zero-Hopf Bifurcation Research Articles

Zero-Hopf Bifurcations of 3D Quadratic Jerk System

This paper is devoted to local bifurcations of three-dimensional (3D) quadratic jerk system. First, we start by analysing the saddle-node bifurcation. Then we introduce the concept of canonical system. Next, we study the transcritial bifurcation of canonical system. Finally we study the zero-Hopf bifurcations of canonical system, which constitutes the core contributions of this paper. By averaging theory of first order, we prove that, at most, one limit cycle bifurcates from the zero-Hopf equilibrium. By averaging theory of second order, third order, and fourth order, we show that, at most, two limit cycles bifurcate from the equilibrium. Overall, this paper can help to increase our understanding of local behaviour in the jerk dynamical system with quadratic non-linearity.

On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in R 3

Here we study 3-dimensional Lotka–Volterra systems. It is known that some of these differential systems can have at least four periodic orbits bifurcating from one of their equilibrium points. Here we prove that there are some of these differential systems exhibiting at least six periodic orbits bifurcating from one of their equilibrium points. We remark that these systems with such six periodic orbits are non-competitive Lotka–Volterra systems. The proof is done using the algorithm that we provide for computing the periodic solutions that bifurcate from a zero-Hopf equilibrium based in the averaging theory of third order. This algorithm can be applied to any differential system having a zero-Hopf equilibrium.

Hopf–Zero bifurcation in an age-dependent predator–prey system with Monod–Haldane functional response comprising strong Allee effect

The article is intended to study the Hopf−Zero bifurcation of a novel age-dependent predator−prey system with Monod−Haldane functional response comprising strong Allee effect. Here in this system the predators fertility function f(x) is regarded as a piecewise function with respect to their maturation period τ. The system is interpreted as a non-densely defined abstract Cauchy problem, and the condition of the existence and uniqueness of the non-negative steady state for a coupled dynamic system both a partial differential equation and an ordinary differential equation is yielded. Via employing the center manifold theorem and normal form theory for semilinear equations with non-dense domain, we discover much richer fresh dynamical behavior in an age-dependent predator−prey system than the existing ones.

N-Dimensional Zero-Hopf Bifurcation of Polynomial Differential Systems via Averaging Theory of Second Order

Using the averaging theory of second order, we study the limit cycles which bifurcate from a zero-Hopf equilibrium point of polynomial vector fields with cubic nonlinearities in $\mathbb {R}^{n}$ . We prove that there are at least 3n− 2 limit cycles bifurcating from such zero-Hopf equilibrium points. Moreover, we provide an example in dimension 6 showing that this number of limit cycles is reached.

Integrability and zero-Hopf bifurcation in the Sprott A system

The first objective of this paper is to study the Darboux integrability of the polynomial differential systemx˙=y,y˙=−x−yz,z˙=y2−a and the second one is to show that for a>0 sufficiently small this model exhibits one small amplitude periodic solution that bifurcates from the origin of coordinates when a=0. This model was introduced by Hoover as the first example of a differential equation with a hidden attractor and it was used by Sprott to illustrate a differential equation having a chaotic behavior without equilibrium points, and now this system is known as the Sprott A system.

Periodic solutions for a four-dimensional hyperchaotic system

In this paper, we show a zero-Hopf bifurcation in a four-dimensional smooth quadratic autonomous hyperchaotic system. Using averaging theory, we prove the existence of periodic orbits bifurcating from the zero-Hopf equilibrium located at the origin of the hyperchaotic system, and the stability conditions of periodic solutions are given.

Host-virus evolutionary dynamics with specialist and generalist infection strategies: Bifurcations, bistability, and chaos.

