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.     

Codimension-2 bifurcations of the Kaldor model of business cycle

In this paper, complete analysis is presented to study codimension-2 bifurcations for the nonlinear Kaldor model of business cycle. Sufficient conditions are given for the model to demonstrate Bautin and Bogdanov–Takens (BT) bifurcations. By computing the first and second Lyapunov coefficients and performing nonlinear transformation, the normal forms are derived to obtain the bifurcation diagrams such as Hopf, homoclinic and double limit cycle bifurcations. Some examples are given to confirm the theoretical results.     

Bogdanov–Takens bifurcation in an oscillator with negative damping and delayed position feedback

In this paper, Bogdanov–Takens bifurcation occurring in an oscillator with negative damping and delayed position feedback is investigated. By using center manifold reduction and normal form theory, dynamical classification near Bogdanov–Takens point can be completely figured out in terms of the second and third derivatives of delayed feedback term evaluated at the zero equilibrium. The obtained normal form and numerical simulations show that multistability, heteroclinic orbits, stable double homoclinic orbits, large amplitude periodic oscillation, and subcritical Hopf bifurcation occur in an oscillator with negative damping and delayed position feedback. The results indicate that negative damping and delayed position feedback can make the system produce more complicated dynamics.     

Complex Dynamics in a Unified SIR and HIV Disease Model: A Bifurcation Theory Approach

This paper is concerned with complex dynamical behaviors of a simple unified SIR and HIV disease model with a convex incidence and four real parameters. Due to the complex nature of the disease dynamics, our goal is to explore bifurcations including multistable states, limit cycles, and homoclinic loops in the whole parameter space. The first contribution is the proof of the existence of multiple limit cycles giving rise from Hopf bifurcation, which further induces bistable or tristable states because of the coexistence of stable equilibria and periodic motion. Next, we propose that the existence of Bogdanov–Takens (BT) bifurcation yields the bifurcation of homoclinic loops, which provides a new mechanism for generating disease recurrence, for example, the relapse–remission, viral blip cycles in HIV infection. Last, we present a novel method for the derivation of the normal forms of codimension two and three BT bifurcations. The method is based on the simplest normal form theory from Yu’s previous works.

     

Bogdanov‐Takens bifurcations of codimensions 2 and 3 in a Leslie‐Gower predator‐prey model with Michaelis‐Menten–type prey harvesting

The Bogdanov‐Takens bifurcations of a Leslie‐Gower predator‐prey model with Michaelis‐Menten–type prey harvesting were studied. In the paper “Diff. Equ. Dyn. Syst. 20(2012), 339‐366,” Gupta et al proved that the Leslie‐Gower predator‐prey model with Michaelis‐Menten–type prey harvesting has rich dynamics. Some equilibria of codimension 1 and their bifurcations were discussed. In this paper, we find that the model has an equilibrium of codimensions 2 and 3. We also prove analytically that the model undergoes Bogdanov‐Takens bifurcations (cusp cases) of codimensions 2 and 3. Hence, the model can have 2 limit cycles, coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1 as the values of parameters vary. Moreover, several numerical simulations are conducted to illustrate the validity of our results.     

Bifurcation analysis of a Leslie-type predator–prey system with simplified Holling type IV functional response and strong Allee effect on prey

In this paper, a Leslie-type predator–prey system with simplified Holling type IV functional response and strong Allee effect on prey is proposed. The dissipativity of the system and the existence of all possible equilibria are investigated. The investigation emphasizes the exploring of bifurcation. It is shown that the system exists several non-hyperbolic positive equilibria, such as a weak focus of multiplicities one and two, (degenerate) saddle–nodes and Bogdanov–Takens singularities (cusp case) of codimensions 2 and 3. At these equilibria, it is proved that the system undergoes various kinds of bifurcations, such as saddle–node bifurcation, Hopf bifurcation, degenerate Hopf bifurcation and Bogdanov–Takens bifurcation of codimensions 2 and 3. With the parameters selected properly, there exhibits a limit cycle, a homoclinic loop, two limit cycles, a semistable limit cycle, or the simultaneous occurrence of a homoclinic loop and a limit cycle in the system. Moreover, it is also proved that the system has a cusp of codimension at least 4. Hence, there may exist three limit cycles generated from Hopf bifurcation of codimension 3. Numerical simulations are done to support the theoretical results.     

