Bogdanov-Takens Bifurcation Point Research Articles

Modeling the Dynamics of a Ratio-Dependent Leslie–Gower Predator–Prey System with Strong Allee Effect

In this paper, the dynamics of a ratio-dependent predator–prey model with strong Allee effect and Holling IV functional response is investigated by using dynamical analysis. The model is shown to have complex dynamical behaviors including subcritical or supercritical Hopf bifurcation, saddle-node bifurcation, Bogdanov–Takens bifurcation of codimension-2, a nilpotent focus or cusp of codimension-2. The codimension-2 Bogdanov–Takens bifurcation point acts as an organizing center for the whole bifurcation set. The coexistence of stable and unstable positive equilibria, homoclinic cycle is also found. Our analysis shows that the ratio-dependent model may collapse suddenly due to certain parameter variation, i.e. the numbers of predator and prey will decrease sharply to zeroes after undergoing a short time of sustained oscillations with small amplitudes. Of particular interest is that the coalescence of saddle-node bifurcation point and Hopf bifurcation point may indicate the occurrence of relaxation oscillations and the critical state of extinction of predator and prey. Numerical simulations and phase portraits including one-parameter bifurcation curve and two-parameter bifurcation curves are given to illustrate the theoretical results.

Modelling HIV dynamics with cell‐to‐cell transmission and CTL response

In most HIV models, the emergence of backward bifurcation means that the control for basic reproduction number less than one is no longer effective for HIV treatment. In this paper, we study an HIV model with CTL response and cell‐to‐cell transmission by using the dynamical approach. The local and global stability of equilibria is investigated, the relations of subcritical Hopf bifurcation and supercritical bifurcation points are revealed, especially, the so‐called new type bifurcation is also found with two Hopf bifurcation curves meeting at the same Bogdanov–Takens bifurcation point. Forward and backward bifurcation, Hopf bifurcation, saddle‐node bifurcation, Bogdanov–Takens bifurcation are investigated analytically and numerically. Two limit cycles are also found numerically, which indicates that the complex behavior of HIV dynamics. Interestingly, the role of cell‐to‐cell interaction is fully uncovered, it may cause the oscillations to disappear and keep the so‐called new type bifurcation persist. Finally, some conclusions and discussions are also given.

Firing patterns and bifurcation analysis of neurons under electromagnetic induction

<p style='text-indent:20px;'>Based on the three-dimensional endocrine neuron model, a four-dimensional endocrine neuron model was constructed by introducing the magnetic flux variable and induced current according to the law of electromagnetic induction. Firstly, the codimension-one bifurcation and Interspike Intervals (ISIs) analysis were applied to study the bifurcation structure with respect to external stimuli and parameter <inline-formula><tex-math id="M1">$ k_0 $</tex-math></inline-formula>, and two dynamical behaviors were found: period-adding and period-doubling bifurcation leading to chaos. Besides, Hopf bifurcation was specially discussed corresponding to the transformation of the state. Secondly, the different firing patterns such as regular bursting, subthreshold oscillations, fast spiking, mixed-mode oscillations (MMOs) etc. can be observed by changing the external stimuli and the induced current. The neuron model presented more firing activities under strong coupling strength. Finally, the codimension-two bifurcation analysis shown more details of bifurcation. At the same time, the Bogdanov-Takens bifurcation point was also analyzed and three bifurcation curves were derived.</p>

Non-linear stability analysis of supercritical carbon dioxide flow in inclined heated channel