In this work, we have investigated the evolutionary dynamics of a generalist pathogen, e.g., a virus population, that evolves toward specialization in an environment with multiple host types. We have particularly explored under which conditions generalist viral strains may rise in frequency and coexist with specialist strains or even dominate the population. By means of a nonlinear mathematical model and bifurcation analysis, we have determined the theoretical conditions for stability of nine identified equilibria and provided biological interpretation in terms of the infection rates for the viral specialist and generalist strains. By means of a stability diagram, we identified stable fixed points and stable periodic orbits, as well as regions of bistability. For arbitrary biologically feasible initial population sizes, the probability of evolving toward stable solutions is obtained for each point of the analyzed parameter space. This probability map shows combinations of infection rates of the generalist and specialist strains that might lead to equal chances for each type becoming the dominant strategy. Furthermore, we have identified infection rates for which the model predicts the onset of chaotic dynamics. Several degenerate Bogdanov-Takens and zero-Hopf bifurcations are detected along with generalized Hopf and zero-Hopf bifurcations. This manuscript provides additional insights into the dynamical complexity of host-pathogen evolution toward different infection strategies.

Zero-Hopf bifurcation analysis in an inertial two-neural system with delayed Crespi function

In this paper, we study a four-dimensional inertial two-nervous system with delay. By analyzing the distribution of eigenvalues, the critical value of zero-Hopf bifurcation is obtained. Complex dynamic behaviors are considered when two parameters change simultaneously. Pitchfork and Hopf bifurcation critical lines at near the zero-Hopf point are obtained by using the central manifold reduction and the normal form theory. The bifurcation diagram is given, and the results of period-doubling bifurcation into chaotic region in the inertial two-neural system with delayed Crespi function are shown.

Zero-Hopf bifurcation in continuous dynamical systems using multiple scale approach

Abstract An efficient scheme for studying multiparametric Zero-Hopf bifurcation is provided to extend A. Nayfeh multiple-time scale technique of codimension-one bifurcations. The suggested approach treats the cases of high codimension bifurcations successfully. Compared to the well-known averaging method, center manifold reduction method and projection method, the present approach is more simple and can be applied with less computational cost and high efficiency to a wider class of problems involving the cases where purely imaginary, zeros and negative real eigenvalues coexist simultaneously.

Zero-Hopf bifurcations and chaos of quadratic jerk systems

Zero-Hopf bifurcations and chaos of quadratic jerk systems

4-dimensional zero-Hopf bifurcation for polynomial differentials systems with cubic homogeneous nonlinearities via averaging theory

The averaging theory of second order shows that for polynomial differential systems in ℝ4 with cubic homogeneous nonlinearities at least nine limit cycles can be born in a zero-Hopf bifurcation.

4-dimensional zero-Hopf bifurcation for polynomial differentials systems with cubic homogeneous nonlinearities via averaging theory

4-dimensional zero-Hopf bifurcation for polynomial differentials systems with cubic homogeneous nonlinearities via averaging theory

Stability and zero-Hopf bifurcation analysis of a tumour and T-helper cells interaction model in the case of HIV infection

Stability and zero-Hopf bifurcation analysis of a tumour and T-helper cells interaction model in the case of HIV infection

Orbital normal forms for a class of three-dimensional systems with an application to Hopf-zero bifurcation analysis of Fitzhugh–Nagumo system

We consider a class of three-dimensional systems having an equilibrium point at the origin, whose principal part is of the form (−∂h∂y(x,y),∂h∂x(x,y),f(x,y))T. This principal part, which has zero divergence and does not depend on the third variable z, is the coupling of a planar Hamiltonian vector field Xh(x,y):=(−∂h∂y(x,y),∂h∂x(x,y))T with a one-dimensional system.We analyze the quasi-homogeneous orbital normal forms for this kind of systems, by introducing a new splitting for quasi-homogeneous three-dimensional vector fields. The obtained results are applied to the nondegenerate Hopf-zero singularity that falls into this kind of systems. Beyond the Hopf-zero normal form, a parametric normal form is obtained, and the analytic expressions for the normal form coefficients are provided. Finally, the results are applied to a case of the three-dimensional Fitzhugh–Nagumo system.

