Numerical Bifurcation Analysis Research Articles

Disconnected stationary solutions for 3D Kolmogorov flow problem: preliminary results

The extension of the classical A.N. Kolmogorov’s flow problem for the stationary 3D Navier-Stokes equations on a stretched torus for velocity vector function is considered. A spectral Fourier method with the Leray projection is used to solve the problem numerically. The resulting system of nonlinear equations is used to perform numerical bifurcation analysis. The problem is analyzed by constructing solution curves in the parameter-phase space using previously developed deflated pseudo arc-length continuation method. Disconnected solutions from the main solution branch are found. These results are preliminary and shall be generalized elsewhere.

Numerical solution and bifurcation analysis of nonlinear partial differential equations with extreme learning machines

We address a new numerical method based on a class of machine learning methods, the so-called Extreme Learning Machines (ELM) with both sigmoidal and radial-basis functions, for the computation of steady-state solutions and the construction of (one-dimensional) bifurcation diagrams of nonlinear partial differential equations (PDEs). For our illustrations, we considered two benchmark problems, namely (a) the one-dimensional viscous Burgers with both homogeneous (Dirichlet) and non-homogeneous boundary conditions, and, (b) the one- and two-dimensional Liouville–Bratu–Gelfand PDEs with homogeneous Dirichlet boundary conditions. For the one-dimensional Burgers and Bratu PDEs, exact analytical solutions are available and used for comparison purposes against the numerical derived solutions. Furthermore, the numerical efficiency (in terms of numerical accuracy, size of the grid and execution times) of the proposed numerical machine-learning method is compared against central finite differences (FD) and Galerkin weighted-residuals finite-element (FEM) methods. We show that the proposed numerical machine learning method outperforms in terms of numerical accuracy both FD and FEM methods for medium to large sized grids, while provides equivalent results with the FEM for low to medium sized grids; both methods (ELM and FEM) outperform the FD scheme. Furthermore, the computational times required with the proposed machine learning scheme were comparable and in particular slightly smaller than the ones required with FEM.

The effect of characteristic times on collective modes of two quorum sensing coupled identical ring oscillators

There has long been interest in methods to achieve variability of collective modes in systems of coupled identical oscillators. Extensive multistability and a dynamically rich parameter space exist for two identical ring oscillators mean-field coupled by the quorum sensing mechanism when there is significant heterogeneity of the timescales of the ring elements’ variables. These findings have been shown in numerical and electrical circuit models based on the Repressilator, a ring oscillator using repressor genes as the inverter elements. Here we investigate the effect of reducing the degree of timescale heterogeneity. We find the 2D parameter map of dynamical regimes for the previously established heterogeneous timescale, using the coupling strength and the isolated oscillator strength for the parameters. The map shows a sea of anti-phase (AP) oscillation, with a large island of complex oscillations bounded by Neimark-Sacker torus bifurcation of the AP. The island is dominated by chaos produced by period doubling cascades of two prominent limit cycles (an unusual resonant cycle and a highly asymmetric cycle) and by torus destruction. Extensive regions of multistability exist, including between different oscillatory states, and, notably, coexistence of the steady state region and oscillatory states. Using numerical bifurcation analysis with timescale heterogeneity as a continuation parameter, we follow the evolution of the collective modes, reveal the appearance of new dynamical regimes, and determine changes in the multistability. We find that decreasing the heterogeneity causes: period-doubling cascades of the AP outside of the island, thereby creating new regimes of complexity; an increase in the dominance of chaos due to a reduction in the size of regions dominated by limit cycles; loss of stable oscillatory states inside steady state regions, converting those regions from multistable to sole dynamic steady state regions; and the appearance of in-phase oscillations and its sole dominance.

Analytical and Numerical Bifurcation Analysis of Circuits Based on Nonlinear Resonators

This work presents new frequency-domain methods for the analysis and simulation of circuits based on a nonlinear resonator, with operation ranges delimited by turning points. Insightful analytical conditions fulfilled at these turning points are derived, which will enable an identification of the effect of each circuit element on their location in the solution curve. The cusp points, or co-dimension two bifurcations at which two turning points merge into one, thus delimiting the multivalued intervals, are directly calculated for the first time to our knowledge. In addition, a new numerical method, compatible with the use of commercial harmonic balance, is presented for the straightforward tracing of the multivalued solution curves, together with a new procedure to determine the locus of turning points in terms of any two analysis parameters. This relies on the use of a new mathematical condition to obtain a surface of turning points in the space defined by the two parameters and the input power. The methods have been applied to a wireless power-transfer system based on a recently proposed configuration, obtaining very good agreement with the experimental results.

