systems of invariant circles with as many as 120 components. We also employ a numerical continuation scheme (where the rotation number is the continuation parameter) to compute large numbers of quasiperiodic circles in these systems. During the continuation we monitor the Sobolev norm of the parameterization, as explained in [R. Calleja and R. de la Llave, Nonlinearity, 23 (2010), pp. 2029--2058], to automatically detect the breakdown of the family. [ABSTRACT FROM AUTHOR]

The fractional homotopy method, involving the incorporation of the homotopic parameter into the derivatives of differential equations in nonlinear optimal control problems, has been plagued by significant computational inefficiency and reduced solution accuracy. In response to these challenges, this paper introduces a novel and improved approach that not only overcomes these limitations but also offers several notable advantages. The proposed method begins by establishing a more generalized form of fractional differential equations, where the left-hand sides comprise combinations of first-order derivatives and fractional-order derivatives of state variables. This is achieved through insights from fractional optimal control theory, ensuring alignment with established principles and greatly enhancing the method's efficiency and ease of implementation. Furthermore, the homotopic parameter is integrated into the left-hand sides of these generalized fractional differential equations, resulting in fractional embedded problems. Additionally, the fractional formulations are transformed into their equivalent integer-order counterparts. This transformation allows the utilization of well-established numerical techniques, resulting in significantly faster computations and a marked improvement in precision. Numerical demonstrations presented in this paper serve to underscore the superior performance of the proposed method, showcasing its efficiency, accuracy, and its potential to address the limitations that have hindered the original fractional homotopy approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider the piecewise-linear Chua's circuit, which is well known for its rich variety of bifurcation, chaotic and other nonlinear phenomena. Suitable switching boundaries are introduced based on the piecewise-linear representation of Chua's diode. In this way, we derive analytical conditions for a grazing bifurcation to occur, when one or two families of periodic orbits have a zero-velocity contact with the switching boundaries. In connection to this phenomenon, we also study the focus-center-limit cycle bifurcation and its implications regarding the system dynamics, from both analytical and numerical points of view. Furthermore, a detailed parametric study of Chua's circuit is carried out via path-following techniques for nonsmooth dynamical systems, implemented via the continuation software COCO. This study reveals the presence of codimension-one bifurcations of limit cycles, such as those mentioned above, as well as classical (fold and period-doubling) bifurcations. The analysis confirms the presence of coexisting attractors, which are produced by a hysteresis loop induced by the interaction of a fold and a focus-center-limit cycle bifurcation. [ABSTRACT FROM AUTHOR]

Estimating model parameters is a crucial step in mathematical modelling and typically involves minimizing the disagreement between model predictions and experimental data. This calibration data can change throughout a study, particularly if modelling is performed simultaneously with the calibration experiments, or during an on-going public health crisis as in the case of the COVID-19 pandemic. Consequently, the optimal parameter set, or maximal likelihood estimator (MLE), is a function of the experimental data set. Here, we develop a numerical technique to predict the evolution of the MLE as a function of the experimental data. We show that, when considering perturbations from an initial data set, our approach is significantly more computationally efficient that re-fitting model parameters while producing acceptable model fits to the updated data. We use the continuation technique to develop an explicit functional relationship between fit model parameters and experimental data that can be used to measure the sensitivity of the MLE to experimental data. We then leverage this technique to select between model fits with similar information criteria, a priori determine the experimental measurements to which the MLE is most sensitive, and suggest additional experiment measurements that can resolve parameter uncertainty. [ABSTRACT FROM AUTHOR]

The time-delayed Moran-Ricker population model is investigated in this paper with an aim to identify some of its unknown features. In this model, the decline of the essential resources arising from the previous generation emerges as a delay in the density dependency of the population. The random fluctuations in population size may cause the model’s dynamics to change. In this study, we aim to scrutinize the model thoroughly and reveal more properties of the model. A discussion about the fixed points and their stability is presented in a brief way. By studying the normal form of the model through the reduction of the model to the associated center manifold, we show that the model will experience flip (period-doubling), Neimark–Sacker, strong resonances, and period-doubling-Neimark Sacker bifurcations. The bifurcation conditions are extracted with their critical coefficients. Numerical bifurcation analysis confirms the validity of theoretical findings. [ABSTRACT FROM AUTHOR]