Predator interference in a Leslie–Gower intraguild predation model

The aim of this paper is to analyze general dynamics features of a new Intraguild Predation (IGP) model where the top predator feeds only on the mesopredator and also affects its consumption rate. Important dynamical aspects of the model are described. Specifically, we prove that the trajectories of the associated system are bounded and defined for all positive time; there is a trapping domain; there are open subsets of parameters, such that the system in the first octant has at most five equilibrium solutions and at most three of them are of co-existence. Here we characterize the existence of Hopf bifurcations and we prove that this model exhibits either one, two or three small amplitude periodic solutions which arise from a zero-Hopf bifurcation. We prove the existence of alternative stable states. Finally, some numerical computations have been given in order to support our analytical results. The importance of some parameters of the model is discussed, in particular the role of the interference rate. The numerical exploration suggests that the equilibrium biomass of each of the three species grows as the level of interference grows.

New Symmetric Periodic Solutions for the Maxwell-Bloch Differential System

We provide sufficient conditions for the existence of a pair of symmetric periodic solutions in the Maxwell-Bloch differential equations modeling laser systems. These periodic solutions come from a zero-Hopf bifurcation studied using recent results in averaging theory.

Study of a dynamical system with a strange attractor and invariant tori

A simple three-dimensional time-reversible system of ODEs with quadratic nonlinearities is considered in a recent paper by Sprott (2014). The author finds in this system, that has no equilibria, the coexistence of a strange attractor and invariant tori. The goal of this letter is to justify theoretically the existence of infinite invariant tori and chaotic attractors. For this purpose we embed the original system in a one-parameter family of reversible systems. This allows to demonstrate the presence of a Hopf-zero bifurcation that implies the birth of an elliptic periodic orbit. Thus, the application of the KAM theory guarantees the existence of an extremely complex dynamics with periodic, quasiperiodic and chaotic motions. Our theoretical study is complemented with some numerical results. Several bifurcation diagrams make clear the rich dynamics organized around a so-called noose bifurcation where, among other scenarios, cascades of period-doubling bifurcations also originate chaotic attractors. Moreover, a cross section and other numerical simulations are also presented to illustrate the KAM dynamics exhibited by this system.

Theoretical conditions for the coexistence of viral strains with differences in phenotypic traits: a bifurcation analysis.

We investigate the dynamics of a wild-type viral strain which generates mutant strains differing in phenotypic properties for infectivity, virulence and mutation rates. We study, by means of a mathematical model and bifurcation analysis, conditions under which the wild-type and mutant viruses, which compete for the same host cells, can coexist. The coexistence conditions are formulated in terms of the basic reproductive numbers of the strains, a maximum value of the mutation rate and the virulence of the pathogens. The analysis reveals that parameter space can be divided into five regions, each with distinct dynamics, that are organized around degenerate Bogdanov–Takens and zero-Hopf bifurcations, the latter of which gives rise to a curve of transcritical bifurcations of periodic orbits. These results provide new insights into the conditions by which viral populations may contain multiple coexisting strains in a stable manner.

Hopf-zero bifurcation of Oregonator oscillator with delay

In this paper, we study the Hopf-zero bifurcation of Oregonator oscillator with delay. The interaction coefficient and time delay are taken as two bifurcation parameters. Firstly, we get the normal form by performing a center manifold reduction and using the normal form theory developed by Faria and Magalhães. Secondly, we obtain a critical value to predict the bifurcation diagrams and phase portraits. Under some conditions, saddle-node bifurcation and pitchfork bifurcation occur along M and N, respectively; Hopf bifurcation and heteroclinic bifurcation occur along H and S, respectively. Finally, we use numerical simulations to support theoretical analysis.

Dynamics of a general jerky equation

Some classic nonlinear dynamical systems, such as Rössler's toroidal model, the Genesio model, and 19 Sprott's models, can be classified into seven distinct basic classes of jerky dynamics, labeled by . This paper is devoted to the dynamics of a general jerky equation which contains as parameters vary. It is shown that the system undergoes fold, Hopf, zero-Hopf, and Bogdanov–Takens bifurcations based on the center manifold theorem and normal form theory. Numerical simulations are also given to make the theoretical results visible and to detect more complicated dynamical behaviors, including degenerate Hopf bifurcation, fold bifurcation of cycle, and limit cycles. Especially, an apple-like attractive portrait is discovered near the zero-Hopf bifurcation point for the first time. Finally, according to the conclusions of the general jerky equation, exact conditions are summarized by two tables on how bifurcations will occur for , respectively.