In this paper, the nonlinear stability analysis of supercritical carbon dioxide (SC−CO2) has been carried out using a reduced order nodalized model for a single inclined heated channel. The primary objective of the present study is to portray the generic view of linear as well as nonlinear stability characteristics of SC−CO2 channel along with the prospect of different types of bifurcation phenomena. The linear stability analysis is carried out with the help of eigenvalues of the Jacobian at steady state conditions, and stability boundary is shown in the parameter plane of pseudo-sub-cooling (Nspc) and pseudo-phase-change numbers (Ntpc). The non-linear analysis includes the detailed study of dynamic and static instabilities. Different types of bifurcation phenomena namely; sub-critical, super-critical and Generalized Hopf are observed; indicating various features of the dynamic instabilities. The first Lyapunov coefficient has been calculated to distinguish between sub-critical and super-critical Hopf bifurcations. Whereas in static instability; Ledinegg excursive phenomena, which is characterized as a saddle-node bifurcation is observed. Additionally, on saddle-node bifurcation curve, Bogdanov-Takens bifurcation points (as interaction with Hopf bifurcation) appear. These bifurcations lead to complex dynamics in the system therefore, several numerical simulations have been carried out around the stability threshold. This type of non-linear analysis is rarely reported for SC−CO2 in the existing literature. To extend this analysis, the dependence of various system design parameters on the bifurcation curve has been investigated along with the shifting of Generalized Hopf (GH) bifurcation point. Furthermore, the effect of the inclination angle of the channel on the stability threshold in parametric space is investigated.

Characterization of stability limits of Ledinegg instability and density wave oscillations for two-phase flow in natural circulation loops

A detailed study of excursive (or Ledinegg) instability and density wave oscillations (DWOs) is carried out for two-phase flow in a natural circulation loop. The maps in the parametric space have been obtained, which indicate excursive instability (Ledinegg instability) as well as density wave oscillations (DWOs). The dynamic (DWOs) and static (Ledinegg) stability boundary on these maps have been drawn. These diagrams dealing with the interaction of Ledinegg instability and DWOs have rarely been reported, and detailed mathematical analysis of these maps is lacking. In the present work, a detailed study of such interactions is carried out. It is also noted that the Ledinegg instability is a manifestation of saddle node bifurcation or limit point (LP) for the dynamical system of natural circulation loop. Density wave oscillations in the dynamical system of natural circulation loop can be observed with the Hopf bifurcation. The detailed bifurcation analysis of these instabilities shows that stability maps can be divided into three broad regions, the first region can only have DWOs (Hopf bifurcation), and the second region can have Ledinegg as well as DWOs (Hopf bifurcation and saddle node bifurcation co-exists), while the third region can have only Ledinegg instability (saddle node bifurcation). The first and second region are separated by a Cusp point (CP) and it is found that between CP and Bogdanov-Takens (BT) bifurcation point both Hopf and LP exist, due to presence of both DWOs and Ledinegg instability. The region beyond BT bifurcation point only has Ledinegg or excursive instability. Moreover, DWOs in the first region have two types of Hopf bifurcation depending on the nature of limit cycles. These subcritical and supercritical Hopf bifurcations, are separated by the Generalized Hopf (GH) bifurcation. Abovementioned three regions are observed only for the Type II DWOs stability boundary, whereas Type I DWOs have only subcritical and supercritical Hopf bifurcations. The impact of design parameters on the saddle-node curve along with the shifting of interaction point (Bogdanov-Takens bifurcation) have been investigated as well.

Bursting Types and Bifurcation Analysis in the Pre-Bötzinger Complex Respiratory Rhythm Neuron

Many types of neurons and excitable cells could intrinsically generate bursting activity, even in an isolated case, which plays a vital role in neuronal signaling and synaptic plasticity. In this paper, we have mainly investigated bursting types and corresponding bifurcations in the pre-Bötzinger complex respiratory rhythm neuron by using fast–slow dynamical analysis. The numerical simulation results have showed that for some appropriate parameters, the neuron model could exhibit four distinct types of fast–slow bursters. We also explored the bifurcation mechanisms related to these four types of bursters through the analysis of phase plane. Moreover, the first Lyapunov coefficient of the Hopf bifurcation, which can decide whether it is supercritical or subcritical, was calculated with the aid of MAPLE software. In addition, we analyzed the codimension-two bifurcation for equilibria of the whole system and gave a detailed theoretical derivation of the Bogdanov–Takens bifurcation. Finally, we obtained expressions for a fold bifurcation curve, a nondegenerate Hopf bifurcation curve, and a saddle homoclinic bifurcation curve near the Bogdanov–Takens bifurcation point.

Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds

In this paper we establish an SIR model with a standard incidence rate and a nonlinear recovery rate, formulated to consider the impact of available resource of the public health system especially the number of hospital beds. For the three dimensional model with total population regulated by both demographics and diseases incidence, we prove that the model can undergo backward bifurcation, saddle-node bifurcation, Hopf bifurcation and cusp type of Bogdanov–Takens bifurcation of codimension 3. We present the bifurcation diagram near the cusp type of Bogdanov–Takens bifurcation point of codimension 3 and give epidemiological interpretation of the complex dynamical behaviors of endemic due to the variation of the number of hospital beds. This study suggests that maintaining enough number of hospital beds is crucial for the control of the infectious diseases.

Delay-induced Bogdanov–Takens bifurcation in a Leslie–Gower predator–prey model with nonmonotonic functional response

This work is concerned with the dynamics of a Leslie–Gower predator–prey model with nonmonotonic functional response near the Bogdanov–Takens bifurcation point. By analyzing the characteristic equation associated with the nonhyperbolic equilibrium, the critical value of the delay inducing the Bogdanov–Takens bifurcation is obtained. In this case, the dynamics near this nonhyperbolic equilibrium can be reduced to the study of the dynamics of the corresponding normal form restricted to the associated two-dimensional center manifold. The bifurcation diagram near the Bogdanov–Takens bifurcation point is drawn according to the obtained normal form. We show that the change of delay can result in heteroclinic orbit, homoclinic orbit and unstable limit cycle.

Bifurcation analysis on a generalized recurrent neural network with two interconnected three-neuron components

A class of recurrent neural networks is constructed by generalizing a specific class of n-neuron networks. It is shown that the newly constructed network experiences generic pitchfork and Hopf codimension one bifurcations. It is also proved that the emergence of generic Bogdanov–Takens, pitchfork–Hopf and Hopf–Hopf codimension two, and the degenerate Bogdanov–Takens bifurcation points in the parameter space is possible due to the intersections of codimension one bifurcation curves. The occurrence of bifurcations of higher codimensions significantly increases the capability of the newly constructed recurrent neural network to learn broader families of periodic signals.

Periodic orbits and chaos in fast–slow systems with Bogdanov–Takens type fold points

The existence of stable periodic orbits and chaotic invariant sets of singularly perturbed problems of fast–slow type having Bogdanov–Takens bifurcation points in its fast subsystem is proved by means of the geometric singular perturbation method and the blow-up method. In particular, the blow-up method is effectively used for analyzing the flow near the Bogdanov–Takens type fold point in order to show that a slow manifold near the fold point is extended along the Boutroux's tritronquée solution of the first Painlevé equation in the blow-up space.

The Bogdanov–Takens bifurcation analysis on a three dimensional recurrent neural network

A class of recurrent neural networks is investigated in the vicinity of the Bogdanov–Takens bifurcation point in the parameter space when the slope of the transfer function of the neurons at the origin is not equal to one. It will be shown that two different bifurcation diagrams can be constructed. In each bifurcation diagram, there are critical values for the parameters of the network for which curves of pitchfork and Hopf bifurcation intersect each other at a point where the linear part of the system that describes the network, has a pair of simple zero eigenvalues. As curves of homoclinic and heteroclinic bifurcation emanate from the Bogdanov–Takens point, a complicated behavior is observed by the variation of weights in the recurrent neural network.