Axle tramp is a self-sustaining vibration in the driven axle of a vehicle with a beam axle layout, known to occur under heavy braking or acceleration. A 6DOF mathematical model of this phenomenon is used to identify how the key parameters of driveline stiffness, axle mass and fore/aft stiffness change the system's dynamics. A bifurcation analysis is performed to study this nonlinear system's dynamics. Four Hopf bifurcations in the underlying equilibria, along with a fold bifurcation in the outermost limit cycle branch, are shown to bound the parameter space where tramp occurs. The severity of tramp was found to be minimised by increasing drivetrain stiffness, reducing axle mass or increasing fore/aft stiffness: the trade-off for minimising tramp severity is that it may be easier to excite tramp when the drivetrain stiffness is increased, and the speed range over which tramp can occur is increased as fore/aft stiffness is increased. A key outcome from this work is that future electrified powertrains may experience more tramp, albeit at a reduced magnitude, than their combustion-powered counterparts. [ABSTRACT FROM AUTHOR]

In this paper, we use new methods to investigate different bifurcations of fixed points in a discrete-time toxic-phytoplankton–zooplankton model with Allee effect. The nonstandard discretization scheme produces a discrete analog of the continuous-time toxic-phytoplankton–zooplankton model with Allee effect. The local stability for proposed system around all of its fixed points is derived. We obtain the codimension-1 conditions of various bifurcations such as period doubling and Neimark–Sacker. Moreover, the system produces codimension-2 bifurcations such as resonance 1:1, 1:2, 1:3, and 1:4. Furthermore, the system can produce very rich dynamics, such as the existence of a semi-stable limit cycle, multiple coexisting periodic orbits, and chaotic behavior. Theoretical analysis is validated by numerical methods. [ABSTRACT FROM AUTHOR]

In the industry field, the increasingly stringent requirements of lightweight structures are exposing the ultimately nonlinear nature of mechanical systems. This is extremely true for systems with moving parts and loose fixtures which show piecewise stiffness behaviours. Nevertheless, the numerical solution of systems with ideal piecewise mathematical characteristics is associated with time-consuming procedures and a high computational burden. Smoothing functions can conveniently simplify the mathematical form of such systems, but little research has been carried out to evaluate their effect on the mechanical response of multi-degree-of-freedom systems. To investigate this problem, a slightly damped mechanical two-degree-of-freedom system with soft piecewise constraints is studied via numerical continuation and numerical integration procedures. Sigmoid functions are adopted to approximate the constraints, and the effect of such approximation is explored by comparing the results of the approximate system with the ones of the ideal piecewise counter-part. The numerical results show that the sigmoid functions can correctly catch the very complex dynamics of the proposed system when both the above-mentioned techniques are adopted. Moreover, a reduction in the computational burden, as well as an increase in numerical robustness, is observed in the approximate case. [ABSTRACT FROM AUTHOR]

We review some recent progress on the research of the periodic orbits of the N-body problem, and numerically study the spatial doubly symmetric periodic orbits (SDSPs for short). Both comet- and lunar-type SDSPs in the circular restricted three-body problem are computed, as well as the Hill-type SDSPs in Hill's lunar problem. Double symmetries are exploited so that the SDSPs can be computed efficiently. The monodromy matrix can be calculated by the information of one fourth period. The periodicity conditions are solved by Broyden's method with a line-search, and some numerical examples show that the scheme is very efficient. For a fixed period ratio and a given acute angle, there exist sixteen cases of initial values. For the restricted three-body problem, the cases of "Copenhagen problem" and the Sun–Jupiter–asteroid model are considered. New SDSPs are also numerically found in Hill's lunar problem. Though the period ratio should be small theoretically, some new periodic orbits are found when the ratio is not too small, and the linear stability of the searched SDSPs is numerically determined. [ABSTRACT FROM AUTHOR]

