Takens Bifurcation Research Articles

Dynamical analysis and simulation of a 2-dimensional disease model with convex incidence

In this paper, a previously developed 2-dimensional disease model is studied, which can be used for both epidemiologic modeling and in-host disease modeling. The main attention of this paper is focused on various dynamical behaviors of the system, including Hopf and generalized Hopf bifurcations which yield bistability and tristability, Bogdanov–Takens bifurcation, and homoclinic bifurcation. It is shown that the Bogdanov–Takens bifurcation and homoclinic bifurcation provide a new mechanism for generating disease recurrence, that is, cycles of remission and relapse such as the viral blips observed in HIV infection.

The Bogdanov–Takens Normal Form: A Minimal Model for Single Neuron Dynamics

Conductance-based (CB) models are a class of high dimensional dynamical systems derived from biophysical principles to describe in detail the electrical dynamics of single neurons. Despite the high dimensionality of these models, the dynamics observed for realistic parameter values is generically planar and can be minimally described by two equations. In this work, we derive the conditions to have a Bogdanov–Takens (BT) bifurcation in CB models, and we argue that it is plausible that these conditions are verified for experimentally-sensible values of the parameters. We show numerically that the cubic BT normal form, a two-variable dynamical system, exhibits all of the diversity of bifurcations generically observed in single neuron models. We show that the Morris–Lecar model is approximately equivalent to the cubic Bogdanov–Takens normal form for realistic values of parameters. Furthermore, we explicitly calculate the quadratic coefficient of the BT normal form for a generic CB model, obtaining that by constraining the theoretical I-V curve’s curvature to match experimental observations, the normal form appears to be naturally cubic. We propose the cubic BT normal form as a robust minimal model for single neuron dynamics that can be derived from biophysically-realistic CB models.

Canard Explosion in Delay Differential Equations

We analyze canard explosions in delay differential equations with a one-dimensional slow manifold. This study is applied to explore the dynamics of the van der Pol slow–fast system with delayed self-coupling. In the absence of delays, this system provides a canonical example of a canard explosion. We show that as the delay is increased a family of ‘classical’ canard explosions ends as a Bogdanov–Takens bifurcation occurs at the folds points of the S-shaped critical manifold.

Delay-Induced Bogdanov–Takens Bifurcation and Dynamical Classifications in a Slow-Fast Flexible Joint System

A slow-fast delay-coupled flexible joint system is investigated in this paper. To understand the effects of time delay on the stability and oscillation of the manipulator, the geometric singular perturbation method is extended in dealing with delay differential equations. Bogdanov–Takens (BT) bifurcation of the fast subsystem is obtained, which leads to the existence of homoclinic orbits and is proved to be related to the formation of spiking. After the break of homoclinic orbits, Melnikov theory is introduced to predict the threshold curve indicating the occurrence of chaos. Numerical results show that with the increase of time delay, the stability of the system gets worse, and complicated oscillations including bursting, chaotic-bursting and complete chaos turn up. Besides, it is briefly summarized that the effect of the small parameter in the slow-fast system is to influence the convergence rate of solution trajectories, which is widely neglected in previous works.

Bifurcation and Firing Patterns of the Pancreatic β-Cell

Using a model of individual isolated pancreatic β-cells, we investigated bifurcation diagrams of interspike intervals (ISIs) and largest Lyapunov exponents (LLE), which clearly demonstrated a wide range of transitions between different firing patterns. The numerical simulation results revealed the effect of different time constants and ion channels on the neuronal discharge rhythm. Furthermore, an individual cell exhibited tonic spiking, square-wave bursting, and tapered bursting. Additionally, several bifurcation phenomena can be observed in this paper, such as period-doubling, period-adding, inverse period-doubling and inverse period-adding scenarios. In addition, we researched the mechanisms underlying two kinds of bursting (tapered and square-wave bursting) by use of fast-slow dynamics analysis. Finally, we analyzed the codimension-two bifurcation of the fast subsystem and studied cusp bifurcation, generalized Hopf (or Bautin) bifurcation and Bogdanov–Takens bifurcation.

Accurate Approximation of Homoclinic Solutions in Gray–Scott Kinetic Model

The second-order predictor for the homoclinic orbit is applied to the Gray–Scott model. The problem is used to illustrate the approximation of the homoclinic orbits near a generic Bogdanov–Takens bifurcation in n-dimensional systems of differential equations. In the process, we show that it is necessary to take (usually ignored) cubic terms in the Bogdanov–Takens normal form into account to derive a correct second-order prediction for the homoclinic bifurcation curve. The analytic solutions are compared with those obtained by numerical continuation.