Internal phosphorus recycling promotes rich and complex dynamics in an algae-phosphorus model: Implications for eutrophication management

Internal phosphorus recycling in lakes is an important nutrient source that promotes algal growth. Its persistence impedes the effort to improve water quality and thus poses a challenge to the management of eutrophication in lakes, especially in shallow lakes where the occurrence of internal phosphorus recycling is reportedly more common. This paper aims to provide crucial insights on the effects of internal phosphorus recycling on eutrophication dynamics for effective management of lake eutrophication. For this purpose, a mathematical model for lake eutrophication, comprising two compartments of algae and phosphorus, is first formulated for application to a eutrophic tropical lake named Tasik Harapan in Universiti Sains Malaysia. Numerical bifurcation analysis of the model is then performed to assess the combined influences of internal phosphorus recycling, algal mortality and external phosphorus loading on Tasik Harapan eutrophication dynamics. Specifically, co-dimension one bifurcation analysis of algal mortality rate is carried out by means of XPPAUT for various external phosphorus loading rates. The emergence of limit cycle for a certain range of algal mortality rate could be related to the hydra effect (i.e., algal concentration increases in response to greater algal mortality) and the paradox of enrichment (i.e., destabilization of algae in nutrient rich environment). To trace the locus of co-dimension one bifurcation, co-dimension two bifurcation analysis is performed by means of MatCont. The analysis demonstrated that the inclusion of the internal phosphorus recycling term induces rich and complex dynamics of the model. These dynamics include saddle-node bifurcation, cusp, Bogdanov-Takens bifurcation, Generalized Hopf bifurcation and limit point bifurcation of cycles. The results suggest that high internal phosphorus recycling rate promotes bistability and catastrophic shift in a shallow and tropical lake like Tasik Harapan. Hence, the key to effective management of eutrophication in shallow and tropical lakes is the control of internal phosphorus recycling.

Data-Driven Mathematical Modeling and Dynamical Analysis for SARS-CoV-2 Coinfection with Bacteria

The ongoing coronavirus disease-2019 (COVID-19) pandemic caused by severe acute respiratory syndrome-coronavirus 2 (SARS-CoV-2) has serious influences on human health and economy. The available clinical data suggest that patients infected by SARS-CoV-2 have the possibility of simultaneous infection of bacteria. In this study, we present a data-driven mathematical model for coinfection of SARS-CoV-2 and bacteria to investigate the dynamics of COVID-19 progress. Specifically, based on the statistical analysis of different clinical data from China and some other countries, a system model with ordinary differential equations (ODEs) in four variables, i.e. SARS-CoV-2, bacteria, neutrophils and lymphocytes, is established. We further validate our model through theoretical analysis and fitting of different clinical data. Moreover, through numerical simulations and bifurcation analysis, we find that bacterial infection and immune-related parameters in certain ranges lead to the system transitions among three steady states, i.e. mild, severe and death. We also analyzed the influence of the time it takes for patients to switch from a high-risk area to a low-risk area on the recovery time. These results reveal that the coinfection of viruses and bacteria can explain the changes in neutrophils and lymphocytes, and that initial bacterial infection and immune-related parameters have great influences on the severity degree and mortality in COVID-19 patients. Together, our model and quantitative analysis suggest that preventing bacterial infection and enhancing immune ability during the early phase of infections could be a potential treatment option for high-risk COVID-19 patients.