This work is concerned with the analysis of vibro-impact responses observed in large-scale nonlinear geared systems. Emphasis is laid on the interactions between the high-frequency internal excitation generated by the meshing process, i.e. the static transmission error and time-varying mesh stiffness, and low-frequency external excitations. To this end, a three-dimensional finite element model of a pump equipped with a reverse spur gear pair (gear ratio 1 : 1 ) is built. The model takes into account the flexibility of the kinematic chain, the bearings and the housing and the gear backlash nonlinearity. A reduced-order model is solved with the Harmonic Balance Method coupled to an arc-length continuation algorithm which allows one to compute the periodic solutions of the system. The onset and disappearance of vibro-impact responses is studied through the computation of grazing bifurcations. Results show that the coupling between the external excitation and the time-varying mesh stiffness term greatly modifies the characteristics of the responses in terms of number and periodicity of impacts and contact loss duration. [ABSTRACT FROM AUTHOR]

Some comet- and Hill-type families of nearly circular symmetric periodic orbits of the elliptic restricted three-body problem in the inertial frame are numerically explored by Broyden's method with a line search. Some basic knowledge is introduced for self-consistency. Set j / k as the period ratio between the inner and the outer orbits. The values of j / k are mainly 1 / j with 2 ≤ j ≤ 10 and j = 15 , 20 , 98 , 100 , 102 . Many sets of the initial values of these periodic orbits are given when the orbital eccentricity e p of the primaries equals 0.05. When the mass ratio μ = 0.5 , both spatial and planar doubly-symmetric periodic orbits are numerically investigated. The spacial orbits are almost perpendicular to the orbital plane of the primaries. Generally, these orbits are linearly stable when the j / k is small enough, and there exist linearly stable orbits when j / k is not small. If μ ≠ 0.5 , there is only one symmetry for the high-inclination periodic orbits, and the accuracy of the periodic orbits is determined after one period. Some diagrams between the stability index and e p or μ are supplied. For μ = 0.5 , we set j / k = 1 / 2 , 1 / 4 , 1 / 6 , 1 / 8 and e p ∈ [ 0 , 0.95 ] . For e p = 0.05 and 0.0489, we fix j / k = 1 / 8 and set μ ∈ [ 0 , 0.5 ] . Some Hill-type high-inclination periodic orbits are numerically studied. When the mass of the central primary is very small, the elliptic Hill lunar model is suggested, and some numerical examples are given. [ABSTRACT FROM AUTHOR]

In this study, the dynamic response behavior of a generalized nonlinear dynamic system is investigated using a newly proposed extended Galerkin method. The algebraic equations of vibration amplitudes are obtained through an integration of the weighted functions. The new method is equivalent to the harmonic balance method but with a much simpler calculation procedure and a higher efficiency. This is the first time to use the method for the analysis of nonlinear systems with high number of modes, manifesting that the method is applicable to forced vibrations of nonlinear behavior. The method is further validated by the numerical Runge-Kutta method. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamical innovations occur in an upgraded chaotic Pan system. The local dynamics includes the stability analysis, types of attractor and Hopf bifurcation analysis. The proposed system undergoes periodic, chaotic and hyperchaotic orbits with the variation of bifurcation and control parameter. The hidden self-excited chaotic attractors are localized within the parametric boundaries; also, proposed system is initialized in the self-excited attractor zone where all the unstable equilibria exist. The hidden and self-existing attractor of hyperchaotic Pan system is revealed with the help of multidimensional transient dynamical system. Moreover, a complete set of conditions are derived that guarantee the existence of supercritical Hopf bifurcation in the upgraded chaotic Pan system. The Hopf bifurcation diagram is also investigated for numerical confirmation of analytical results. [ABSTRACT FROM AUTHOR]

Nonlinear problems involving phases occur ubiquitously throughout applied mathematics andphysics, ranging from neuronal models to the search for elementary particles. The phase variables present in such models usually enter as harmonic terms and, being unbounded, pose an open challenge for studying bifurcations in these systems through standard numerical continuation techniques. Here, we propose to transform and embed the original model equations involving phases into structurally stable generalized systems that are more suitable for analysis via standard predictor–corrector numerical continuation methods. The structural stability of the generalized system is achieved by replacing each harmonic term in the original system by a supercritical Hopf bifurcation normal form subsystem. As an illustration of this general approach, specific details are provided for the ac-driven, Stewart–McCumber model of a single Josephson junction. It is found that the dynamics of the junction is underpinned by a two-parameter Hopf–Hopf bifurcation, detected in the generalized system. The Hopf–Hopf bifurcation gives birth to an invariant torus through Neimark–Sacker bifurcation of limit cycles. Continuation of the Neimark–Sacker bifurcation of limit cycles in the two-parameter space provides a complete picture of the overlapping Arnold tongues (regions of frequency-locked periodic solutions), which are in precise agreement with the widths of the Shapiro steps that can be measured along the current–voltage characteristics of the junction at various fixed values of the ac-drive amplitude. [ABSTRACT FROM AUTHOR]

