Numerical Bifurcation Research Articles

Connecting Gaits in Energetically Conservative Legged Systems

In this work, we present a nonlinear dynamics perspective on generating and connecting gaits for energetically conservative models of legged systems. In particular, we show that the set of conservative gaits constitutes a connected space of locally defined 1D submanifolds in the gait space. These manifolds are coordinate-free parameterized by energy level. We present algorithms for identifying such families of gaits through the use of numerical continuation methods, generating sets and bifurcation points. To this end, we also introduce several details for the numerical implementation. Most importantly, we establish the necessary condition for the Delassus’ matrix to preserve energy across impacts. An important application of our work is with simple models of legged locomotion that are often able to capture the complexity of legged locomotion with just a few degrees of freedom and a small number of physical parameters. We demonstrate the efficacy of our framework on a one-legged hopper with four degrees of freedom.

Numerical Bifurcation Analysis of PDEs From Lattice Boltzmann Model Simulations: a Parsimonious Machine Learning Approach

We address a three-tier data-driven approach for the numerical solution of the inverse problem in Partial Differential Equations (PDEs) and for their numerical bifurcation analysis from spatio-temporal data produced by Lattice Boltzmann model simulations using machine learning. In the first step, we exploit manifold learning and in particular parsimonious Diffusion Maps using leave-one-out cross-validation (LOOCV) to both identify the intrinsic dimension of the manifold where the emergent dynamics evolve and for feature selection over the parameter space. In the second step, based on the selected features, we learn the right-hand-side of the effective PDEs using two machine learning schemes, namely shallow Feedforward Neural Networks (FNNs) with two hidden layers and single-layer Random Projection Networks (RPNNs), which basis functions are constructed using an appropriate random sampling approach. Finally, based on the learned black-box PDE model, we construct the corresponding bifurcation diagram, thus exploiting the numerical bifurcation analysis toolkit. For our illustrations, we implemented the proposed method to perform numerical bifurcation analysis of the 1D FitzHugh-Nagumo PDEs from data generated by D1Q3 Lattice Boltzmann simulations. The proposed method was quite effective in terms of numerical accuracy regarding the construction of the coarse-scale bifurcation diagram. Furthermore, the proposed RPNN scheme was sim 20 to 30 times less costly regarding the training phase than the traditional shallow FNNs, thus arising as a promising alternative to deep learning for the data-driven numerical solution of the inverse problem for high-dimensional PDEs.

Numerical analysis of the behavioral response of pump-modulated linear and ring erbium-doped fiber lasers

ObjectiveThe behavioral response of an erbium-doped fiber laser having two different configurations (linear cavity and ring cavity) and subjected to pump modulation is analyzed numerically. These same laser configurations with the same initial conditions have previously been studied experimentally. MethodologyFor the numerical analysis, a modified laser model based on the rate equations of a three-level class B laser which is quite similar to that of a non-linear oscillator is presented. ResultsThe numerical model generates all the linear and non-linear spectral features observed during the experimental analysis. The experimental and numerical bifurcation, time domain and frequency domain diagrams match reasonably well. The dynamic behavior of the EDFRL is numerically analyzed at low modulation frequencies (below 7 kHz) when the resonance frequency ranges from 2 kHz to 10 kHz. The results for the descending chirp of the EDRFL under certain pump modulation conditions showed a new strange dynamic behavior requiring further studies. SignificanceThe EDFRL can be used for sensing purposes in locations with high background noise as it is sensitive even at low frequencies irrespective of the noise. Numerical analysis is a fast and reliable way to obtain remarkable information on the spectral behavior of the EDFL.

A dynamic interplay between Allee effect and time delay in a mathematical model with weakening memory

This paper deals with a model of population dynamics comprising Allee effect and weakening memory with constant time delays. Since predator density depends on the prey density in current time and in past, a two-component model of prey-predator interactions is complemented with a third differential equation for the influence of recent past. The role of constant time delays incorporated in the functional form of Allee effect (referred as delays in competition and cooperation) is investigated analytically and numerically. Steady states of the model are obtained and the local stability analysis around the coexisting state is calculated in the presence of both delays. The critical threshold for time delays, above which the stability of the system switches from stable (unstable) to unstable (stable), is computed for various cases. Analytical findings of this paper are supported with numerical simulations, where time evolution as well as numerical bifurcation diagrams are presented. The results of this paper demonstrate that the influence of past on the prey-predator density in the present of time delay may have a considerable effect upon the system behaviour and can give important insights into underlying biological mechanism.

