Numerical Bifurcation Research Articles

Dynamics of Three Coupled Generators of Quasi-periodic Oscillations

The dynamics of three coupled generators capable of demonstrating autonomous quasi-periodic oscillations is studied. The complex structure of the Lyapunov charts of the system is discussed, which reveals invariant tori of different dimensions, quasi-periodic bifurcations of tori, and Arnold’s resonant web based on tori of different dimensions. Cases of different types of tuning of individual generators (periodic oscillations, quasi-periodic oscillations) are considered. A detailed numerical bifurcation analysis of the equilibrium state and limit cycles, which form a complex picture of dynamic regimes, is carried out.Common features and differences compared to the case of three coupled van der Pol oscillators are discussed.

Global dynamics and bifurcation for a discontinuous oscillator with irrational nonlinearity

The objective is to carry out thorough analysis of global dynamics and bifurcation for a discontinuous oscillator, showing the transition of dynamics between the piecewise smooth system (with irrational nonlinearity) and the piecewise linear system. The existence of one or three equilibrium sets is proven by the approach of Filippov. Based on non-smooth Lyapunov functions and their set-valued derivatives, asymptotical stability especially finite-time asymptotical stability is shown for the equilibrium sets. In particular, some conditions are established guaranteeing the uniqueness of an equilibrium set and its global finite-time asymptotical stability. Making use of the parametric representation, codimension-one bifurcation curves and a codimension-two bifurcation point are obtained. We give numerical simulations and bifurcation diagram to illustrate the transition of dynamics and verify the results. We further clarify some previous misconceptions in Li et al. (2016).

Nigel Oscar Weiss. 16 December 1936—24 June 2020

Nigel Weiss was an astrophysicist and applied mathematician, born in South Africa but based in Cambridge for most of his career. He made a number of important contributions to the study of sunspots, where he produced a detailed model of their complex structure. He also made important contributions to theory of the solar 22-year magnetic cycle, using pioneering numerical modelling and drawing attention to the dynamical effects of symmetry breaking and of historic proxy data through observations of radioactive isotopes in tree rings and ice cores. A major part of his work was the numerical and theoretical investigation of many aspects of the interaction of magnetic fields with thermal convection in the Sun and stars. The numerical studies in particular were pioneering in the context of computer resources at that time. He led a talented group of students and postdocs, many of whom have held senior positions in UK and US universities. He combined great physical insight with expertise in numerical computation, nonlinear dynamics and bifurcation theory, and was an inspirational supervisor and mentor, combining high academic standards with an open generous nature. Apart from his work he had a strong interest in twentieth-century art and served on the National Gallery Consultative Group for many years.

New results for periodic solution in Liebau phenomenon

This paper is devoted to the existence and bifurcations of positive periodic solutions of nonlinear differential equation related to the Liebau phenomenon in rigid pipe-tank flow configuration. We obtain some more general requirements on the existence of a positive harmonic solution which improves the results in literatures. It is revealed that the system undergoes saddle node bifurcation, period doubling bifurcation and Neimark–Sacker bifurcation generating various periodic solution of different stability types. The multiplicity of both the positive harmonic solution and positive second-order subharmonic solution is first detected in this equation by numerical bifurcation analysis. Moreover, this work reveals that the positive subharmonic solutions of the nonlinear differential equation will also lead to the Liebau phenomenon in the model.

Mathematical modeling of the population dynamics of a distinct interactions type system with local dispersal

Distinct biotic interactions in multi-species communities are a ubiquitous force in the natural ecosystem, and this force is an essential determinant of community stability and species coexistence outcomes. We conduct numerical simulations and bifurcation analysis of partial differential equations to gain better understanding and ecological insights into how predation (a), predator handling time (h), and local dispersal affect multi-species community dynamics. This system consists of resource-mutualist-exploiter-competitor interactions and local dispersal. From the inspection of our numerical simulations and co-dimension one bifurcation analysis findings, we discover several critical values that correspond to transcritical bifurcation, subcritical and supercritical Hopf bifurcations. This occurs as we vary the bifurcation parameters a and h in this complex ecological system under symmetric and asymmetric dispersal scenarios. Furthermore, the interplay between these local bifurcation points results in an exciting co-dimension two bifurcations, i.e., Bogdanov-Takens and cusp bifurcation points, respectively, which act as the synchronization points in this complex ecological system. From an ecological viewpoint, we find that (i) the effect of the no-dispersal scenario supports the maintenance of species biodiversity when the predation strength is moderate; (ii) symmetric dispersal induces both subcritical and supercritical Hopf bifurcation and support species diversity for moderate predation strength; and (iii) asymmetric dispersal promotes species diversity as it simplifies the bifurcation changes in dynamics by eliminating the subcritical bifurcations that trigger uncertainty, and this dispersal mechanism mediates species coexistence outcomes. Fundamentally, stable limit cycles have been reported as predator handling time varies in some ecological models; however, we observed in our bifurcation analysis the emergence of the unstable limit cycle as predator handling time changes. We discover that intense predator handling time destabilizes this complex ecological community. In general, our results demonstrate the influential roles of predation, predator handling time, and local dispersal in determining this system’s coexistence dynamics. This knowledge provides a better understanding of species conservation and biological control management.