The nonlinear dynamics of turbine generator shaft trains for power generation are investigated in this paper. Realistic models of rotors, pedestals, and nonlinear bearings of partial arc and lemon bore configuration are implemented to compose a nonlinear set of differential equations for autonomous (balanced) and nonautonomous (unbalanced--per ISO) cases. The solution branches of the dynamic system are evaluated with the pseudo-arc length continuation programed by the authors, and the respective limit cycles are evaluated by an orthogonal collocation method and investigated on their stability properties and quality of motion for the respective key design parameters for the rotor dynamic design of such systems, namely, bearing profile and respective pad length, preload and offset, pedestal stiffness and elevation (misalignment), and rotor slenderness. Model order reduction is applied to the finite element rotor model and the reduced system is validated in terms of unbalanced response and stability characteristics. The main conclusion of the current investigation is that the system has the potential to develop instabilities at rotating speeds lower than the threshold speed of instability (evaluated by the linear approach) for specific unbalance magnitude and design properties. Unbalance response (with stable and unstable branches) is evaluated in severely reduced time compared to this applying time integration methods, enabling nonlinear rotor dynamic design of such systems as a standard procedure, and revealing the complete potential of motions (not only local). [ABSTRACT FROM AUTHOR]

High-speed rotor systems mounted on gas foil bearings present bifurcations which change the quality of stability, and may compromise the operability of rotating systems, or increase noise level when amplitude of specific harmonics drastically increases. The paper identifies the dissipating work in the gas film to be the source of self-excited motions driving the rotor whirling close to bearing's surface. The energy flow among the components of a rotor gas foil bearing system with unbalance is evaluated for various design sets of bump foil properties, rotor stiffness and unbalance magnitude. The paper presents a methodology to retain the dissipating work at positive values during the periodic limit cycle motions caused by unbalance. An optimization technique is embedded in the pseudo-arc length continuation of limit cycles, those evaluated (when exist) utilizing an orthogonal collocation method. The optimization scheme considers the bump foil stiffness and damping as the variables for which bifurcations do not appear in a certain speed range. It is found that secondary Hopf (Neimark–Sacker) bifurcations, which trigger large limit cycle motions, do not exist in the unbalanced rotors when bump foil properties follow the optimization pattern. Period-doubling (flip) bifurcations are possible to occur, without driving the rotor in high response amplitude. Different design sets of rotor stiffness and unbalance magnitude are investigated for the efficiency of the method to eliminate bifurcations. The quality of the optimization pattern allows optimization in real time, and gas foil bearing properties shift values during operation, eliminating bifurcations and allowing operation at higher speed margins. Compliant bump foil is found to enhance the stability of the system. [ABSTRACT FROM AUTHOR]

In this research, we offer a bifurcation analysis to describe impacting, stick, and grazing between a particle and the piston to better understand the nonlinear dynamics of a Fermi oscillator. The principles of hybrid dynamical systems will be utilized to explain the moving process in such a Fermi oscillator. The motion complexity and stick mechanism of such an oscillator are demonstrated using periodic and chaotic motions. We introduce the hybrid dynamical system and numerical continuation detection tool COCO. Physical and mathematical models are used to construct the obtained bounded mathematical model. The frequency, amplitude in oscillating base displacement, and the gap between the stationary boundary and the piston's equilibrium position are the major parameters. We employ path-following analysis to illustrate the bifurcation that leads to solution destabilization. From the viewpoint of eigenvalue analysis, the essence of period-doubling and Fold bifurcation is revealed. Numerical continuation methods are used to perform a complete one-parameter bifurcation analysis of the Fermi oscillator. The presence of codimension-1 bifurcations of limit cycles, like period-doubling, fold, and grazing bifurcations, is shown in this work. Then, the one-parameter continuation analysis is tracked in two-parameter bifurcation diagrams that include a second important factor of interest. Bifurcations cause all solutions to lose stability, according to our findings. The acquired results provide a better understanding of the Fermi oscillator mechanism and demonstrate that we may control the system dynamics by modifying the parameters. [ABSTRACT FROM AUTHOR]