An SIRS model with a nonlinear incidence rate

The global dynamics of an SIRS model with a nonlinear incidence rate is investigated. We establish a threshold for a disease to be extinct or endemic, analyze the existence and asymptotic stability of equilibria, and verify the existence of bistable states, i.e., a stable disease free equilibrium and a stable endemic equilibrium or a stable limit cycle. In particular, we find that the model admits stability switches as a parameter changes. We also investigate the backward bifurcation, the Hopf bifurcation and Bogdanov–Takens bifurcation and obtain the Hopf bifurcation criteria and Bogdanov–Takens bifurcation curves, which are important for making strategies for controlling a disease.     

Dynamics and bifurcations of a modified Leslie‐Gower–type model considering a Beddington‐DeAngelis functional response

In this paper, a planar system of ordinary differential equations is considered, which is a modified Leslie‐Gower model, considering a Beddington‐DeAngelis functional response. It generates a complex dynamics of the predator‐prey interactions according to the associated parameters. From the system obtained, we characterize all the equilibria and its local behavior, and the existence of a trapping set is proved. We describe different types of bifurcations (such as Hopf, Bogdanov‐Takens, and homoclinic bifurcation), and the existence of limit cycles is shown. Analytic proofs are provided for all results. Ecological implications and a set of numerical simulations supporting the mathematical results are also presented.     

Bogdanov–Takens bifurcation in a predator–prey model

In this paper, we investigate a class of predator–prey model with age structure and discuss whether the model can undergo Bogdanov–Takens bifurcation. The analysis is based on the normal form theory and the center manifold theory for semilinear equations with non-dense domain combined with integrated semigroup theory. Qualitative analysis indicates that there exist some parameter values such that this predator–prey model has an unique positive equilibrium which is Bogdanov–Takens singularity. Moreover, it is shown that under suitable small perturbation, the system undergoes the Bogdanov–Takens bifurcation in a small neighborhood of this positive equilibrium.     

Mixed-mode oscillation genealogy in a compartmental model of bone mineral metabolism

Summary A well-supported self-oscillating eight-compartment model has been proposed by Staub et al. to account for thein vivo rat calcium metabolism (Staub et al.,Am. J. Physiol. 254, R134–139, 1988). The nonlinear nucleus of this model is a three-compartment subunit which represents the dynamic autocatalytic processes of phase transition at the interface between bone and extracellular fluids. The organization of the temporal mixed-mode oscillations which successively appear as the calcium input is varied is analyzed. On one side of the bifurcation diagram, the generation of periodic trajectories with a single large amplitude oscillation is governed by homoclinic tangencies to small amplitude limit cycles and follows the universal sequence (U-sequence) given for the periodic solutions of unimodal transformations of the unit interval into itself. On the other side, the progressive appearance and interweaving of trajectories with multiple large amplitude oscillations per period is linked to homoclinic tangencies to large amplitude unstable cycles. The bifurcation sequence responsible for the temporal pattern generation has been analyzed by modeling the first return map of the differential system associated with the compartmental subunit. We establish that this genealogy does not follow the usual Farey treelike organization and that a comprehensive view of the resulting fractal bifurcation structure can be obtained from the unfolding of singular points of bimodal maps. These theoretical features can be compared with those reported in experiments on dissolution processes, and the extent to which the knowledge of the subunit bifurcation structure provides new conceptual insights in the field of bone and calcium metabolism is discussed.     

Global indeterminacy of the equilibrium in the Chamley model of endogenous growth in the vicinity of a Bogdanov–Takens bifurcation

This paper studies the dynamics implied by the Chamley (1993) model, a variant of the two-sector model with an implicit characterization of the learning function. We first show that under some “regularity” conditions regarding the learning function, the model has (a) one steady state, (b) no steady states or (c) two steady states (one saddle and one non-saddle). Moreover, via the Bogdanov–Takens theorem, we prove that for critical regions of the parameters space, the dynamics undergoes a particular global phenomenon, namely the homoclinic bifurcation. Because these findings imply the existence of a continuum of equilibrium trajectories, all departing from the same initial value of the predetermined variable, the model exhibits global indeterminacy.     

On cascades of elliptic periodic points in two-dimensional symplectic maps with homoclinic tangencies

We study bifurcations of two-dimensional symplectic maps with quadratic homoclinic tangencies and prove results on the existence of cascade of elliptic periodic points for one and two parameter general unfoldings.      

On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors

We study dynamics and bifurcations of three-dimensional diffeomorphisms with nontransverse heteroclinic cycles. We show that bifurcations under consideration lead to the birth of wild-hyperbolic Lorenz attractors. These attractors can be viewed as periodically perturbed classical Lorenz attractors, however, they allow for the existence of homoclinic tangencies and, hence, wild hyperbolic sets.      

