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We prove that all non-degenerate relative equilibria of the planar Newtonian n–body problem can be continued to spaces of constant curvature κ, positive or negative, for small enough values of this parameter. We also compute the extension of some classical relative equilibria to curved spaces using numerical continuation. In particular, we extend Lagrange's triangle configuration with different masses to both positive and negative curvature spaces.

The non-degeneracy is one of the conditions to check for bifurcation analysis. Therefore, we need to compute the critical normal form coefficients to verify the non-degeneracy of the listed bifurcations. Using the critical normal form coefficients method to examine the bifurcation analysis makes it avoid calculating the central manifold and converting the linear part of the map into Jordan form. This is one of the most effective methods in the bifurcation analysis that has not received much attention so far. So in this article, we turn our attention to this method. In this study, the dynamic behaviors of the discrete Bonhoeffer–van der Pol (BVP) model are discussed. It is shown that the BVP model undergoes codimension one (codim-1) bifurcations such as pitchfork, fold, flip (period doubling) and Neimark–Sacker. Besides, codimension two (codim-2) bifurcations including resonance 1:2, 1:3, 1:4 and Chenciner have been achieved. For each bifurcation, normal form coefficients along with its scenario are investigated thoroughly. Bifurcation curves of the fixed points are drawn with the aid of numerical continuation techniques. Besides, a numerical continuation not only confirms our analytical results but also reveals richer dynamics of the model especially in the higher iteration.

We analyze the nonlinear dynamics of an isothermal system involving complex enzymatic reactions for L-DOPA (L-3,4-dihydroxyphenylalanine) production by numerical simulation. The mass action kinetics of the system forms a family of 9ordinary differential equations with 22 parameters. The multiple steady states are calculated by the chemical reaction network toolbox. Starting from one of the steady states, a limit point is guaranteed to be detected due to a change in the system parameters using the numerical continuation software MatCont. Other bifurcations are also obtained via the bifurcation continuations of the limit point, such as cusp bifurcations, Hopf bifurcations, limit cycles, zero Hopf bifurcations, generalized Hopf bifurcations, period-doubling, and so on. A transition of a period-doubling bifurcation to chaos occurs by numerical simulations. Positive values, 0.25~0.71, of Lyapunov exponents are obtained for the chaotic dynamics. Poincare maps and power spectrum densities are also plot. The Feigenbaum constant is computed to be 4.681~4.705.

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.

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.

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.

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.

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.

Machine tool vibrations in turning processes are analyzed by taking into account the nonlinearity of the cutting force characteristics. Unstable limit cycles are computed for the governing nonlinear delay-differential equation in order to determine the bistable technological parameter region where stable stationary cutting and large-amplitude machine tool vibrations coexist. Simple closed-form formulas are derived for the amplitude of limit cycles and for the size of the bistable region considering a general cutting force characteristics. The analytical results are determined by the method of averaging, which can be used to treat the nonlinearities without their third-order approximation. The results are confirmed by numerical continuation and using Melnikov’s integral.

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.

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.

The paper presents linearized coefficients for finite-length journal bearings based on the HD forces with multiplicative correction polynomials applied to the infinitely short and the infinitely long solutions under the assumption of π-film boundary condition. Based on the linear theory, stability thresholds and their dependence on the length-to-diameter ratio are investigated. Moreover, using the numerical continuation method the change of the generalized Hopf bifurcation which separates regions with subcritical and supercritical Hopf is documented. All results are compared with those of the solution obtained using the finite differences method.

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.

A system of ordinary differential equations of a predator–prey type, depending on nine parameters, is studied. We have included in this model a nonmonotonic response function and time periodic perturbation. Using numerical continuation software, we have detected three codimension two bifurcations for the unperturbed system, namely cusp, Bogdanov-Takens and Bautin bifurcations. Furthermore, we concentrate on two regions in the parameter space, the region where the Bogdanov-Takens and the region where Bautin bifurcations occur. As we turn on the time perturbation, we find strange attractors in the neighborhood of invariant tori of the unperturbed system.