The multilinear PageRank model [Gleich et al., SIAM J Matrix Anal Appl, 2015;36(4):1507–41] is a tensor‐based generalization of the PageRank model. Its computation requires solving a system of polynomial equations that contains a parameter α∈[0,1). For α≈1, this computation remains a challenging problem, especially since the solution may be nonunique. Extrapolation strategies that start from smaller values of α and "follow" the solution by slowly increasing this parameter have been suggested; however, there are known cases where these strategies fail, because a globally continuous solution curve cannot be defined as a function of α. In this article, we improve on this idea, by employing a predictor‐corrector continuation algorithm based on a more general representation of the solutions as a curve in ℝn+1. We prove several global properties of this curve that ensure the good behavior of the algorithm, and we show in our numerical experiments that this method is significantly more reliable than the existing alternatives. [ABSTRACT FROM AUTHOR]

This paper presents a nonlinear stability analysis of an unbalanced rotor-bearing system using the numerical continuation method and the numerical integration method. In this study, the effect of unbalance on journal motion is highlighted and a relationship is established between the bifurcation diagram of a balanced rotor and that of an unbalanced rotor. The results show that the stable operating speed range, the shaft motion type, the whirl speed and the chaotic motion occurrence depend on the unbalance level, the bearing geometry, the oil viscosity, and the speed range of unstable limit cycles existence.

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 modeling 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; (i i) 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. • We consider the Allen–Cahn equation with random spatially-heterogeneous coefficients. • We analyze propagation of uncertainty in nonlinear systems focusing on bifurcations. • We show that the expected value of the random coefficient is a bifurcation parameter. • We propose the polynomial chaos method to assess uncertainty in bifurcation diagrams. • We interface stochastic collocation with numerical continuation. [ABSTRACT FROM AUTHOR]

Early afterdepolarizations (EADs) are voltage oscillations that can occur during the plateau phase of a cardiac action potential. EADs at the cellular level have been linked to potentially deadly tissue-level arrhythmias, and the mechanisms for their arisal are not fully understood. There is ongoing debate as to which is the predominant biophysical mechanism of EAD production: imbalanced interactions between voltage-gated transmembrane currents or overactive Ca2+-dependent transmembrane currents brought about by pathological intracellularCa2+-release dynamics. In this article, we address this issue using a foundational 10-dimensional biophysical ventricular action potential model which contains both electrical and intracellularCa2+-components. Surprisingly, we find that the model can produce EADs through both biophysical mechanisms, which hints at a more fundamental dynamical mechanism for EAD production. Fast-slow analysis reveals EADs, in both cases, to be canard-induced mixed-mode oscillations. While the voltage-driven EADs arise from a fast-slow problem with two slow variables, the Ca2+--driven EADs arise from the addition of a third slow variable. Hence, we adapt existing computational methods in order to compute 2D slow manifolds and 1D canard orbits in the reduced 7D model from which voltage-driven EADs arise. Further, we extend these computational methods in order to compute, for the first time, 2D sets of maximal canards which partition the 3D slow manifolds of the 8D problem from which Ca2+--driven EADs arise. The canard viewpoint provides a unifying alternative to the voltage- or Ca2+--driven viewpoints while also providing explanatory and predictive insights that cannot be obtained through the use of the traditional fast-slow approach. [ABSTRACT FROM AUTHOR]