Steady-state,self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations

This paper analyzes a controlled servomechanism with feedback and a cubic nonlinearity by means of the Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations, from which steady-state, self-oscillating and chaotic behaviors will be investigated using the center manifold theorem. The system controller is formed by a Proportional plus Integral plus Derivative action (PID) that allows to stabilize and drive to a prescribed set point a body connected to the shaft of a DC motor. The Bogdanov–Takens bifurcation is analyzed through the second Lyapunov stability method and the harmonic-balance method, whereas the first Lyapunov value is used for the Andronov–Poincaré–Hopf bifurcation. On the basis of the results deduced from the bifurcation analysis, we show a procedure to select the parameters of the PID controller so that an arbitrary steady-state position of the servomechanism can be reached even in presence of noise. We also show how chaotic behavior can be obtained by applying a harmonical external torque to the device in self-oscillating regime. The advantage of achieving chaotic behavior is that it can be used so that the system reaches a set point inside a strange attractor with a small control effort. The analytical calculations have been verified through detailed numerical simulations.     

Multiple bifurcation analysis in a NDDE arising from van der Pol’s equation with extended delay feedback

Van der Pol’s equation with extended delay feedback control is considered, which is equivalent to a system of neutral differential–difference equations (NDDEs). Fold bifurcation and Hopf bifurcation in this NDDE are studied by the formal adjoint theory, the center manifold theorem and the normal form method. These methods are also first employed in studying the Bogdanov–Takens singularity of NDDE. Bifurcation sets theoretically indicate the existence of a homoclinic orbit and the coexistence of three periodic solutions, which are all illustrated by the numerical methods. The coexistence of three stable periodic solutions and the existence of stable torus near the Hopf–fold and Hopf–Hopf bifurcations are also illustrated, respectively.     

Multiple bifurcations in a delayed predator–prey diffusion system with a functional response

The present paper is concerned with a delayed predator–prey diffusion system with a Beddington–DeAngelis functional response and homogeneous Neumann boundary conditions. If the positive constant steady state of the corresponding system without delay is stable, by choosing the delay as the bifurcation parameter, we can show that the increase of the delay can not only cause spatially homogeneous Hopf bifurcation at the positive constant steady state but also give rise to spatially heterogeneous ones. In particular, under appropriate conditions, we find that the system has a Bogdanov–Takens singularity at the positive constant steady state, whereas this singularity does not occur for the corresponding system without diffusion. In addition, by applying the normal form theory and center manifold theorem for partial functional differential equations, we give normal forms of Hopf bifurcation and Bogdanov–Takens bifurcation and the explicit formula for determining the properties of spatial Hopf bifurcations.     

Abundance of generic homoclinic tangencies in real-analytic families of diffeomorphisms

We consider one-parameter families of two-dimensional diffeomorphisms with homoclinic tangencies. Various authors considered the dynamical complexities due to such tangencies satisfying certain nondegeneracy conditions. In this paper we provide methods to actually verify, for real analytic families, that there are homoclinic tangencies which satisfy these (generic) nondegeneracy generic conditions.     

Triple-zero bifurcation in van der Pol’s oscillator with delayed feedback

In this paper, we study a classical van der Pol’s equation with delayed feedback. Triple-zero bifurcation is investigated by using center manifold reduction and the normal form method for retarded functional differential equation. We get the versal unfolding of the norm forms at the triple-zero bifurcation and show that the model can exhibit transcritical bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and zero-Hopf bifurcation. Some numerical simulations are given to support the analytic results.     

Unique normal forms for the Takens–Bogdanov singularity in a special case

We consider further reduction of normal forms for nilpotent planar vector fields. We give a unique normal form for a special case of an open problem for the Takens–Bogdanov singularity.     

Numerical approximation of homoclinic chaos

Transversal homoclinic orbits of maps are known to generate a Cantor set on which a power of the map conjugates to the Bernoulli shift on two symbols. This conjugacy may be regarded as a coding map, which for example assigns to a homoclinic symbol sequence a point in the Cantor set that lies on a homoclinic orbit of the map with a prescribed number of humps. In this paper we develop a numerical method for evaluating the conjugacy at periodic and homoclinic symbol sequences in a systematic way. The approach combines our previous method for computing the primary homoclinic orbit with the constructive proof of Smale's theorem given by Palmer. It is shown that the resulting nonlinear systems are well conditioned uniformly with respect to the characteristic length of the symbol sequence and that Newton's method converges uniformly too when started at a proper pseudo orbit. For the homoclinic symbol sequences an error analysis is given. The method works in arbitrary dimensions and it is illustrated by examples. This revised version was published online in June 2006 with corrections to the Cover Date.