We consider a classical spring-mass model of human running which is built upon an inverted elastic pendulum. Based on previous results concerning asymptotic solutions for large spring constant (or small angle of attack), we introduce an analytical approximation of a reduced mapping. Although approximate solutions already exist in the literature, our results have some benefits over them. They give us an advantage of being able to explicitly control the error of the approximation in terms of the small parameter, which has a physical meaning—the inverse of the square-root of the spring constant. Our approximation also shows how the solutions are asymptotically related to the magnitude of attack angle $\alpha $. The model itself consists of two sets of differential equations—one set describes the motion of the centre of mass of a runner in contact with the ground (support phase), and the second set describes the phase of no contact with the ground (flight phase). By appropriately concatenating asymptotic solutions for the two phases we are able to reduce the dynamics to a one-dimensional apex to apex return map. We find sufficient conditions for this map to have a unique stable fixed point. By numerical continuation of fixed points with respect to energy, we find a transcritical bifurcation in our model system.

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.

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 recently developed generalized Fourier–Galerkin method is complemented by a numerical continuation with respect to the kinetic energy, which extends the framework to the investigation of modal interactions resulting in folds of the nonlinear modes. In order to enhance the practicability regarding the investigation of complex large-scale systems, it is proposed to provide analytical gradients and exploit sparsity of the nonlinear part of the governing algebraic equations. A novel reduced order model (ROM) is developed for those regimes where internal resonances are absent. The approach allows for an accurate approximation of the multi-harmonic content of the resonant mode and accounts for the contributions of the off-resonant modes in their linearized forms. The ROM facilitates the efficient analysis of self-excited limit cycle oscillations, frequency response functions and the direct tracing of forced resonances. The ROM is equipped with a large parameter space including parameters associated with linear damping and near-resonant harmonic forcing terms. An important objective of this paper is to demonstrate the broad applicability of the proposed overall methodology. This is achieved by selected numerical examples including finite element models of structures with strongly nonlinear, non-conservative contact constraints.

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.

A common numerical model for bolted or riveted joints is the Iwan model, which uses a number of discrete Jenkins elements to capture the hysteretic behaviour of the system. Previously, the Iwan model has been primarily implemented within time-domain simulations using the Newmark integrator. However, in many applications it is desirable to predict the nonlinear Frequency Response Functions (FRFs) of a structure that contains joints. In order to do so, the steady-state response must be estimated over a range of frequencies, and it is very time consuming to simply compute the time response until steady-state is reached. Furthermore, the implicit nature of the state variables (i.e. the appearance of “hidden” state variables in the form of slider displacements) makes it non-trivial to use continuation to compute the frequency response using already established techniques such as the shooting method. This paper presents a novel method to numerically compute the non-linear FRFs of a single degree-of-freedom (SDOF) system with an Iwan element. The maximum displacement over the response period is included as a state variable, along with the initial displacement and velocity, and we demonstrate that the position of all sliders can be calculated using these states. The shooting method is modified to account for the added state variable. The method has been tested by computing the FRFs of a SDOF system containing of an Iwan element at multiple force amplitudes. The results show that the proposed method is able to compute the steady-state response even at large force amplitudes, when the system behaves quite non-linearly.

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.

The path synthesis of a four-bar linkage is commonly framed so as to size links such that a coupler curve passes through specified planar precision points. It may instead be framed with consideration of higher order derivative information at each precision point. In this paper, we enumerate all 30 combinations of higher order path synthesis that lead to square polynomial systems for the four-bar. We present in detail the synthesis equations for (1) the case when one precision point is specified along with derivative information up to its eighth derivative, and (2) when three precision points are specified along with derivative information up to the second derivative at each point. For each case, 224 and 227 possible Roberts’ cognate triplets were computed, respectively.

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

We investigate analytically and numerically soliton solutions of the stationary Lugiato–Lefever equation $$\displaystyle -da''-(\mathrm {i}-\zeta )a -|a|{ }^2a +\mathrm {i} f=0 $$ in the case of anomalous dispersion d > 0. By an analytical result based on the Implicit Function Theorem, we prove the existence of two families of soliton solutions on the real line when the parameters lie in certain regions of the ζ-f plane. We then use these ideas to compute numerically 2π-periodic approximate solutions and obtain a map of the ζ-f plane which shows numerical existence regions for soliton solutions. We also indicate the stability/instability of these solutions. The borders of the numerically computed existence regions in the ζ-f plane match very well heuristically obtained borders.

