Numerical Bifurcation Techniques Research Articles

Bifurcation Analysis of an Ecological System with State-Dependent Feedback Control and Periodic Forcing

By assuming a periodic variation in the intrinsic growth rate of the prey, a nonlinear ecological system with periodic forcing and state-dependent feedback control is proposed. The main purpose of the present paper is to study the dynamical behavior generated by periodic forcing and nonlinear impulse perturbations and their effects on pest control. To do this, we first investigate the existence and stability of the boundary periodic solution, and then we employ the numerical bifurcation techniques, mainly including one-dimensional and two-dimensional parameter bifurcation analyses, to reveal that the system exhibits rich and complex dynamic behaviors. Especially, period-adding bifurcation with chaos is found in the two-parameter bifurcation plane. Moreover, we find the periodic structure similar to Arnold tongues, and they are arranged according to the sequence of a Farey tree. In addition, one-dimensional bifurcation diagrams reveal the existence of order-[Formula: see text] periodic, and chaotic solutions, multiple coexisting attractors, period-doubling bifurcations, period-halving bifurcations, and so on. Finally, the effects of the initial population density of pests and natural enemies on the pulse frequency and the biological significance related to the numerical results are studied and discussed.

Symmetries and choreographies in families that bifurcate from the polygonal relative equilibrium of the n-body problem

We use numerical continuation and bifurcation techniques in a boundary value setting to follow Lyapunov families of periodic orbits. These arise from the polygonal system of $n$ bodies in a rotating frame of reference. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship then the orbit is also periodic in the inertial frame. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a diophantine equation, correspond to choreographies. We present a sample of the many choreographies that we have determined numerically along the Lyapunov families and along bifurcating families, namely for the cases $n=4,~6,~7,~8$, and $9$. We also present numerical results for the case where there is a central body that affects the choreography, but that does not participate in it. Animations of the families and the choreographies can be seen at the link: http://mym.iimas.unam.mx/renato/choreographies/index.html

Numerical Bifurcation and Spectral Stability of Wavetrains in Bidirectional Whitham Models

AbstractWe consider several different bidirectional Whitham equations that have recently appeared in the literature. Each of these models combines the full two‐way dispersion relation from the incompressible Euler equations with a canonical shallow water nonlinearity, providing nonlocal model equations that may be expected to exhibit some of the interesting high‐frequency phenomena present in the Euler equations that standard “long‐wave” theories fail to capture. Of particular interest here is the existence and stability of periodic traveling wave solutions in such models. Using numerical bifurcation techniques, we construct global bifurcation diagrams for each system and compare the global structure of branches, together with the possibility of bifurcation branches terminating in a “highest” singular (peaked/cusped) wave. We also numerically approximate the stability spectrum along these bifurcation branches and compare the stability predictions of these models. Our results confirm a number of analytical results concerning the stability of asymptotically small waves in these models and provide new insights into the existence and stability of large amplitude waves.

Periodic solutions in an SIRWS model with immune boosting and cross-immunity

Incidence of whooping cough, an infection caused by Bordetella pertussis and Bordetella parapertussis, has been on the rise since the 1980s in many countries. Immunological interactions, such as immune boosting and cross-immunity between pathogens, have been hypothesised to be important drivers of epidemiological dynamics. We present a two-pathogen model of transmission which examines how immune boosting and cross-immunity can influence the timing and severity of epidemics. We use a combination of numerical simulations and bifurcation techniques to study the dynamical properties of the system, particularly the conditions under which stable periodic solutions are present. We derive analytic expressions for the steady state of the single-pathogen model, and give a condition for the presence of periodic solutions. A key result from our two-pathogen model is that, while studies have shown that immune boosting at relatively strong levels can independently generate periodic solutions, cross-immunity allows for the presence of periodic solutions even when the level of immune boosting is weak. Asymmetric cross-immunity can produce striking increases in the incidence and period. Our study underscores the importance of developing a better understanding of the immunological interactions between pathogens in order to improve model-based interpretations of epidemiological data.