Initial source of heterogeneity in a model for cell fate decision in the early mammalian embryo.

During development, cells from a population of common progenitors evolve towards different fates characterized by distinct levels of specific transcription factors, a process known as cell differentiation. This evolution is governed by gene regulatory networks modulated by intercellular signalling. In order to evolve towards distinct fates, cells forming the population of common progenitors must display some heterogeneity. We applied a modelling approach to obtain insights into the possible sources of cell-to-cell variability initiating the specification of cells of the inner cell mass into epiblast or primitive endoderm cells in early mammalian embryo. At the single-cell level, these cell fates correspond to three possible steady states of the model. A combination of numerical simulations and bifurcation analyses predicts that the behaviour of the model is preserved with respect to the source of variability and that cell-cell coupling induces the emergence of multiple steady states associated with various cell fate configurations, and to a distribution of the levels of expression of key transcription factors. Statistical analysis of these time-dependent distributions reveals differences in the evolutions of the variance-to-mean ratios of key variables of the system, depending on the simulated source of variability, and, by comparison with experimental data, points to the rate of synthesis of the key transcription factor NANOG as a likely initial source of heterogeneity.

Three-Variable Reduction of the Spin-Flip Model for Spin-VCSELs

Recent increased interest in spin-polarised vertical-cavity surface-emitting lasers (spin-VCSELs) as potential high-speed sources has spurred research into the analysis of their dynamics. This has often been explored by combining the spin-flip model (SFM) with numerical methods. However, numerical simulation does not readily expose key dependencies and to date there is a lack of accessible closed-form analytical results for the steady-state solutions and dynamic stability boundaries. Thus in the present contribution we address this and show that, for zero dichroism, the five rate equations of the SFM can be reduced to a set of three. These can be solved in the steady-state in terms of the intensity and polarisation of the pump (optical or electrical), and the ellipticity of the output. Additionally, a small-signal analysis leads to analytic results for the boundaries between stable and unstable operation in the plane of pump ellipticity versus pump intensity. Comparison of the results from these expressions with those from numerical bifurcation and continuation methods shows very good agreement. The accuracy of the reduced set of equations is confirmed by comparing the results with those from the full set of SFM equations. In the limiting case of very high birefringence, as would be required for a potential THz source, a simple algebraic relation is derived for the spin relaxation rate in terms of other parameters. Hence we find that the range of spin relaxation rates to achieve THz oscillations is very limited. The relative simplicity of the present approach thus offers a rapid, intuitive and convenient route to study the dynamics of spin-VCSELs.

Oscillatory wave patterns and spiral breakup in the Brusselator model using numerical bifurcation analysis

This article addresses the core breakup analysis of the spiral dynamics of an oscillatory system of chemical reactions, the so-called Brusselator model. We first obtain stable oscillatory periodic solutions corresponding to the stable limit cycle near a Hopf bifurcation point when the diffusion terms are neglected. Then, we investigate the occurrence of periodic traveling wave solutions of the model and perform the stability analysis of these solutions on the parameter plane. A stability boundary is also identified on the parameter plane. In the two-dimensional spatial domain, we illustrate rotating spiral waves and their instability that leads to a spiral breakup from the center of rotation.

A Discrete Dynamics Approach to a Tumor System

In this paper, we present a cancer system in a continuous state as well as some numerical results. We present discretization methods, e.g., the Euler method, the Taylor series expansion method, and the Runge–Kutta method, and apply them to the cancer system. We studied the stability of the fixed points in the discrete cancer system using the new version of Marotto’s theorem at a fixed point; we prove that the discrete cancer system is chaotic. Finally, we present numerical simulations, e.g., Lyapunov exponents and bifurcations diagrams.