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

Heteroclinic-induced spiral waves may arise in systems of partial differential equations that exhibit robust heteroclinic cycles between spatially uniform equilibria. Robust heteroclinic cycles arise naturally in systems with invariant subspaces, and their robustness is considered with respect to perturbations that preserve these invariances. We make use of particular symmetries in the system to formulate a relatively low-dimensional spatial two-point boundary-value problem in Fourier space that can be solved efficiently in conjunction with numerical continuation. The standard numerical set-up is formulated on an annulus with small inner radius, and Neumann boundary conditions are used on both inner and outer radial boundaries. We derive and implement alternative boundary conditions that allow for continuing the inner radius to zero and so compute spiral waves on a full disk. As our primary example, we investigate the formation of heteroclinic-induced spiral waves in a reaction–diffusion model that describes the spatiotemporal evolution of three competing populations in a 2D spatial domain—much like the Rock–Paper–Scissors game. We further illustrate the efficiency of our method with the computation of spiral waves in a larger network of cyclic dominance between five competing species, which describes the so-called Rock–Paper–Scissors–Lizard–Spock game.

In order to determine the frequency response curve and its unstable region of a strongly nonlinear oscillator, a new method is proposed. This method is based on splitting the system parameters and introducing some unknown parameters into the system. The evaluation of the introduced parameters is done by optimizing the cumulative equation error induced by multiple-scales solution. The Duffing oscillator, the Helmholtz–Duffing oscillator, and an oscillator with both nonlinear restoring and nonlinear inertial forces are analyzed as examples to reveal the validity of the proposed method. The frequency response curves obtained by numerical continuation method are adopted to compare with those obtained by the proposed method and the conventional multiple-scales method. The unstable regions obtained by the harmonic balance method are adopted to examine those obtained by the conventional multiple-scales method and the proposed method. The efficiency of the proposed method is tested by comparing the computational time of each method.

A general-purpose method is presented and implemented to express analytically one stationary configuration of an ideal 3D elastic rod when the end-to-end relative position and orientation are imposed. The mechanical equilibrium of such a rod is described by ordinary differential equations and parametrized by six scalar quantities. When one end of the rod is anchored, the analytical integration of these equations lead to one unique solution for given values of these six parameters. When the second end is also anchored, six additional nonlinear equations must be resolved to obtain parameter values that fit the targeted boundary conditions. We find one solution of these equations with a zero-finding algorithm, by taking initial guesses from a grid of potential candidates. We exhibit the symmetries of the problem, which reduces drastically the size of this grid and shortens the time of selection of an initial guess. The six variables used in the search algorithm, forces and moments at one end of the rod, are particularly adapted due to their unbounded definition domain. More than 850 000 tests are performed in a large region of configurational space, and in 99.9% of cases the targeted boundary conditions are reached with short computation time and a precision better than 10 − 5 . We propose extensions of the method to obtain many solutions instead of only one, using numerical continuation or starting from different initial guesses.

This paper is focused on the pseudo bifurcations and the variety of periodic ratio in the periodic orbits near the primary of binary irregular asteroid system (22) Kalliope, which would help on trajectory design for asteroid missions and give a practical insight into the generation and dynamic behaviour of binary asteroid systems. In this paper, we find three basic pseudo bifurcations in the periodic orbit families near (22) Kalliope during the numerical continuation with the variation of Jacobian constant. We also discover a nonuple mixed bifurcations which possess the highest multiplicity of bifurcations ever found and consist of three pseudo tangent bifurcations, two period-doubling bifurcations, one pseudo period-doubling bifurcation, one Neimark-Sacker bifurcation, one pseudo Neimark-Sacker bifurcation, and one real saddle bifurcation. Moreover, we find that the periodic ratio in the periodic orbit family may change during the continuation. Based on plenty of numerical evidences, we summarize three astonishing and exciting conclusions about the relationship of the periodic ratio and (pseudo) bifurcations in the periodic motion near (22) Kalliope. Firstly, the pseudo period-doubling bifurcation shows up when the periodic ratio comes near a half-integer (i.e. 1.5:1, 2.5:1, 3.5:1). Furthermore, almost all the cuspidal changes of the periodic ratio are accompanied with tangent bifurcation or pseudo tangent bifurcation. In addition, an integer can be admitted as the asymptotic value of the periodic ratio at the end of continuation, if the Jacobian constant isn't stuck into its local extremum, yet a half-integer can not.

This paper is concerned with the vibration-driven system which can move due to the periodic motion of the internal mass and the dry friction; the system can be modeled as Filippov system and has the property of stick-slip motion. Different periodic solutions of stick-slip motion can be analyzed through sliding bifurcation, two-parameter numerical continuation for sliding bifurcation is carried out to get the different bifurcation curves, and the bifurcation curves divide the parameters plane into different regions which stand for different stick-slip motion of the periodic solution. Furthermore, continuations with additional condition v=0 are carried out for the directional control of the vibration-driven system in one period; the curves divide the parameter plane into different progressions.