Spike-adding in parabolic bursters: The role of folded-saddle canards

The present work develops a new approach to studying parabolic bursting, and also proposes a novel four-dimensional canonical and polynomial-based parabolic burster. In addition to this new polynomial system, we also consider the conductance-based model of the Aplysia R15 neuron known as the Plant model, and a reduction of this prototypical biophysical parabolic burster to three variables, including one phase variable, namely the Baer–Rinzel–Carillo (BRC) phase model. Revisiting these models from the perspective of slow–fast dynamics reveals that the number of spikes per burst may vary upon parameter changes, however the spike-adding process occurs in an explosive fashion that involves special solutions called canards. This spike-adding canard explosion phenomenon is analysed by using tools from geometric singular perturbation theory in tandem with numerical bifurcation techniques. We find that the bifurcation structure persists across all considered systems, that is, spikes within the burst are incremented via the crossing of an excitability threshold given by a particular type of canard orbit, namely the true canard of a folded-saddle singularity. However there can be a difference in the spike-adding transitions in parameter space from one case to another, according to whether the process is continuous or discontinuous, which depends upon the geometry of the folded-saddle canard. Using these findings, we construct a new polynomial approximation of the Plant model, which retains all the key elements for parabolic bursting, including the spike-adding transitions mediated by folded-saddle canards. Finally, we briefly investigate the presence of spike-adding via canards in planar phase models of parabolic bursting, namely the theta model by Ermentrout and Kopell.

On the nonlinear dynamics of a rotor in autorotation: a combined experimental and numerical approach.

This article presents a systematic assessment of the use of numerical continuation and bifurcation techniques in investigating the nonlinear periodic behaviour of a teetering rotor operating in forward autorotation. The aim is to illustrate the potential of these tools in revealing complex blade dynamics, when used in combination (not necessarily at the same time) with physical testing. We show a simple procedure to promote understanding of an existing but not fully understood engineering instability problem, when uncertainties in the numerical modelling and constraints in the experimental testing are present. It is proposed that continuation and bifurcation methods can play a significant role in developing numerical and experimental techniques for studying the nonlinear dynamics not only for rotating blades but also for other engineering systems with uncertainties and constraints.

Dynamical crises, multistability and the influence of the duration of immunity in a seasonally-forced model of disease transmission.

BackgroundHighly successful strategies to make populations more resilient to infectious diseases, such as childhood vaccinations programs, may nonetheless lead to unpredictable outcomes due to the interplay between seasonal variations in transmission and a population’s immune status.MethodsMotivated by the study of diseases such as pertussis we introduce a seasonally-forced susceptible-infectious-recovered model of disease transmission with waning and boosting of immunity. We study the system’s dynamical properties using a combination of numerical simulations and bifurcation techniques, paying particular attention to the properties of the initial condition space.ResultsWe find that highly unpredictable behaviour can be triggered by changes in biologically relevant model parameters such as the duration of immunity. In the particular system we analyse — previously used in the literature to study pertussis dynamics — we identify the presence of an initial-condition landscape containing three coexisting attractors. The system’s response to interventions which perturb population immunity (e.g. vaccination "catch-up" campaigns) is therefore difficult to predict.ConclusionGiven the increasing use of models to inform policy decisions regarding vaccine introduction and scheduling and infectious diseases intervention policy more generally, our findings highlight the importance of thoroughly investigating the dynamical properties of those models to identify key areas of uncertainty. Our findings suggest that the often stated tension between capturing biological complexity and utilising mathematically simple models is perhaps more nuanced than generally suggested. Simple dynamical models, particularly those which include forcing terms, can give rise to incredibly complex behaviour.

Numerical Bifurcation Methods and their Application to Fluid Dynamics: Analysis beyond Simulation

AbstractWe provide an overview of current techniques and typical applications of numerical bifurcation analysis in fluid dynamical problems. Many of these problems are characterized by high-dimensional dynamical systems which undergo transitions as parameters are changed. The computation of the critical conditions associated with these transitions, popularly referred to as ‘tipping points’, is important for understanding the transition mechanisms. We describe the two basic classes of methods of numerical bifurcation analysis, which differ in the explicit or implicit use of the Jacobian matrix of the dynamical system. The numerical challenges involved in both methods arementioned and possible solutions to current bottlenecks are given. To demonstrate that numerical bifurcation techniques are not restricted to relatively low-dimensional dynamical systems, we provide several examples of the application of the modern techniques to a diverse set of fluid mechanical problems.