Nonlinear dynamics of fully developed swirling jets

This paper characterises the steady and time-periodic behaviour of swirling jets using numerical bifurcation analysis. Its objective is to elucidate the dynamics of fully developed, unconfined, laminar swirling jets under variations in the Reynolds number $\textit {Re}$ and swirl ratio $S$ . Within the $(0,0)\leq (\textit {Re},S)\leq (300,3)$ range, the steady, axisymmetric flow exhibits several distinct patterns ranging from a quasi-columnar jet along the central axis at low $S$ to a radial jet attached to the containing wall at high $S$ with various forms of vortex breakdown in between. A cusp bifurcation appears in the steady solution manifold which triggers bistable behaviour due to a competition between inner and outer low pressure regions associated with vortex breakdown and entrainment of the ambient fluid, respectively. Instability of the steady flow is linked to eigenmodes which are singly ( $|m|=1$ ) or doubly ( $|m|=2$ ) azimuthally periodic, although additional instabilities with other azimuthal wavenumbers occur at $(\textit {Re},S)$ -values beyond the leading neutral curves. The various branches of limit cycle solutions stemming from these neutral curves are associated with both super- and sub-critical Hopf bifurcations. The resulting unsteady flow fields exhibit a wide array of rotating, three-dimensional flow structures, and comparisons between the time-averaged and steady flow patterns highlight the role of these unsteady nonlinear interactions on the overall behaviour of swirling jets. Similarities and differences between this laterally unconfined jet and broader classes of swirling flows, including confined swirling jets and unconfined vortex models, are also discussed.

Noise-induced dynamical regimes in a system of globally coupled excitable units.

We study the interplay of global attractive coupling and individual noise in a system of identical active rotators in the excitable regime. Performing a numerical bifurcation analysis of the nonlocal nonlinear Fokker-Planck equation for the thermodynamic limit, we identify a complex bifurcation scenario with regions of different dynamical regimes, including collective oscillations and coexistence of states with different levels of activity. In systems of finite size, this leads to additional dynamical features, such as collective excitability of different types and noise-induced switching and bursting. Moreover, we show how characteristic quantities such as macroscopic and microscopic variability of interspike intervals can depend in a non-monotonous way on the noise level.

Assessing the Impacts of Competition and Dispersal on a Multiple Interactions Type Model

Multiple interactions (e.g., mutualist-resource-competitor-exploiter interactions) type models are known to exhibit oscillatory behaviour as a result of their complexity. This large-amplitude oscillation often de-stabilises multispecies communities and increases the chances of species extinction. What mechanisms help species in a complex ecological system to persist? Some studies show that dispersal can stabilise an ecological community and permit multi-species coexistence. However, previous empirical and theoretical studies often focused on one- or two-species systems, and in real life, we have more than two-species coexisting together in nature. Here, we employ a (four-species) multiple interactions type model to investigate how competition interacts with other biotic factors and dispersal to shape multi-species communities. Our results reveal that dispersal has (de-)stabilising effects on the formation of multi-species communities, and this phenomenon shapes coexistence mechanisms of interacting species. These contrasting effects of dispersal can best be illustrated through its combined influences with the competition. To do this, we employ numerical simulation and bifurcation analysis techniques to track the stable and unstable attractors of the system. Results show the presence of Hopf bifurcations, transcritical bifurcations, period-doubling bifurcations and limit point bifurcations of cycles as we vary the competitive strength in the system. Furthermore, our bifurcation analysis findings show that stable coexistence of multiple species is possible for some threshold values of ecologically-relevant parameters in this complex system. Overall, we discover that the stability and coexistence mechanisms of multiple species depend greatly on the interplay between competition, other biotic components and dispersal in multi-species ecological systems.

Localized and extended patterns in the cubic-quintic Swift-Hohenberg equation on a disk.

Axisymmetric and nonaxisymmetric patterns in the cubic-quintic Swift-Hohenberg equation posed on a disk with Neumann boundary conditions are studied via numerical continuation and bifurcation analysis. Axisymmetric localized solutions in the form of spots and rings known from earlier studies persist and snake in the usual fashion until they begin to interact with the boundary. Depending on parameters, including the disk radius, these states may or may not connect to the branch of domain-filling target states. Secondary instabilities of localized axisymmetric states may create multiarm localized structures that grow and interact with the boundary before broadening into domain-filling states. High azimuthal wave number wall states referred to as daisy states are also found. Secondary bifurcations from these states include localized daisies, i.e., wall states localized in both radius and angle. Depending on parameters, these states may snake much as in the one-dimensional Swift-Hohenberg equation, or invade the interior of the domain, yielding states referred to as worms, or domain-filling stripes.