Solitary routes to chimera states.

We show how solitary states in a system of globally coupled FitzHugh-Nagumo oscillators can lead to the emergence of chimera states. By a numerical bifurcation analysis of a suitable reduced system in the thermodynamic limit we demonstrate how solitary states, after emerging from the synchronous state, become chaotic in a period-doubling cascade. Subsequently, states with a single chaotic oscillator give rise to states with an increasing number of incoherent chaotic oscillators. In large systems, these chimera states show extensive chaos. We demonstrate the coexistence of many of such chaotic attractors with different Lyapunov dimensions, due to different numbers of incoherent oscillators.

Stability and Hopf bifurcation of a love model with two delays

We analyze the stability and Hopf bifurcation of a proposed nonlinear love model with two delays. The nonlinearity stems from a logistic term injected to meticulously depict a couple’s cautious love affairs. The analysis is based on the linearization theory and starts with a single-delay case assuming that one of the two characters is always impulsive. The critical delay values and existence conditions for the Hopf bifurcation are obtained. Subsequently, we extend to the more practical but also more involved dual-delay case where the delays affect the emotions of both individuals. Accordingly, a non-trivial resultant-based analysis framework is proposed, and the corresponding Hopf bifurcation is detected intuitively and exhaustively by an obtained stability map in the delay domain. The results explain an interesting and well-known feature in our daily life, i.e., proper time delays stabilize the love dynamics. Furthermore, the spectral analysis based on a numerical case reveals another common emotional experience, that proper delays not only lead to a smoother transition process but also yield a shorter settling time for the love convergence. The numerical bifurcation diagram and computational cost are also considered to demonstrate the effectiveness and efficiency of the presented analysis framework. Finally, some additional suggestions are provided for the studied couple to earn a sweet love relationship.

Exploring the constituent mechanisms of hepatitis: a dynamical systems approach.

Hepatitis is the term used to describe inflammation in the liver. It is associated with a high rate of mortality, but the underlying disease mechanisms are not completely understood and treatment options are limited. We present a mathematical model of hepatitis that captures the complex interactions between hepatocytes (liver cells), hepatic stellate cells (cells in the liver that produce hepatitis-associated fibrosis) and the immune components that mediate inflammation. The model is in the form of a system of ordinary differential equations. We use numerical techniques and bifurcation analysis to characterize and elucidate the physiological mechanisms that dominate liver injury and its outcome to a healthy or unhealthy, chronic state. This study reveals the complex interactions between the multiple cell types and mediators involved in this complex disease and highlights potential problems in targeting inflammation in the liver therapeutically.

Global modelling of nonlinear spatiotemporal dynamics of a drill‐string with multiple regenerative effects

We develop an integrated model with spatially distributed inertia to study the axial and torsional dynamics of a drill-string. This model involves nonlinearities that arise due to regenerative cutting and frictional effects in drill-bit/rock interaction, which couple the axial and torsional dynamics with each other. Moreover, the multiple regenerative effects arising from bit-bounce are also taken into account by modelling the evolution of the cutting surface profile. The developed dynamic model is tested by checking its convergence and prediction ability, showing that the model is robust. Then, the linear eigenvalue analysis for drilling stability is performed, and a special attention is devoted to the influence of spatial discretization resolution of the drill-string. Numerical bifurcation analysis reveals both subcritical and supercritical types of instability near the stability boundaries, which trigger various co-existing attractors and nonlinear drilling dynamics, such as stable limit-cycle motion, periodic stick-slip vibration and bit-bounce. Delving further into the unstable region outside the left stability boundary, we capture more complex dynamics including periodic-two and even chaotic chatters with bit-bounce.