We study the existence and stability of stationary solutions of Poisson-Nernst- Planck equations with steric effects (PNP-steric equations) with two counter-charged species. These equations describe steady current through open ionic channels quite well. The current levels in open ionic channels are known to switch between `open' or `closed' states in a spontaneous stochastic process called gating, suggesting that their governing equations should give rise to multiple stationary solutions that enable such multi-stable behavior. We show that within a range of parameters, steric effects give rise to multiple stationary solutions that are smooth. These solutions, however, are all unstable under PNP-steric dynamics. Following these findings, we introduce a novel PNP-Cahn-Hilliard model, and show that it admits multiple stationary solutions that are smooth and stable. The various branches of stationary solutions and their stability are mapped utilizing bifurcation analysis and numerical continuation methods.

We adopt a robust numerical continuation scheme to examine the global bifurcation of periodic traveling waves of the capillary-gravity Whitham equation, which combines the dispersion in the linear theory of capillary-gravity waves and a shallow water nonlinearity. We employ a highly accurate numerical method for space discretization and time stepping, to address orbital stability and instability for a rich variety of the solutions. Our findings can help classify capillary-gravity waves and understand their long-term dynamics., 24 pages, 22 figures

A boundary crisis occurs when a chaotic attractor outgrows its basin of attraction and suddenly disappears. As previously reported, the locus of a boundary crisis is organised by homo- or heteroclinic tangencies between the stable and unstable manifolds of saddle periodic orbits. In two parameters, such tangencies lead to curves, but the locus of boundary crisis along those curves exhibits gaps or channels, in which other non-chaotic attractors persist. These attractors are stable periodic orbits which themselves can undergo a cascade of period-doubling bifurcations culminating in multi-component chaotic attractors. The canonical diffeomorphic two-dimensional Henon map exhibits such periodic channels, which are structured in a particular ordered way: each channel is bounded on one side by a saddle-node bifurcation and on the other by a period-doubling cascade to chaos; furthermore, all channels seem to have the same orientation, with the saddle-node bifurcation always on the same side. We investigate the locus of boundary crisis in the Ikeda map, which models the dynamics of energy levels in a laser ring cavity. We find that the Ikeda map features periodic channels with a richer and more general organisation than for the Henon map. Using numerical continuation, we investigate how the periodic channels depend on a third parameter and characterise how they split into multiple channels with different properties.

This paper derives nonlinear plate equations for electroelastic crystals using both power series and trigonometric expansions of the three-dimensional equations. Unlike existing theories, material nonlinearities are included to cubic order in the gradients of the field variables, which allows Duffing behavior to be properly modeled. Moreover, inconsistencies in existing nonlinear power series expansions are revealed, and a consistent expansion is given. Next, a Galerkin truncation is applied to the variational formulation of the plate equations to give a very general reduced-order model of its dynamics near primary resonance. By comparison with the Galerkin discretization of the exact equations, nonlinear correction factors are derived for both power series and trigonometric expansions. Numerical continuation of the resulting nonlinear ODEs demonstrates the effect of lateral eigenmodes on the Duffing behavior of the frequency response. Both power series and trigonometric expansions produce results in close agreement. In the limit of purely thickness vibrations, the nonlinear plate equations reduce to the Galerkin truncation of the exact equations.

The present study concerns the development of a new iterative method applied to a numerical continuation procedure for parameterized scalar nonlinear equations. Combining both a modified Newton’s technique and a stationary-type numerical procedure, the proposed method is able to provide suitable approximate solutions associated with scalar nonlinear equations. A numerical analysis of predictive capabilities of this new iterative algorithm is addressed, assessed, and discussed on some specific examples.

In this paper we investigate the bifurcation structure of the cross-diffusion Shigesada–Kawasaki–Teramoto model (SKT) in the triangular form and in the weak competition regime, and of a corresponding fast-reaction system in 1D and 2D domains via numerical continuation methods. We show that the software can be exploited to treat cross-diffusion systems, reproducing the already computed bifurcation diagrams on 1D domains. We show the convergence of the bifurcation structure obtained selecting the growth rate as bifurcation parameter. Then, we compute the bifurcation diagram on a 2D rectangular domain providing the shape of the solutions along the branches and linking the results with the linearized analysis. In 1D and 2D, we pay particular attention to the fast-reaction limit by always computing sequences of bifurcation diagrams as the time-scale separation parameter tends to zero. We show that the bifurcation diagram undergoes major deformations once the fast-reaction systems limits onto the cross-diffusion singular limit. Furthermore, we find evidence for time-periodic solutions by detecting Hopf bifurcations, we characterize several regions of multi-stability, and improve our understanding of the shape of patterns in 2D for the SKT model.