On numerical bifurcation analysis of periodic motions of autonomous Hamiltonian systems with two degrees of freedom

In this work we consider bifurcation problem for natural families of periodic motions of autonomous Hamiltonian systems with two degrees of freedom. While there exists a well-developed analytical approach to this problem, it is limited to small neighborhoods of known equilibria and stationary solutions. To explore the bifurcations of periodic motions for all admissible values of the problem’s parameters it is necessary to employ numerical methods. We propose an approach combining analytical and numerical computation of the natural families with numerical bifurcation analysis. We obtain the so-called base solutions either analytically or numerically for particular values of the problem’s parameters and then employ a numerical method to continue the base solutions to the borders of their existence domains and to identify bifurcation points. Linear orbital stability domains are also obtained in course of the continuation. To illustrate the proposed approach we analyze bifurcation of periodic motions emanating from Cylindrical precession of a dynamically-symmetric satellite on a circular orbit.

Numerical bifurcation analysis of renewal equations via pseudospectral approximation

We propose an approximation of nonlinear renewal equations by means of ordinary differential equations. We consider the integrated state, which is absolutely continuous and satisfies a delay differential equation. By applying the pseudospectral approach to the abstract formulation of the differential equation, we obtain an approximating system of ordinary differential equations. We present convergence proofs for equilibria and the associated characteristic roots, and we use some models from ecology and epidemiology to illustrate the benefits of the approach to perform numerical bifurcation analyses of equilibria and periodic solutions. The numerical simulations show that the implementation of the new approximating system can be about ten times more efficient in terms of computational times than the one originally proposed in Breda et al. (2016), as it avoids the numerical inversion of an algebraic equation.

Numerical bifurcation and stability analysis of variational gradient-damage models for phase-field fracture

Gradient damage models used in phase-field approaches to brittle fracture are characterised by material softening and instabilities. We present novel numerical techniques for the bifurcation and stability analysis along quasi-static evolution paths as well as practical tools to select stable evolutions. Our approach stems from the variational approach to fracture and the theory of rate-independent irreversible processes whereby a quasi-static evolution is formulated in terms of incremental energy minimisation under unilateral constraints. Focusing on the discrete setting obtained with finite elements techniques, we discuss the links between bifurcation criteria for an evolution and stability of equilibrium states. Key concepts are presented through the analytical solution of a two-degrees-of-freedom model featuring a continuum family of bifurcation branches. We introduce numerical methods to (i) assess (second-order) stability conditions for time-discrete evolutions subject to damage irreversibility, and (ii) to select possible stable evolutions based on an energetic criterion. Our approach is based on the solution of a coupled eigenvalue problem which accounts for the time-discrete irreversibility constraint on damage. Several numerical examples illustrate that this approach allows us to filter out unstable solutions provided by standard (first-order) minimisation algorithms as well as to effectively compute stable evolution paths. We demonstrate our purpose on a multifissuration problem featuring complex fracture patterns, to show how the eigenvalue analysis enables to compute and retrieve morphological properties of emerging cracks.

Multiple solutions and hysteresis in the flows driven by surface with antisymmetric velocity profile* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11902043 and 11772065), and the Science Challenge Project (Grant No. TZ2016001).

Multiple steady solutions and hysteresis phenomenon in the square cavity flows driven by the surface with antisymmetric velocity profile are investigated by numerical simulation and bifurcation analysis. A high order spectral element method with the matrix-free pseudo-arclength technique is used for the steady-state solution and numerical continuation. The complex flow patterns beyond the symmetry-breaking at Re ≃ 320 are presented by a bifurcation diagram for Re < 2500. The results of stable symmetric and asymmetric solutions are consistent with those reported in literature, and a new unstable asymmetric branch is obtained besides the stable branches. A novel hysteresis phenomenon is observed in the range of 2208 < Re < 2262, where two pairs of stable and two pairs of unstable asymmetric steady solutions beyond the stable symmetric state coexist. The vortices near the sidewall appear when the Reynolds number increases, which correspond to the bifurcation of topology structure, but not the bifurcation of Navier–Stokes equations. The hysteresis is proposed to be the result of the combined mechanisms of the competition and coalescence of secondary vortices.

Modelling the impact of plasma therapy and immunotherapy for recovery of COVID-19 infected individuals.