Global bifurcation and pattern formation for a reaction–diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses

Of concern in this paper is to propose an accurate description for the global bifurcation structure of the nonconstant steady states for a reaction–diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses systematically. By treating the coefficient of nonlinear prey-taxis as a bifurcation parameter and utilizing the user-friendly version of Crandall–Rabinowitz bifurcation theory, we study the global bifurcation theory of the system. Meanwhile, the existence of nonconstant steady states will be offered by the exported global bifurcation theorem under a rather natural condition. In the proof, a priori estimate of steady states will play an important role. The local stability analysis with a numerical simulation and bifurcation analysis are given.

Washout filter control technique for a fractional-order chaotic finance model

On the basis of the previous publications, we set up a new fractional-order chaotic finance model. Taking advantage of washout filter controller, we are committed to the study on the chaotic control for the considered fractional-order chaotic finance model. Applying the stability criterion and Hopf bifurcation knowledge of fractional order differential dynamical system, a series of delay-independent stability and bifurcation conditions are established to assure the stability and the occurrence of Hopf bifurcation of the fractional-order controlled chaotic finance model. The investigation indicates that the role of delay in controlling the stability and Hopf bifurcation for the fractional-order controlled chaotic finance model is fully reflected. In the end, numerical simulation figures and bifurcation plots are presented to sustain the established primary conclusions. Our work supplements the earlier studies and enriches the chaos control theory of fractional-order dynamical system.

Medial Gap: A Structural Factor at the Arterial Bifurcation Aggravating Hemodynamic Insult.

Previous studies have reported that intracranial aneurysms frequently occur adjacent to the medial gap. However, the role of the medial gap in aneurysm formation is controversial. We designed this study to explore the potential role of the medial gap in aneurysm formation. Widened artery bifurcations with or without medial gaps were microsurgically created and pathologically stained in the carotid arteries of 30 rats. Numerical artery bifurcation models were constructed, and bidirectional fluid-solid interaction analyses were performed. Animal experiments showed that the apexes of widened bifurcations with a medial gap were prone to being insulted by blood flow compared to those without a medial gap. The bidirectional fluid-solid interaction analyses indicated that artery bifurcations with the medial gap exhibited higher wall shear stress (WSS) and von Mises stress (VMS) at the apex of the bifurcation. The disparity of stress between the gap and no-gap model was larger for widened bifurcations, peaking at 180° with a maximum of 1.9 folds. The maximum VMS and relatively high WSS were located at the junction between the medial gap and the adjacent arterial wall. Our results suggest that the medial gap at the widened arterial bifurcation may promote aneurysm formation.

Stability and Bifurcation of Mixing in the Kuramoto Model with Inertia

The Kuramoto model of coupled second order damped oscillators on convergent sequences of graphs is analyzed in this work. The oscillators in this model have random intrinsic frequencies and interact with each other via nonlinear coupling. The connectivity of the coupled system is assigned by a graph which may be random as well. In the thermodynamic limit the behavior of the system is captured by the Vlasov equation, a hyperbolic partial differential equation for the probability distribution of the oscillators in the phase space. We study stability of mixing, a steady state solution of the Vlasov equation, corresponding to the uniform distribution of phases. Specifically, we identify a critical value of the strength of coupling, at which the system undergoes a pitchfork bifurcation. It corresponds to the loss of stability of mixing and marks the onset of synchronization. As for the classical Kuramoto model, the presence of the continuous spectrum on the imaginary axis poses the main difficulty for the stability analysis. To overcome this problem, we use the methods from the generalized spectral theory developed for the original Kuramoto model. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model on Erd\H{o}s--R\'enyi and small-world graphs. Applications of the second-order Kuramoto model include power networks, coupled pendula, and various biological networks. The analysis in this paper provides a mathematical description of the onset of synchronization in these systems.