We perform both analytical and numerical bifurcation analysis of an alternating forest and grassland ecosystem model coupled with human interaction. The model consists of two nonlinear ordinary differential equations incorporating the human perception of the value of the forest. The system displays multiple steady states corresponding to different forest densities as well as regimes characterized by both stable and unstable limit cycles. We derive analytically the conditions with respect to the model parameters that give rise to various types of codimension-one criticalities such as transcritical, saddle-node, and Andronov–Hopf bifurcations and codimension-two criticalities such as cusp and Bogdanov–Takens bifurcations at which homoclinic orbits occur. We also perform a numerical continuation of the branches of limit cycles. By doing so, we reveal turning points of limit cycles marking the appearance/disappearance of sustained oscillations. Such critical points that cannot be detected analytically give rise to the abrupt loss of the sustained oscillations, thus leading to another mechanism of catastrophic shifts.

We present results of the application of the numerical continuation and bifurcation package pde2path to the 3D Brusselator model, focusing on snaking branches of planar fronts between body centered cubes (BCCs) and the spatial homogeneous solution, and on planar fronts between BCCs and tubes (also called square prisms). These solutions also yield approximations of localized BCCs, and of BCCs embedded in a background of tubes (or vice versa). Additionally, we compute some moving fronts between lamellas and tubes. To give some theoretical background, and to aid the numerics for the full system, we use the Maxwell points for the cubic amplitude system over the BCC lattice., 15 pages, 7 figures, accepted by Physica D

We describe a method for computing an atlas for the stable or unstable manifold attached to an equilibrium point and implement the method for the saddle-focus libration points of the planar equilateral restricted four-body problem. We employ the method at the maximally symmetric case of equal masses, where we compute atlases for both the stable and unstable manifolds. The resulting atlases are comprised of thousands of individual chart maps, with each chart represented by a two-variable Taylor polynomial. Post-processing the atlas data yields approximate intersections of the invariant manifolds, which we refine via a shooting method for an appropriate two-point boundary value problem. Finally, we apply numerical continuation to some of the BVP problems. This breaks the symmetries and leads to connecting orbits for some nonequal values of the primary masses.

A high order unstructured spectral element method with a domain decomposition Stokes solver is presented for the hydrodynamic instability and bifurcation analysis. A Jacobian-Free Inexact-Newton-Krylov algorithm with a Stokes time-stepping preconditioning technique is introduced for the steady-state solution of incompressible flow. A matrix-free arc-length approach with Householder transformation is used for the numerical continuation near a turning point. An Arnoldi method is utilized to calculate the leading eigenvalues and their corresponding eigenvectors for the big system of linearized incompressible Navier-Stokes equations, which are responsible for initiating the hydrodynamic instability. The new method can do the steady and unsteady simulations in the similar way without time-splitting divergence error, it do not form the Jacobian matrix, which can reduce the memory allocation, decrease the computation cost, and speed up the convergence rate. The symmetric-breaking Hopf and Pitchfork bifurcations are considered in the flow passed a circular cylinder between two parallel plates. An antisymmetric sinusoidal velocity driven cavity problem is considered and the stable and unstable patterns are analyzed by checking the leading eigenvalues of their steady states. Besides the stable patterns of steady symmetric and steady asymmetric solutions£a new pair of unsteady asymmetric solutions are found depending on the different initial conditions.

This paper considers the extreme type-II Ginzburg-Landau equations, a nonlinear PDE model that describes the states of a wide range of superconductors. For two-dimensional domains, a robust method is developed that performs a numerical continuation of the equations, automatically exploring the whole solution landscape. The strength of the applied magnetic field is used as the bifurcation parameter. Our branch switching algorithm is based on Lyapunov-Schmidt reduction, but we will show that for an important class of domains an alternative method based on the equivariant branching lemma can be applied as well. The complete algorithm has been implemented in Python and tested for multiple examples. For each example a complete solution landscape was constructed, showing the robustness of the algorithm.