Floating ring bearings have been widely used, over the last decades, in rotors of automotive turbochargers because of their improved damping behavior and their good emergency-operating capabilities. They also offer a cost-effective design and have good assembly properties. Nevertheless, rotors with floating ring bearings show vibration effects of nonlinear nature induced by self-excited oscillations originating from the bearing oil films (oil whirl/whip phenomena) and may exhibit various nonlinear vibration effects which may cause damage to the rotor. In order to investigate these dynamic phenomena, this paper has developed a nonlinear model of a perfectly balanced rigid rotor supported by two identical floating ring bearings with consideration of their vibration behavior mainly governed by fluid dynamics. The dimensionless hydrodynamic forces of floating ring bearings have been derived based on the short bearing theory and the half Sommerfeld solution. Using the numerical continuation approach, different bifurcations are detected when a control parameter, the journal speed, is varied. Depending on the system's physical parameters, the rotor can show stable or unstable limit cycles which themselves may collapse beyond a certain rotor speed to exhibit a fold bifurcation. Bifurcation analysis is performed to investigate the occurring instabilities and nonlinear phenomena. Such results explain the instabilities characteristics of the floating ring bearing in high-speed applications. It has also been found that the selection of the bearing modulus plays an important role in the characterization of the rotor stability threshold speed and bifurcation sequences. An understanding of the system's nonlinear behavior serves as the basis for new and rational criteria for the design and the safe operation of rotating machines. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Hydrodynamic journal bearings are used in many applications which involve high speeds and loads. However, they are susceptible to oil whirl instability, which may cause bearing failure. In this work, a flexible Jeffcott rotor supported by two identical journal bearings is used to investigate the stability and bifurcations of rotor bearing system. Since a closed form for the finite bearing forces is not exist, nonlinear bearing stiffness and damping coefficients are used to represent the bearing forces. The bearing forces are approximated to the third order using Taylor expansion, and infinitesimal perturbation method is used to evaluate the nonlinear bearing coefficients. The mesh sensitivity on the bearing coefficients is investigated. Then, the equations of motion based on bearing coefficients are used to investigate the dynamics and stability of the rotor-bearing system. The effect of rotor stiffness ratio and applied load on the Hopf bifurcation stability and limit cycle continuation of the system are investigated. The results of this work show that evaluating the bearing forces using Taylor's expansion up to the third-order bearing coefficients can be used to profoundly investigate the rich dynamics of rotor-bearing systems. [ABSTRACT FROM AUTHOR]

This paper studies the bifurcation analysis of the discrete time Lorenz system considering its generalization for two control parameters. The one‐ and two‐parameter bifurcations of the system, including pitchfork, period‐doubling, Neimark–Sacker, 1:2, 1:3, and 1:4 resonances, are surveyed thoroughly. The critical coefficients are computed to analyze the nondegeneracy of listed bifurcations and predict their bifurcation scenarios. The numerical continuation method reveals complex dynamics including bifurcations up to 16th iterations. The results show an excellent agreement between the analytical predictions and the numerical observations. [ABSTRACT FROM AUTHOR]

This paper investigates the unlocking of a non-conventional nose landing gear mechanism that uses a single lock to fix the landing gear in both its downlocked and uplocked states (as opposed to having two separate locks as in most present nose landing gears in operation today). More specifically, we present a bifurcation analysis of a parameterized mathematical model for this mechanical system that features elastic constraints and takes into account internal and external forces. This formulation makes it possible to employ numerical continuation techniques to determine the robustness of the proposed unlocking strategy with respect to changing aircraft attitude. In this way, we identify as a function of several parameters the steady-state solutions of the system, as well as their bifurcations: fold bifurcations where two steady states coalesce, cusp points on curves of fold bifurcations, and a swallowtail bifurcation that generates two cusp points. Our results are presented as surfaces of steady states, joined by curves of fold bifurcations, over the plane of retraction actuator force and unlock actuator force, where we consider four scenarios of the aircraft: level flight; steep climb; steep descent; intermediate descent. A crucial cusp point is found to exist irrespective of aircraft attitude: it corresponds to the mechanism being at overcentre, which is a position that creates a mechanical singularity with respect to the effect of forces applied by the actuators. Furthermore, two cusps on a key fold locus are unfolded in a (codimension-three) swallowtail bifurcation as the aircraft attitude is changed: physical factors that create these bifurcations are presented. A practical outcome of this research is the realization that the design of this and other types of landing gear mechanism should be undertaken by considering the effects of forces over considerable ranges, with a special focus on the overcentre position, to ensure a smooth retraction occurs. More generally, continuation methods are shown to be a valuable tool for determining the overall geometric structure of steady states of mechanisms subject to (external) forces. [ABSTRACT FROM AUTHOR]