Codimension-Two Bifurcations and Bifurcation Controls in a Discrete Biological System with Weak Allee Effect

In this paper, the codimension-two bifurcations and bifurcation controls of a discrete predatory system with weak Allee effect on prey are investigated. After analyzing the qualitative properties of all the fixed points, we explore the existences of 1:2 and 1:3 strong resonances at the unique interior fixed point. Combining the adjoint operator method with an approximate identity transformation, we obtain the normal forms of these strong resonances. Meanwhile, we calculate the approximate expressions of bifurcation curves around the resonance point, which include Neimark–Sacker and period doubling bifurcation curves. And we give the conditions under which these bifurcations are supercritical or subcritical. To study global bifurcations near the resonance point, we use an ordinary differential equation to approximate the original discrete system by the time-one approximation flow theory. In biology, bifurcations and unstable fluctuations may be harmful for the breeding of biological population. Then we extend the hybrid control strategy to control 1:2 and 1:3 strong resonances. Numerical simulations are in good agreement with our theoretical results. By using the MatContM package, we perform numerical bifurcation analyses to further support our theoretical results, in which the bifurcation points and the normal form coefficients are completely consistent with our theories. According to the results, we finally analyze the impact of weak Allee effect on the system. And a reasonable biological explanation is given for these conclusions.

Grazing and Symmetry-Breaking Bifurcations Induced Oscillations in a Switched System Composed of Duffing and van der Pol Oscillators

By introducing a switching scheme related to the state and time, a typical switched model alternating between a Duffing oscillator and van der Pol oscillator is established to explore the typical dynamical behaviors as well as the mechanism of the switched system. Shooting methods to locate the limit cycle and specify bifurcation sets are described by defining an appropriate Poincaré map. Different types of multiple-Focus/Cycle and single-Focus/Cycle period oscillations in the system can be observed. Symmetry-breaking, period-doubling, and grazing bifurcation curves are obtained in the plane of bifurcation parameters, dividing the parameters plane into several regions corresponding to different kinds of oscillations. Meanwhile, based on the numerical simulation and bifurcation analysis, the mechanisms of several typical dynamical behaviors observed in different regions are presented.

Joint Effects of Dispersal and Trophic Interactions on the Biodiversity of Species

Local dispersal and trophic interactions are thought to have important influences on species range limits (i.e., the boundaries of locations where species is found) across geographical regions. To better understand this problem, we perform a numerical bifurcation analysis of a partial differential equations model. This system consists of prey-predator interactions, environmental suitability, and local dispersal terms. The main objective of this investigation is to examine the combined effects of distinct ecological factors on species population dynamics. In the absence of dispersal, our results show that variations in abiotic environments and handling times of predator lead to different outcomes of species interactions (i.e., stable limit cycles, species coexistence, and species exclusion). The inclusion of dispersal into this system reduces the predation pressure and stabilizes this ecological community to mediate species biodiversity. Local dispersal also induces intriguing dynamical behaviour in the system such as bistability and unstable limit cycles.

Dynamical System Analysis of the Prey-predator Interactions involving Prey Refuge and Herd Behaviors in Preys

By employing a prey refuge mechanism, more preys can be protected from predation. Prey species are also better protected from predation when they congregate in herds. However, what if the prey refuge and herd behavior mechanisms were combined in a system? To investigate this phenomenon, we consider two different prey-predator systems with prey refuge capacity. The first system is a simple prey-predator with prey refuge, whereas the second system considers prey refuge and prey herd behavior mechanisms. Using these models, we explore how different prey refuge strategies affect species interactions in both systems. To accomplish this, we use theoretical techniques (e.g., computing steady states and performing the stability analysis) and numerical bifurcation analysis to demonstrate various dynamical behaviors of these two prey-predator systems. Once prey refuge is treated as a bifurcation parameter, we observe the occurrence of supercritical Hopf and transcritical bifurcations in both systems. Furthermore, we explore the dynamic effects of prey refuge and predator handling time on species population interactions: our findings reveal that using both prey refuge and herd behavior as escape strategies; it is possible to dilute the predation pressure and ensure species biodiversity.