Phase description of nonlinear dissipative waves under space–time-dependent external forcing

Based on the model system undergoing phase separation and chemical reactions, we investigate the dynamics of propagating dissipative waves under external forcing which is periodic both in space and time. A phase diagram for the entrained and non-entrained states under the external forcing is obtained numerically. Theoretical analysis in terms of phase description of the traveling waves is carried out to show that the transition between the entrained and the non-entrained states by changing the external frequency occurs either through a saddle–node bifurcation or through a Hopf bifurcation and that these two bifurcation lines are connected at a Bogdanov–Takens bifurcation point.

NOTES ON MULTIPLE BIFURCATIONS IN A DELAYED PREDATOR–PREY MODEL WITH NONMONOTONIC FUNCTIONAL RESPONSE

By using the normal form theory for RFDE developed by Faria and Magalhaes, the multi-bifurcations for a delayed predator–prey model with nonmonotonic functional response [Formula: see text] have been studied, where a, b, μ, c, m, d, and τ are all nonnegative constants. Our results show that there exists a Bogdanov–Takens bifurcation point of codimension three when τ varies and extend some conclusions in two related articles.

Bifurcations and Chaos in a Discrete Predator-prey System with Holling Type-IV Functional Response

A discrete predator-prey system with Holling type-IV functional response obtained by the Euler method is first investigated. The conditions of existence for fold bifurcation, flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory. Furthermore, we give the condition for the occurrence of codimension-two bifurcation called the Bogdanov-Takens bifurcation for fixed points and present approximate expressions for saddle-node, Hopf and homoclinic bifurcation sets near the Bogdanov-Takens bifurcation point. We also show the existence of degenerated fixed point with codimension three at least. The numerical simulations, including bifurcation diagrams, phase portraits, and computation of maximum Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behaviors such as the attracting invariant circle, period-doubling bifurcation from period-2,3,4 orbits, interior crisis, intermittency mechanic, and sudden disappearance of chaotic dynamic.

Melnikov's Method and Codimension-Two Bifurcations in Forced Oscillations

Using a Melnikov-type technique, we study codimension-two bifurcations called the Bogdanov–Takens bifurcations for subharmonics in periodic perturbations of planar Hamiltonian systems. We give a criterion for the occurrence of the Bogdanov–Takens bifurcations and present approximate expressions for saddle-node, Hopf and homoclinic bifurcation sets near the Bogdanov–Takens bifurcation points. We illustrate the theoretical result with an example.

Bogdanov-Takens bifurcation points and Sil'nikov homoclinicity in a simple power-system model of voltage collapse

The bifurcation structure of a simple power-system model is investigated, with respect to changes to both the real and reactive loads. Numerical methods for this bifurcation analysis are presented and discussed. The model is shown to have a Bogdanov-Takens bifurcation point and hence homoclinic orbits; these orbits can be of Sil'nikov type with many coexisting periodic solutions. We may use the bifurcation calculations to divide the two-parameter plane into a number of regions, for which there are qualitatively different dynamics. We classify and further investigate the dynamical behavior in each of these regions, using a Monte Carlo method to investigate basins of attraction of various stable states. We then show how this classification can be used to denote each regions as either safe or unsafe with respect to the likelihood of voltage collapse.

HOMOCLINIC BIFURCATIONS IN A PLANAR DYNAMICAL SYSTEM

The homoclinic bifurcation properties of a planar dynamical system are analyzed and the corresponding bifurcation diagram is presented. The occurrence of two Bogdanov–Takens bifurcation points provides two local existing curves of homoclinic orbits to a saddle excluding the separatrices not belonging to the homoclinic orbits. Using numerical techniques, these curves are continued in the parameter space. Two further curves of homoclinic orbits to a saddle including the separatrices not belonging to the homoclinic orbits are calculated by numerical methods. All these curves of homoclinic orbits have a unique intersection point, at which there exists a double homoclinic orbit. The local homoclinic bifurcation diagram of both the double homoclinic orbit point and the points of homoclinic orbits to a saddle-node are also gained by numerical computation and simulation.