Assessing the role of internal phosphorus recycling on eutrophication in four lakes in China and Malaysia

Internal phosphorus recycling (IPR) is an important nutrient source driving algal growth and eutrophication in lakes. The complexity of eutrophication behaviours caused by high IPR complicates lake management and undermines restoration efforts. Hence, knowledge about the possible types of bifurcation behaviours caused by high IPR is essential for effective and sustainable lake eutrophication management. For this purpose, numerical bifurcation analysis is performed on an algae‑phosphorus model to investigate how IPR drives complex and rich eutrophication behaviours in two tropical and two subtropical lakes. The two tropical lakes are Tasik Harapan and Sunway Lagoon in Malaysia, while the two subtropical lakes are Lake Fuxian and Lake Taihu in China. For each specified level of IPR, co-dimension one bifurcation analysis is performed by means of XPPAUT. Co-dimension two bifurcation analysis is then carried out by means of MatCont. At low IPR, Lake Fuxian exhibits reversible behaviour, accompanied by higher external phosphorus loading (EPL) thresholds. Lake Fuxian is also more conducive to stable equilibrium and its lake dynamics are easily predictable. At moderate IPR, Sunway Lagoon is likely to exhibit stable equilibrium, accompanied by possible shifting between two stable steady states (hysteresis behaviour) and oscillations. With higher IPR, Lake Taihu and Tasik Harapan are prone to irreversibility, accompanied by lower EPL thresholds. Because of increased complexity in lake dynamics in Lake Taihu and Tasik Harapan, small changes in EPL or in algal mortality rates could trigger various transitions in lake dynamics. Overall, high IPR can trigger unexpected sharp increases in algal concentration and can reduce the resilience of an oligotrophic lake. For shallow lakes, high IPR would cause unexpected sharp increases in algal concentrations, undermine resilience of lakes, complicate lake management, and delay lake recovery process.

Spatio-temporal modelling of phenotypic heterogeneity in tumour tissues and its impact on radiotherapy treatment

We present a mathematical model that describes how tumour heterogeneity evolves in a tissue slice that is oxygenated by a single blood vessel. Phenotype is identified with the stemness level of a cell and determines its proliferative capacity, apoptosis propensity and response to treatment. Our study is based on numerical bifurcation analysis and dynamical simulations of a system of coupled, non-local (in phenotypic “space”) partial differential equations that link the phenotypic evolution of the tumour cells to local tissue oxygen levels. In our formulation, we consider a 1D geometry where oxygen is supplied by a blood vessel located on the domain boundary and consumed by the tumour cells as it diffuses through the tissue. For biologically relevant parameter values, the system exhibits multiple steady states; in particular, depending on the initial conditions, the tumour is either eliminated (“tumour-extinction”) or it persists (“tumour-invasion”). We conclude by using the model to investigate tumour responses to radiotherapy, and focus on identifying radiotherapy strategies which can eliminate the tumour. Numerical simulations reveal how phenotypic heterogeneity evolves during treatment and highlight the critical role of tissue oxygen levels on the efficacy of radiation protocols that are commonly used in the clinic.

Bifurcation Analysis and Spatio-Temporal Patterns of a Prey–Predator Model with Hunting Cooperation

Understanding the effects of cooperative hunting among the predators is an important topic in ecology. We explore a prey–predator model with cooperative hunting among predators modeled by a Holling type-II functional response having saturated encounter rate. Bifurcation analysis is carried out on the temporal model to detect local bifurcations such as transcritical, saddle-node, Hopf bifurcation and global homoclinic bifurcation. A comprehensive numerical bifurcation analysis is performed on the Turing solutions of the corresponding diffusive model. Stable and unstable branches of mode specific Turing solutions and their dependence on parameters are identified. For suitable values of parameters, spatio-temporal chaos via period doubling cascade is observed when a given parameter traverses the Turing–Hopf region.