Bogdanov–Takens bifurcation in a Hopfield network model with bidirectional connection and multiple delays

In this paper, a Hopfield network model with three neurons, bidirectional connection and multiple delays is considered. We aim to show that the connection topology is very important. In fact, we consider the connected weights of this system as bifurcation parameters, and the critical values where a Bogdanov–Takens bifurcation occurs are derived. Using the normal form method and the centre manifold theory, we obtain the normal form of our system and study its dynamical behaviours. Finally, a numerical example is given to demonstrate the validity of the results.

Bifurcation of an SIS model with nonlinear contact rate

The bifurcation and dynamics of SIS models with nonlinear effective contact rate can be very complicated. We study an SIS model with nonlinear contact rate describing behavior change effect of susceptible individual when infectious population increases. By the qualitative and bifurcation analyses, we show that the maximal multiplicity of weak focus is 2, i.e. at most 2 limit cycles can arise from this weak focus. In the meanwhile, we also prove that the model can undergo a Bogdanov–Takens bifurcation of codimension 2, while the responding parameter bifurcation diagrams are presented. These results illustrate that the behavior change of the susceptible individuals may affect the final spread level.

Bifurcation analysis method of nonlinear traffic phenomena

A new bifurcation analysis method for analyzing and predicting the complex nonlinear traffic phenomena based on the macroscopic traffic flow model is presented in this paper. This method makes use of variable substitution to transform a traditional traffic flow model into a new model which is suitable for the stability analysis. Although the substitution seems to be simple, it can extend the range of the variable to infinity and build a relationship between the traffic congestion and the unstable system in the phase plane. So the problem of traffic flow could be converted into that of system stability. The analysis identifies the types and stabilities of the equilibrium solutions of the new model and gives the overall distribution structure of the nearby equilibrium solutions in the phase plane. Then we deduce the existence conditions of the models Hopf bifurcation and saddle-node bifurcation and find some bifurcations such as Hopf bifurcation, saddle-node bifurcation, Limit Point bifurcation of cycles and Bogdanov–Takens bifurcation. Furthermore, the Hopf bifurcation and saddle-node bifurcation are selected as the starting point of density temporal evolution and it will be helpful for improving our understanding of stop-and-go wave and local cluster effects observed in the free-way traffic.

Bogdanov–Takens Bifurcation of a Class of Delayed Reaction–Diffusion System

In this paper, a class of reaction–diffusion system with Neumann boundary condition is considered. By analyzing the generalized eigenvector associated with zero eigenvalue, an equivalent condition for the determination of Bogdonov–Takens (B–T) singularity is obtained. Next, by using center manifold theorem and normal form method, we have a two-dimension ordinary differential system on its center manifold. Finally, two examples show that the given algorithm is effective.

Bifurcation analysis of a speed gradient continuum traffic flow model

A bifurcation analysis approach is presented based on the macroscopic traffic flow model. This method can be used to describe and predict the nonlinear traffic phenomena on the highway from a system global stability perspective. Based on a recently proposed speed gradient continuum traffic flow model, the types and stabilities of the equilibrium solutions are discussed and the existence of Hopf bifurcation and saddle–node bifurcation is proved. Then various bifurcations such as Hopf bifurcation, saddle–node bifurcation, Limit Point bifurcation of cycles, Cusp bifurcation and Bogdanov–Takens bifurcation are found and the traffic flow behaviors at some of them are analyzed. When the Hopf bifurcation is selected as the starting point of density temporal evolution, it may help to explain the stop-and-go traffic phenomena.

Consequences of weak Allee effect on prey in the May–Holling–Tanner predator–prey model

In this work, a modified Holling–Tanner predator–prey model is analyzed, considering important aspects describing the interaction such as the predator growth function is of a logistic type; a weak Allee effect acting in the prey growth function, and the functional response is of hyperbolic type. Making a change of variables and time rescaling, we obtain a polynomial differential equations system topologically equivalent to the original one in which the non‐hyperbolic equilibrium point (0,0) is an attractor for all parameter values. An important consequence of this property is the existence of a separatrix curve dividing the behavior of trajectories in the phase plane, and the system exhibits the bistability phenomenon, because the trajectories can have different ω − limit sets; as example, the origin (0,0) or a stable limit cycle surrounding an unstable positive equilibrium point. We show that, under certain parameter conditions, a positive equilibrium may undergo saddle‐node, Hopf, and Bogdanov–Takens bifurcations; the existence of a homoclinic curve on the phase plane is also proved, which breaks in an unstable limit cycle. Some simulations to reinforce our results are also shown. Copyright © 2015 John Wiley & Sons, Ltd.