Control of Turing patterns and their usage as sensors, memory arrays, and logic gates

We study a model system of three diffusively coupled reaction cells arranged in a linear array that display Turing patterns with special focus on the case of equal coupling strength for all components. As a suitable model reaction we consider a two-variable core model of glycolysis. Using numerical continuation and bifurcation techniques we analyze the dependence of the system's steady states on varying rate coefficient of the recycling step while the coupling coefficients of the inhibitor and activator are fixed and set at the ratios 100:1, 1:1, and 4:5. We show that stable Turing patterns occur at all three ratios but, as expected, spontaneous transition from the spatially uniform steady state to the spatially nonuniform Turing patterns occurs only in the first case. The other two cases possess multiple Turing patterns, which are stabilized by secondary bifurcations and coexist with stable uniform periodic oscillations. For the 1:1 ratio we examine modular spatiotemporal perturbations, which allow for controllable switching between the uniform oscillations and various Turing patterns. Such modular perturbations are then used to construct chemical computing devices utilizing the multiple Turing patterns. By classifying various responses we propose: (a) a single-input resettable sensor capable of reading certain value of concentration, (b) two-input and three-input memory arrays capable of storing logic information, (c) three-input, three-output logic gates performing combinations of logical functions OR, XOR, AND, and NAND.

Bifurcations of quasi-periodic dynamics: torus breakdown

To study the dynamics of quasi-periodic bifurcations, we consider a system of two nonlinearly coupled oscillators using averaging, continuation and numerical bifurcation techniques. This relatively simple system displays considerable complexity. Assuming the internal resonance to be 1:2, we find a 2π-periodic solution which undergoes a supercritical Neimark–Sacker bifurcation, yielding a stable torus. Choosing a route in parameter space, we show by numerical bifurcation techniques how the torus gets destroyed by dynamical and topological changes in the involved manifolds (Krauskopf and Osinga in J Comput Phys 146:404–419, 1998). The 1:6 resonance turns out to be prominent in parameter space, and we detected a cascade of period doublings within the corresponding resonance tongue yielding a strange attractor. The phenomena agree with the Ruelle–Takens (Commun. Math. Phys. 20:167–192, 1971, Commun. Math. Phys. 23:343–344, 1971) scenario leading to strange attractors. Other periodic regimes are present in this system, and there is interesting evidence that two different regimes interact with each other, yielding yet another type of strange attractor. In this context, certain π-periodic solutions emerge that are studied by continuation following the Poincare–Lindstedt method using Mathieu functions; when the implicit function theorem breaks down, the analysis is supplemented by numerical bifurcation techniques.

Continuation and bifurcation analysis of a periodically forced excitable system

The response of an excitable cell to periodic electrical stimulation is modeled using the FitzHugh–Nagumo (FHN) system submitted to a gaussian-shaped pacing, the width of which is small compared with the action potential duration. The influence of the amplitude and the period of the stimulation is studied using numerical continuation and bifurcation techniques (AUTO97 software). Results are discussed in the light of prior experimental and theoretical findings. In particular, agreement with the documented behavior of periodically stimulated cardiac cells and squid axons is discussed. As previously reported, we find many different “ M : N ” periodic solutions, period-doubling sequences leading to seemingly chaotic regimes, and bistability phenomena. In addition, the use of continuation techniques has allowed us to track unstable solutions of the system and thus to determine how the different stable rhythms are connected with each other in a bifurcation diagram. Depending on the stimulus amplitude, the aspect of the bifurcation diagram with the stimulus period as main varying parameter can vary from very simple to very complex. In its most developed structure, this bifurcation diagram consists of a main “tree” of period- 2 P branches, where the 1:1, 1:0, 2:2, 2:1, … rhythms are located, and of several closed loops made up of period- { N × 2 P } branches ( N > 2 ) , isolated from each other and from the main tree. It is mainly on such loops that N:1 rhythms ( N > 2 ) on one hand, and N: N - 1 or Wenckebach rhythms ( N > 2 ) on the other hand, are located. Stable M : N and M: N - 1 rhythms ( M ⩾ N ) can be found on the same branch of solutions. They are separated by a region of unstable solutions at small stimulus amplitudes, but this region shrinks gradually as the stimulus amplitude is raised, until it finally disappears. We believe that this property is related to the excitability characteristics of the FHN system. It would be interesting to know if it has any correspondence in the behavior of real excitable cells.