Nonlinear analysis of the delayed tyre model with control-based continuation

In this study, the numerical bifurcation analysis of a shimmying wheel is performed with a non-smooth, time-delayed model of the tyre-ground contact. This model is capable of reproducing the bistable behaviour often observed in experiments: a stable equilibrium and a stable periodic orbit coexisting for the same set of system parameters, that the simpler quasi-steady tyre models fail to capture. In the bistable parameter domain, there also exists an unstable periodic orbit within the separatrix between the domains of attractions of the two stable steady-state solutions. Although this solution never appears in a real-life system, one may still gain valuable information from tracing it as it gives an indication about the level of perturbation that would drive the system from one stable solution to the other. However, the complexity of the laws governing partial sticking and sliding in the tyre-ground contact makes the numerical bifurcation analysis with the traditional, collocation-based techniques infeasible. Instead, this study is based on numerical simulations and the technique of control-based continuation (CBC) to track the stable and unstable periodic solutions of the system allowing for the assessment of the accuracy of the non-smooth, delayed tyre model in replicating the dynamics observed in experiments. In the meantime, the physics-based model provides an insight into the relationship between the sticking and sliding regions appearing in the tyre-ground contact and the global dynamics of the system.

A feedback linearisation algorithm for single-track models with structural stability properties

To support the design of trajectory tracking controllers, by way of linear tools, for vehicles characterised by nonholonomic constraints, this paper proposes a feedback linearising law for single-track dynamic models. The main characteristics of this algorithm are its simplicity, its independence from any vehicle model parameter, apart from the position of the center of mass, and its robustness. A numerical bifurcation analysis demonstrates that, for physically meaningful values of the center of mass deviation, the equilibrium is structurally asymptotically stable. Experimental results, concerning the linearising law and its application as inner loop of a trajectory tracking controller, are also presented, confirming the efficacy of the proposal.

Stability analysis of fractional order memristor synapse-coupled hopfield neural network with ring structure.

A memristor is a nonlinear two-terminal electrical element that incorporates memory features and nanoscale properties, enabling us to design very high-density artificial neural networks. To enhance the memory property, we should use mathematical frameworks like fractional calculus, which is capable of doing so. Here, we first present a fractional-order memristor synapse-coupling Hopfield neural network on two neurons and then extend the model to a neural network with a ring structure that consists of n sub-network neurons, increasing the synchronization in the network. Necessary and sufficient conditions for the stability of equilibrium points are investigated, highlighting the dependency of the stability on the fractional-order value and the number of neurons. Numerical simulations and bifurcation analysis, along with Lyapunov exponents, are given in the two-neuron case that substantiates the theoretical findings, suggesting possible routes towards chaos when the fractional order of the system increases. In the n-neuron case also, it is revealed that the stability depends on the structure and number of sub-networks.

Modeling and bifurcation analysis of a hydraulic actuator system with different valve spool geometry

The valve spools of hydraulic actuator systems are usually designed into various geometries to achieve different flow characters. Because of the coupling nonlinear flow rate-pressure character and dynamic motion of the valve spool, self-excited vibration can happen in some working conditions of the system, which can lead to unexpected dynamic performance. The object of this paper is to investigate the stability of a hydraulic system with three different spool geometries, a normal sharp edge spool, a spool with chamfer, and a spool with u shape notches. Computational fluid dynamics (CFD) simulations of flow force and flow rate under different pressure difference and valve openings are proposed to obtain the pressure-flow characters of flow field inside the valves with different spool geometries. A nonlinear dynamic model of the actuator system dynamics is established basing on the CFD results of pressure control valve and the system. Numerical Bifurcation analyses are carried out to study the effect of spool geometries on the bifurcation point of the system. Simulation results suggest that the spool geometries can affect the flow characters of the valve and decrease the stable region of the parameter plane.

Bifurcations of Sleep Patterns due to Homeostatic and Circadian Variation in a Sleep-Wake Flip-Flop Model

Differential equation-based physiological models of sleep-wake networks describe sleep-wake regulation by simulating the activity of wake- and sleep-promoting neuronal populations and the modulation of these populations by homeostatic and circadian ($\sim24$ h) drives. Here, we consider a sleep-wake flip-flop network model consisting of mutually inhibitory interactions between wake- and sleep-promoting neuronal populations. Motivated by changes in sleep behavior during early childhood as babies transition from napping to non-napping behavior, we vary homeostatic and circadian modulation and analyze effects on resulting sleep-wake patterns. To identify the types and sequences of bifurcations leading to changes in stable sleep-wake patterns in this piecewise-smooth model, we employ multiple mathematical methods, including fast-slow decomposition and numerical computation of circle maps. We find that the average daily number of sleeps exhibits a period adding sequence as the homeostatic time constants are reduced, and that the temporal circadian profile influences the number of observed solutions in the sequence. These solutions emerge through sequences of saddle-node and border collision bifurcations, where the particular sequence depends on parameter values. When the temporal circadian profile is steep, as is observed with long day lengths, some sleep patterns are lost and bistability of other patterns may occur. We analyze a limiting case of the temporal circadian waveform, a circadian hard switch model, to understand this loss of solutions. Generally, our holistic analysis approach provides an alternative analysis method for model systems that defy conventional numerical bifurcation analysis techniques.