Under compressive creep, viscoplastic solids experiencing internal mass transfer processes can accommodate singular cnoidal wave solutions as material instabilities at the stationary wave limit. These instabilities appear when the loading rate is significantly faster than the material's capacity to diffusive internal perturbations, leading to localized failure features (e.g., cracks and compaction bands). These cnoidal waves, generally found in fluids, have strong nonlinearities that produce periodic patterns. Due to the singular nature of the solutions, the applicability of the theory is currently limited. Additionally, practical simulation tools require proper regularization to overcome the challenges that singularity induces. We focus on the numerical treatment of the governing equation using a nonlinear approach building on a recent adaptive stabilized finite element method. This automatic refinement method provides an error estimate that drives mesh adaptivity, a crucial feature for the problem at hand. We compare the performance of this adaptive strategy against analytical and standard finite element solutions. We then investigate the sensitivity of the diffusivity ratio, the parameter controlling the process, and identify multiple possible solutions with several stress peaks. Finally, we show the evolution of the spacing between peaks for all solutions as a function of that parameter. [ABSTRACT FROM AUTHOR]

The emergency extension system for nose landing gear (NLG) is vitally important for safe landing in the event of hydraulic source failure. To study the dynamic properties of NLG emergency extension, it is necessary to explore the aerodynamic load in detail, which is a potential adverse factor that hinders the successful deployment of NLG. The emergency extension failure of a door-linkage NLG mechanism was studied from a novel bifurcation perspective in this paper. NLG models (excluding and including aerodynamic loads) formulating the mechanism as a series of steady-state constraint equations were derived. Solutions to these equations were continued numerically, and the NLG retraction–extension cycles in different models were compared. The critical condition for smooth extension was obtained based on the bifurcation analysis. The failure principle was explained, that is, the disappearance of bifurcation points characterizing the successful locking. The variations of bifurcation point as a function of spring parameters and door aerodynamic loads were investigated. An effective solution to the failure was put forward in which the unlocking force is reduced by 33.0%. [ABSTRACT FROM AUTHOR]

Using rigorous numerical methods, we prove the existence of 608 isolated periodic orbits in a gravitational billiard in a vibrating unbounded parabolic domain. We then perform pseudo-arclength continuation in the amplitude of the parabolic surface's oscillation to compute large, global branches of periodic orbits. These branches are themselves proven rigorously using computer-assisted methods. Our numerical investigations strongly suggest the existence of multiple pitchfork bifurcations in the billiard model. Based on the numerics, physical intuition and existing results for a simplified model, we conjecture that for any pair (k , p) , there is a constant ξ for which periodic orbits consisting of k impacts per period p cannot be sustained for amplitudes of oscillation below ξ. We compute a verified upper bound for the conjectured critical amplitude for (k , p) = (2 , 2) using our rigorous pseudo-arclength continuation. [ABSTRACT FROM AUTHOR]

Designing and analyzing large cable-driven parallel robots (CDPRs) for precision tasks can be challenging, as the position kinematics are governed by kineto-statics and cable sag equations. Our aim is to find all equilibria for a given set of unstrained cable lengths using numerical continuation techniques. The Irvine sagging cable model contains both non-algebraic and multi-valued functions. The former removes the guarantee of finiteness on the number of isolated solutions, making homotopy start system construction less clear. The latter introduces branch cuts, which could lead to failures during path tracking. We reformulate the Irvine model to eliminate multi-valued functions and propose a heuristic numerical continuation method based on monodromy, removing the reliance on a start system. We demonstrate this method on an eight-cable spatial CDPR, resulting in a well-constrained non-algebraic system with 31 equations. The method is applied to four examples from literature that were previously solved in bounded regions. Our method computes the previously reported solutions along with new solutions outside those bounds much faster, showing that this numerical method enhances existing approaches for comprehensively analyzing CDPR kineto-statics. • We introduce a numerical continuation method for solving the kinetostatics of CDPRs. • The Irvine sagging cable model is reformulated to eliminate multi-valued functions. • We propose a new heuristic monodromy algorithm to solve non-algebraic systems. • We solve an eight-cable spatial CDPR, a well-constrained system with 31 equations. • The new approach finds solutions more quickly than previously published methods. [ABSTRACT FROM AUTHOR]