In this article multiple and generic bifurcations of planar discrete-time Hindmarsh-Rose oscillator are investigated in detail by bifurcation theory and numerical continuation techniques. Three kinds of one-parameter bifurcation and five kinds of two-parameter bifurcation are studied. Different kinds of critical cases of each bifurcation are computed by inner product method and their corresponding scenario are presented, such as possible transitions between different one-parameter bifurcation points derived from two-parameter bifurcation point. Especially, the bifurcations of higher iterations and the bifurcation distributions of two-parameter bifurcation point are observed.

In this paper, the resonance behavior of initial imperfect functionally graded material (FGM) cylindrical shells with the clamped–clamped boundary condition at two ends is investigated. The material properties considered here are graded in the thickness direction according to a simple power law in terms of the volume fractions and are affected by temperature. The classical deformation theory, von-Karman type nonlinear geometric relations and Hamilton’s principle are employed to derive nonlinear partial differential equation of the clamped circular cylindrical shell at two ends. The partial differential governing equations are truncated by the Galerkin technique. Under the effect of transverse external excitation and in-plane force, averaged equations in term of polar coordinate are obtained by using multiple scales method. It is assumed that the dynamic system is in the case of primary resonance and 1:2 internal resonance. The effects of the volume fraction, temperature, damping, imperfection, transverse external excitation and in-plane force on amplitude frequency response of FGM cylindrical shell with initial imperfection are studied in detail by the application of numerical continuation method. With varying the tuning parameter of the excitation frequency, saddle node bifurcation, hopf bifurcation, saddle node bifurcation and hopf bifurcation are detected.

An initially curved beam is considered and its motion is constrained using two elastic constraints; the corresponding non-smooth nonlinear transverse dynamics is investigated for the first time. A clamped-clamped beam with one axially movable end is modelled via Bernoulli-Euler beam theory together with the inextensibility condition, giving rise to nonlinear inertial terms along with nonlinear geometric terms. Furthermore, the damping is modelled via Kelvin-Voigt internal damping model. The proposed model is verified for linear and nonlinear behaviours via comparison to a finite element model. The impact between beam and constraints is incorporated via calculating its work contribution. The nonlinear equation of motion is derived while incorporating geometric, damping, inertial, and constraints nonlinearities. A series of spatial basis functions together with corresponding vibration modes are used as the proposed solution of the transverse displacement. A modal discretisation is performed via the weighted-residual method of Galerkin and the corresponding non-smooth terms are kept intact while conducting numerical integration. A numerical continuation technique is utilised to solve the resultant equations. The non-smooth response is obtained for various cases and the effects of several parameters are studied thoroughly.

In this paper, a change in stability (bifurcation) of a harmonically excited impact oscillator interacting with an elastic constraint is used to determine the stiffness of constraint. For this purpose, detailed one- and two-parameter bifurcation analyzes of the impacting system are carried out by means of experiments and numerical methods. This study reveals the presence of codimension-one bifurcations of limit cycles, such as grazing, period-doubling and fold bifurcations, as well as a cusp singularity and hysteretic effects. Particularly, the two-parameter continuation of the obtained codimension-one bifurcations (including both period-doubling and fold bifurcations) indicates a strong correlation between the stiffness of the impacted constraint and the frequency at which a certain bifurcation appear. The undertaken approach may prove to be useful for condition monitoring of dynamical systems by identifying mechanical properties through bifurcation analysis. The theoretical predictions for the impact oscillator are verified by a number of experimental observations.

This paper is devoted to cable–flexible support coupled nonlinear vibrations using a asymptotic boundary modulation technique. Asymptotic/reduced cable–support coupled nonlinear models are established first using the boundary modulation concept, after a proper scaling analysis at the cable–support interface. The cable and the support turn out to be coupled through cable-induced and support-induced boundary modulations in a rational way, which are derived analytically by asymptotic approximations and multiple scale expansions. Based upon the reduced models, two prototypical kinds of cable–support coupled dynamics are fully investigated, i.e., one with the support excited and the other with the cable excited. Essentially, they correspond to refined versions of two typical degenerate cable dynamics, i.e., cables excited externally with fixed supports and cables excited by ideal moving supports. Applying numerical continuation algorithms to the reduced models, cable–support typical coupled frequency response diagrams are constructed, with their stabilities, bifurcation characteristics, and the coupling’s effects on the cable determined. All these approximate analytical results are verified by the numerical results from the original full cable–support system using the finite difference method.