Bifurcation structure of a wind-driven shallow water model with layer-outcropping

The steady state bifurcation structure of the double-gyre wind-driven ocean circulation is examined in a shallow water model where the upper layer is allowed to outcrop at the sea surface. In addition to the classical jet-up and jet-down multiple equilibria, we find a new regime in which one of the equilibrium solutions has a large outcropping region in the subpolar gyre. Time dependent simulations show that the outcropping solution equilibrates to a stable periodic orbit with a period of 8 months. Co-existing with the periodic solution is a stable steady state solution without outcropping. A numerical scheme that has the unique advantage of being differentiable while still allowing layers to outcrop at the sea surface is used for the analysis. In contrast, standard schemes for solving layered models with outcropping are non-differentiable and have an ill-defined Jacobian making them unsuitable for solution using Newton’s method. As such, our new scheme expands the applicability of numerical bifurcation techniques to an important class of ocean models whose bifurcation structure had hitherto remained unexplored.

Electrohydrodynamic convection in small aspect ratio devices

We present numerical and experimental results on steady electrohydrodynamic convection in nematic liquid crystal flows. Numerical bifurcation techniques have been applied to the Ericksen–Leslie equations in finite two-dimensional domains in order to investigate the interaction between neighbouring instabilities. A cusp has been uncovered in the solution surface in qualitative agreement with complimentary experiments.

Economic development, demographics, and renewable resources: a dynamical systems approach

Many developing economies are faced with growing populations and deteriorating natural resources. It is often argued that economic growth will arrest these trends via cleaner technology and social change accompanied by reduced birth rates. Two questions arise: (1) What general economic, demographic, and ecological conditions favor this scenario? and (2) What adjustments, technological, demographic, or ecological, are more important in realizing this scenario? I address these questions using a two-sector growth model which includes human demographics and a renewable resource base. Using powerful numerical bifurcation techniques and rescaling arguments, I obtain the following general results. If the regeneration rate of the renewable resource base is slow relative to the rate of economic growth, population overshoot and resource collapse is more likely. Demographic adjustments are more important than technological adjustments in avoiding renewable resource degradation. Several related results are presented that support these general findings.

Two-Fluid Marangoni–Bénard Convection with a Deformable Interface

Two immiscible fluid layers that are subjected to a temperature gradient perpendicular to their interface exhibit a range of behaviors that is considerably richer than for the single-fluid case. We describe a numerical technique for calculating thermally driven flows in two fluid layers which uses a simple technique based on a Landau transformation to map the physical domain into a reference domain, enabling the unknown location of the deformable interface to be determined. The coupled system of nonlinear partial differential equations, comprising mapping, continuity, momentum, and energy equations and the appropriate boundary conditions, is solved using the finite-element method in two-dimensional domains. Numerical bifurcation techniques are used to investigate the multiplicity of the solution set. The case of heating from above is considered in some detail and the results of finite-element computations are compared with linear stability calculations performed on unbounded domains. The principal advantages of the finite-element approach are the ability to determine the effect of non-90° contact angles (when the conducting solution no longer exists and traditional linear stability approaches fail), the ability to determine the role of finite aspect ratio domains and the relative volume fractions of the two fluids, and the capability of calculating the nonlinear development of flows beyond the critical temperature gradient.

Dynamical mean-field solution of coupled quantum wells: a bifurcation analysis.