Non-linear modal interactions during rub-impact of a rotating flexible shaft

An understating on interaction between a rotating flexible member and a stationary element has a wide range of industrial applications. The focus of this study was to understand the modal interactions using a conceptual model. The conceptual model of the system consists of multiple disc rotors with a localised conduit. A theoretical model of the system was developed using finite element analysis. An eigenvalue analysis was carried out on the system and a Campbell diagram of the system was developed. The Campbell diagram provided information about the rotor speed wherein synchronization with the whirling modes occur. Further, a reduced order non-linear contact model was developed. A numerical bifurcation analysis with rotor speed as parameter was carried out on the reduced model. Analysis was carried out using randomized initial condition for each rotation speed. During the bifurcation study, certain rotation speed produced a high whirling response at speeds different from the critical speed. The increase in the response was associated with the internal resonance of whirling modes and rotor speed. The participating modes were identified using the Campbell diagram and full spectrum. The results from the theoretical model were validated using an experimental test rig. The identification of these internal resonance rotor speed can be useful in identifying the rotor speed exclusion zone and thereby improves the rotor life.

Robust nonlinear model predictive control of cascade of fermenters with recycle for efficient bioethanol production

In this paper, dynamical behavior and control of the continuous ethanol fermentation process are studied. The process productivity is improved by using two fermenters connected in series with recycle of cells. First, dynamical behavior and productivity of the cascade system are studied by using numerical bifurcation analysis. Then, multi-stage nonlinear model predictive control (MS-NMPC), classical and offset-free NMPCs algorithms are designed to eliminate the undesired oscillations. To ensure acceptable efficiency of the cascade process, a minimal ethanol concentration is defined and included as additional constraint in the controller design. The unmeasured state variables are estimated by using the asymptotic observer. It is shown that for smaller sampling times the predictive controllers are very effective, even in presence of parameter uncertainties. It is also observed that offset-free NMPC algorithm is more sensitive to measurement noise, but MS-NMPC and classical NMPC are more sensitive to estimation errors for longer sampling times.

Chimeras Unfolded

The instability of mixing in the Kuramoto model of coupled phase oscillators is the key to understanding a range of spatiotemporal patterns, which feature prominently in collective dynamics of systems ranging from neuronal networks, to coupled lasers, to power grids. In this paper, we describe a codimension–2 bifurcation of mixing whose unfolding, in addition to the classical scenario of the onset of synchronization, also explains the formation of clusters and chimeras. We use a combination of linear stability analysis and Penrose diagrams to identify and analyze a variety of spatiotemporal patterns including stationary and traveling coherent clusters and twisted states, as well as their combinations with regions of incoherent behavior called chimera states. Penrose diagrams are used to locate the bifurcation of mixing and to determine its type. The linear stability analysis, on the other hand, yields the velocity distribution of the pattern emerging from the bifurcation. Furthermore, we show that network topology can endow chimera states with nontrivial spatial organization. In particular, we present twisted chimera states, whose coherent regions are organized as stationary or traveling twisted states. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model with uni-, bi-, and trimodal frequency distributions and all-to-all and nonlocal nearest-neighbor connectivity.

Generalized Rabinovich–Fabrikant system: equations and its dynamics

The purpose of this work is to numerically study of the generalized Rabinovich–Fabrikant model. This model is obtained using the Lagrange formalism and describing the three-mode interaction in the presence of a general cubic nonlinearity. The model demonstrates very rich dynamics due to the presence of third-order nonlinearity in the equations. Methods. The study is based on the numerical solution of the obtained analytically differential equations, and their numerical bifurcation analysis using the MаtCont program. Results. For the generalized model we present a charts of dynamic regimes in the control parameter plane, Lyapunov exponents depending on parameters, portraits of attractors and their basins. On the plane of control parameters, bifurcation lines and points are numerically found. They are plotted for equilibrium point and period one limit cycle. It is shown that the dynamics of the generalized model depends on the signature of the characteristic expressions presented in the equations. A comparison with the dynamics of the Rabinovich– Fabrikant model is carried out. We indicated a region in the parameter plane in which there is a complete or partial coincidence of dynamics. Conclusion. The generalized model is new and describes the interaction of three modes, in the case when the cubic nonlinearity that determines their interaction is given in a general form. In addition, since the considered model is a certain natural extension of the well-known Rabinovich–Fabrikant model, then it is universal. And it can simulate systems of various physical nature (including radio engineering), in which there is a three-mode interaction and there is a general cubic nonlinearity.