Rich Global Dynamics in a Prey–Predator Model with Allee Effect and Density Dependent Death Rate of Predator

In this work we have considered a prey–predator model with strong Allee effect in the prey growth function, Holling type-II functional response and density dependent death rate for predators. It presents a comprehensive study of the complete global dynamics for the considered system. Especially to see the effect of the density dependent death rate of predator on the system behavior, we have presented the two parametric bifurcation diagrams taking it as one of the bifurcation parameters. In course of that we have explored all possible local and global bifurcations that the system could undergo, namely the existence of transcritical bifurcation, saddle node bifurcation, cusp bifurcation, Hopf-bifurcation, Bogdanov–Takens bifurcation and Bautin bifurcation respectively.

Qualitative analysis of a predator–prey system with double Allee effect in prey

Growing biological evidence from various ecosystems (Berec et al., 2007) suggests that the Allee effect generated by two or more mechanisms can act simultaneously on a single population. Surprisingly, prey–predator system incorporating multiple Allee effect is relatively poorly studied in literature. In this paper, we consider a ratio dependent prey–predator system with a double Allee effect in prey population growth. We have taken into account the case of the strong and the weak Allee effect separately and study the stability and complete bifurcation analysis. It is shown that the model exhibits the bi-stability and there exists separatrix curve(s) in the phase plane implying that dynamics of the system is very sensitive to the variation of the initial conditions. Generic normal forms at double zero singularity are derived by a rigorous mathematical analysis to show the different types of bifurcation of the proposed model system including Bogdanov–Takens bifurcation, saddle–node bifurcation curve, a Hopf bifurcation curve and a homoclinic bifurcation curve. The complete analysis of possible topological structures including elliptic, parabolic, or hyperbolic orbits, and any combination of them in a neighborhood of the complicated singular point (0,0) in the interior of the first quadrant is explored by using a blow up transformation. These structures have important implications for the global behavior of the model. Numerical simulation results are given to support our theoretical results. Finally, the paper concludes with a discussion of the ecological implications of our analytic and numerical findings.

Bifurcation analysis of a predator–prey model with anti-predator behaviour

We investigated a predator–prey model with a nonmonotonic functional response and anti-predator behaviour such that the adult prey can attack vulnerable predators. By analyzing the existence and stability of all possible equilibria and conducting a bifurcation analysis, we obtained the global dynamics of the proposed system. The system could undergo a saddle-node bifurcation, (supercritical and subcritical) Hopf bifurcation, homoclinic bifurcation and a Bogdanov–Takens bifurcation of codimension 2. Further, we obtained a generic family unfolding for the system by choosing the environmental carrying capacity of the prey and the death rate of the predator as bifurcation parameters. Numerical studies showed that anti-predator behaviour not only makes the coexistence of the prey and predator populations less likely, but also damps the predator–prey oscillations. Therefore, anti-predator behaviour helps the prey population to resist predator aggression.

Limit Cycles Sparked by Mutation in the Repeated Prisoner's Dilemma

We explore a replicator–mutator model of the repeated Prisoner's Dilemma involving three strategies: always cooperate (ALLC), always defect (ALLD), and tit-for-tat (TFT). The dynamics resulting from single unidirectional mutations are considered, with detailed results presented for the mutations TFT → ALLC and ALLD → ALLC. For certain combinations of parameters, given by the mutation rate μ and the complexity cost c of playing tit-for-tat, we find that the population settles into limit cycle oscillations, with the relative abundance of ALLC, ALLD, and TFT cycling periodically. Surprisingly, these oscillations can occur for unidirectional mutations between any two strategies. In each case, the limit cycles are created and destroyed by supercritical Hopf and homoclinic bifurcations, organized by a Bogdanov–Takens bifurcation. Our results suggest that stable oscillations are a robust aspect of a world of ALLC, ALLD, and costly TFT; the existence of cycles does not depend on the details of assumptions of how mutation is implemented.

Control exponential growth of tumor cells with slow spread of oncolytic virus.