The time evolution of a discrete model of three quantum wells with a localized mean-field electrostatic interaction has been analyzed making use of numerical simulation and bifurcation techniques. The discrete Schrödinger equation can be written as a classical Hamiltonian system with two constants of motion. The frequency spectrum and the Lyapunov exponents show that the system is chaotic as its continuum counterpart. The organizing centers of the dynamical behavior are bifurcations of rotating periodic solutions whose simple structure allows a thorough analytical investigation as the conserved quantities are varied. The global structure of the periodic behavior is organized via subharmonic bifurcations at which tori of nonsymmetric periodic solutions are born. We have found another kind of bifurcation when two pairs of characteristic multipliers split from the unit circle. The chaotic behavior is related to the nonintegrability of the system.

Characterization of a two-parameter mixed-mode electrochemical behavior regime using neural networks

We use nonlinear signal processing techniques, based on artificial neural networks, to construct a continuous-time model (set of ordinary differential equations, ODEs) from experimental observations of mixed-mode oscillations during the galvanostatic oxidation of hydrogen on platinum in the presence of bismuth and chloride ions. The data was in the form of time-series of the potential for different values of the applied current and chloride ion concentration. We use the model to reconstruct the experimental dynamics and to explore the associated bifurcation structures in phase-space. Using numerical bifurcation techniques we locate stable and unstable periodic solutions, calculate eigenvalues, and identify bifurcation points. This approach constitutes a promising data post-processing procedure for investigating phase-space and parameter space of real experimental systems; it allows us to infer phase-space structures which the experiments can only probe with limited measurement precision and only at a discrete number of operating parameter settings. For example, the fitted model suggests the existence of a sub-critical Hopf bifurcation near the range of parameters probed in the experiments; this might explain the experimental difficulty in locating small amplitude simple oscillations.

Chaotic dynamics of a microbial system of coupled food chains

We analyze a mathematical model of a simple microbial system consisting of two microbial populations competing for a single nutrient and two predator populations, each one feeding upon one competitor, in a chemostat. Monod's model is employed for the specific growth rates of all the microbial populations. We use numerical bifurcation techniques to determine the effect of the operating conditions of the chemostat on the dynamics of the system and construct its operating diagram. We demonstrate that the system exhibits chaotic behavior and multistability. Two different routes to chaos are observed. Chaotic behavior is reached either through a sequence of period doublings or through birth and breaking of quasi-periodic states, as the operating conditions are varied. In some cases, transition from periodic to chaotic behavior is accompanied at certain parameter values by limit-point bifurcations of periodic states, the effect being multistablity, i.e. coexistence of stable periodic states with other stable periodic, quasi-periodic or chaotic states. The results demonstrate the importance of the interaction of food chains with regard to the dynamics exhibited by systems of microbial species inhabiting a common environment.

Numerical bifurcation study of electrohydrodynamic convection in nematic liquid crystals.

We present the results of a numerical investigation of the Ericksen-Leslie equations for the problem of electrohydrodynamic convection in a nematic liquid crystal. The combination of a finite element approach and numerical bifurcation techniques allows us to provide details of the basic flow and include the physically relevant effect of nonslip side walls. We are also able to include material properties as parameters and this permits us to draw comparisons with available experimental data. We then compare and contrast the bifurcation structure with that of Rayleigh-Bénard and Taylor-Couette flows and explore the role of symmetries by including a fringing electric field.

Chaotic dynamics of a food web in a chemostat

We analyze a mathematical model of a simple food web consisting of one predator and two prey populations in a chemostat. Monod’s model is employed for the dependence of the specific growth rates of the two prey populations on the concentration of the rate-limiting substrate and a generalization of Monod’s model for the dependence of the specific growth rate of the predator on the concentrations of the prey populations. We use numerical bifurcation techniques to determine the effect of the operating conditions of the chemostat on the dynamics of the system and construct its operating diagram. Chaotic behavior resulting from successive period doublings is observed. Multistability phenomena of coexistence of steady and periodic states at the same operating conditions are also found.