Since the first case of COVID-19 was detected in Wuhan, China in December 2019, COVID-19 has become a pandemic causing a global economic and public health emergency. There is no known treatment or vaccine available for COVID-19 during the initial period of the outbreak. Immunotherapy and plasma therapy has been used with satisfactory efficacy over the past two decades in many viral infections like SARS (Systemic Acute Respiratory Syndrome), MERS (Middle East Respiratory Syndrome) and H1N1. Limited data from China show clinical benefit, radiological resolution, reduction in viral loads and improved survival. We aim to create a mathematical model for COVID-19 transmission and then apply various control parameters to see their effects on recovery from COVID-19 disease. We have formulated a system of non-linear ordinary differential equations, calculated basic reproduction R_{0} and applied five different controls (self-isolation, quarantine, herd immunity, immunotherapy, plasma therapy) to test the effectiveness of plasma therapy. Control optimality was checked by Lagrangian functions. Numerical simulations and bifurcation analysis were carried out. The study concludes that the COVID-19 outbreak can be controlled up to a significant level in three weeks after applying all the control strategies together. These strategies lead to reduction in hospitalization and a rise in recovery from infection. Immunotherapy is highly effective initially in hospitalized infected individuals however better results were seen in the long term with plasma therapy.

Numerical bifurcation analysis for a prey-predator type interactions with a time lag and habitat complexity

In this paper, a two-component generic prey-predator system incorporated with habitat complexity in predator functional response, and with constant time delay in predator gestation is considered. Although the role of time delay on the system dynamics is widely studied in the literature, only a few researchers have addressed the effect of habitat complexity in the prey-predator type interactions. In the first part of the paper the equilibria and stability analysis of the mathematical model is mentioned. In the second part, particular attention is paid on the numerical bifurcation analysis of the prey and predator densities based on two system parameters:(i) the strength of homogeneous habitat complexity and (ii) predator attack rate with and without time delay. It is found that dynamics with time delay in predator gestation are found to be much richer compared to that without time delay. The system stability may change from stable to unstable through a Hopf bifurcation and the solution branches emanating from these Hopf points are usually stable and supercritical. However, delay driven system may lead unstable orbits arising from Hopf bifurcations. It is also found that increasing the strength of habitat complexity may lead the stability change from unstable to stable.

Bifurcation analysis of the influence of generator excitation forms on system voltage stability

Based on a typical power system model, this paper simulates the scenario of the increase of active power output of a proximal generator around certain bus node and applies bifurcation theory to study the influence of different excitation forms on system voltage stability after reaching generator excitation saturation. With numerical bifurcation analysis software MATCONT, the simulation analysis results show that the smaller the excitation output upper limit value is, the earlier the bus voltage reaches the Hopf bifurcation with the increase of active power output. In the case of self-shunt excitation, the output of the excitation upper limit value is constrained by the generator terminal voltage. Therefore, the bifurcation point in this case occurs earlier than that in the case of separate excitation.

Selective Transport of Airborne Microparticles Through Micro-channels Under Microgravity

This paper analyzes the possibility of obtaining the selective transport of microparticles suspended in air in a microgravity environment through modulated channels without net displacement of air. Using numerical simulation and bifurcation analysis tools, we show the existence of intermittent particle drift under the Stokes assumption of the fluid flow. The particle transport can be selective and the direction of transport is controlled only by the kind of pumping used. The selective transport is interpreted as a deterministic ratchet effect due to spatial variations in the flow and the particle drag. This ratchet phenomenon could be applied to the selective transport of metal particles during the short duration of microgravity experiments.

Analytical and numerical bifurcation analysis of a forest ecosystem model with human interaction

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.

Pseudospectral Approximation of Hopf Bifurcation for Delay Differential Equations

Pseudospectral approximation reduces DDE (delay differential equations) to ODE (ordinary differential equations). Next one can use ODE tools to perform a numerical bifurcation analysis. By way of an example we show that this yields an efficient and reliable method to qualitatively as well as quantitatively analyse certain DDE. To substantiate the method, we next show that the structure of the approximating ODE is reminiscent of the structure of the generator of translation along solutions of the DDE. Concentrating on the Hopf bifurcation, we then exploit this similarity to reveal the connection between DDE and ODE bifurcation coefficients and to prove the convergence of the latter to the former when the dimension approaches infinity.