Great attention has been paid to cancer therapy by means of oncolytic viruses, but the fast virus-spread, which eliminates all tumor cells, cannot be applied to solid tumors. As slow virus-spread is applied, solid tumors are expected to be controlled but complicated dynamical behaviors appear. In this paper we investigate bifurcations of equilibria in the oncolytic virus dynamics model with exponential growth of tumor cells and slow virus-spread. We find conditions of parameters for saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation. Those conditions give thresholds for slow virus-spread to control the population of tumor cells within an appropriate range.

The fast-time instability map of Liñán’s diffusion-flame regime

A detailed spectral map for the fast-time instability in Linan’s diffusion-flame regime is presented in order to clarify the origin of two bifurcations of co-dimension 2, causing the transitions from cellular to uniform-oscillatory instability and from uniform-oscillatory to traveling instability. The role of the real and continuous essential spectrum is found to be pivotal in understanding both transitions. Particular attention is paid to the spectral characteristics in the stable parametric regions, where the interaction with the essential spectrum leads to these transitions. When the Lewis number is increased above unity from below, the discrete real spectrum disappears by submerging below the essential spectrum, and the discrete complex spectrum emerges instead, eventually leading to uniform-oscillatory instability. The transition from uniform-oscillatory to traveling instability, associated with the Bogdanov–Takens bifurcation, involves a phenomenon called gap spectrum. For Lewis numbers slightly greater than unity and Damkohler numbers sufficiently large, the discrete complex spectrum intersects the plane corresponding to the essential spectrum, resulting in a gap in the discrete spectrum for small wave numbers. The discrete complex gap spectrum exhibits a local maximum as the parameter values are modified to approach the Hopf bifurcation boundary. The gap in the discrete complex spectrum disappears and traveling instability emerges when crossing the Hopf bifurcation boundary.

A general class of predation models with multiplicative Allee effect

A class of models of predator–prey interaction with Allee effect on the prey population is presented. Both the Allee effect and the functional response are modelled in the most simple way by means of general terms whose conveniently chosen mathematical properties agree with, and generalise, a number of concrete Leslie–Gower-type models. We show that this class of models is well posed in the sense that any realistic solution is bounded and remains non-negative. By means of topological equivalences and desingularization techniques, we find specific conditions such that there may be extinction of both species. In particular, the local basin boundaries of the origin are found explicitly, which enables one to determine the extinction or survival of species for any given initial condition near this equilibrium point. Furthermore, we give conditions such that an equilibrium point corresponding to a positive steady state may undergo saddle-node, Hopf and Bogdanov–Takens bifurcations. As a consequence, we are able to describe the dynamics governed by the bifurcated limit cycles and homoclinic orbits by means of carefully sketched bifurcation diagrams and suitable illustrations of the relevant invariant manifolds involved in the overall organisation of the phase plane. Finally, these findings are applied to concrete model vector fields; in each case, the particular relevant functions that define the conditions for the associated bifurcations are calculated explicitly.

Bifurcations in a predator–prey system of Leslie type with generalized Holling type III functional response

We consider a predator–prey system of Leslie type with generalized Holling type III functional response p(x)=mx2ax2+bx+1. By allowing b to be negative (b>−2a ), p(x) is monotonic for b>0 and nonmonotonic for b<0 when x≥0. The model has two non-hyperbolic positive equilibria (one is a multiple focus of multiplicity one and the other is a cusp of codimension 2) for some values of parameters and a degenerate Bogdanov–Takens singularity (focus or center case) of codimension 3 for other values of parameters. When there exist a multiple focus of multiplicity one and a cusp of codimension 2, we show that the model exhibits subcritical Hopf bifurcation and Bogdanov–Takens bifurcation simultaneously in the corresponding small neighborhoods of the two degenerate equilibria, respectively. Different phase portraits of the model are obtained by computer numerical simulations which demonstrate that the model can have: (i) a stable limit cycle enclosing two non-hyperbolic positive equilibria; (ii) a stable limit cycle enclosing an unstable homoclinic loop; (iii) two limit cycles enclosing a hyperbolic positive equilibrium; (iv) one stable limit cycle enclosing three hyperbolic positive equilibria; or (v) the coexistence of three stable states (two stable equilibria and a stable limit cycle). When the model has a Bogdanov–Takens singularity of codimension 3, we prove that the model exhibits degenerate focus type Bogdanov–Takens bifurcation of codimension 3. These results not only demonstrate that the dynamics of this model when b>−2a are much more complex and far richer than the case when b>0 but also provide new bifurcation phenomena for predator–prey systems.