Exact mean-field models for spiking neural networks with adaptation.

Networks of spiking neurons with adaption have been shown to be able to reproduce a wide range of neural activities, including the emergent population bursting and spike synchrony that underpin brain disorders and normal function. Exact mean-field models derived from spiking neural networks are extremely valuable, as such models can be used to determine how individual neurons and the network they reside within interact to produce macroscopic network behaviours. In the paper, we derive and analyze a set of exact mean-field equations for the neural network with spike frequency adaptation. Specifically, our model is a network of Izhikevich neurons, where each neuron is modeled by a two dimensional system consisting of a quadratic integrate and fire equation plus an equation which implements spike frequency adaptation. Previous work deriving a mean-field model for this type of network, relied on the assumption of sufficiently slow dynamics of the adaptation variable. However, this approximation did not succeed in establishing an exact correspondence between the macroscopic description and the realistic neural network, especially when the adaptation time constant was not large. The challenge lies in how to achieve a closed set of mean-field equations with the inclusion of the mean-field dynamics of the adaptation variable. We address this problem by using a Lorentzian ansatz combined with the moment closure approach to arrive at a mean-field system in the thermodynamic limit. The resulting macroscopic description is capable of qualitatively and quantitatively describing the collective dynamics of the neural network, including transition between states where the individual neurons exhibit asynchronous tonic firing and synchronous bursting. We extend the approach to a network of two populations of neurons and discuss the accuracy and efficacy of our mean-field approximations by examining all assumptions that are imposed during the derivation. Numerical bifurcation analysis of our mean-field models reveals bifurcations not previously observed in the models, including a novel mechanism for emergence of bursting in the network. We anticipate our results will provide a tractable and reliable tool to investigate the underlying mechanism of brain function and dysfunction from the perspective of computational neuroscience.

Spiral patterns and numerical bifurcation analysis in a three-component Brusselator model for chemical reactions

Pattern formation is a natural phenomenon that can be modeled by reaction–diffusion systems. Regular rotating spiral pattern formation and the instability of the spiral dynamics are very closely related to morphogenesis processes, cardiac arrhythmias, and chemical reactions. Since regular rotating spirals can emit periodic traveling waves from their core, hence the analysis of periodic traveling wave solutions of an oscillatory dynamical system is crucial to understand the instability of spiral patterns. To investigate the spiral patterns for chemical reactions, we consider a three-component Brusselator model. We first examine the local behavior of solutions of the model. Then we analyze the occurrence of periodic traveling wave solutions in a two-dimensional parameter plane. Next, we explore the stability outcome of periodic traveling wave solutions. It is observed that periodic traveling wave solutions change their stability, which is the Eckhaus type. The results are justified by a direct numerical simulation in one and two space dimensions. The regular and oscillatory periodic traveling wave solutions are revealed in the one-dimensional space dimension of the model by considering an Eckhaus bifurcation point. Moreover, we illustrate regular rotating spiral waves and the phenomenon for far-field breakup from the spiral waves in the two-dimensional space.

A Model for the Proliferation–Quiescence Transition in Human Cells

The process of revitalising quiescent cells in order for them to proliferate plays a pivotal role in the repair of worn-out tissues as well as for tissue homeostasis. This process is also crucial in the growth, development and well-being of higher multi-cellular organisms such as mammals. Deregulation of proliferation-quiescence transition is related to many diseases, such as cancer. Recent studies have revealed that this proliferation–quiescence process is regulated tightly by the Rb−E2F bistable switch mechanism. Based on experimental observations, in this study, we formulate a mathematical model to examine the effect of the growth factor concentration on the proliferation–quiescence transition in human cells. Working with a non-dimensionalised model, we prove the positivity, boundedness and uniqueness of solutions. To understand model solution behaviour close to bifurcation points, we carry out bifurcation analysis, which is further illustrated by the use of numerical bifurcation analysis, sensitivity analysis and numerical simulations. Indeed, bifurcation and numerical analysis of the model predicted a transition between bistable and stable states, which are dependent on the growth factor concentration parameter (GF). The derived predictions confirm experimental observations.