zero-Hopf Bifurcation Research Articles

Zero-Hopf bifurcation and Hopf bifurcation for smooth Chua\u2019s system

Based on the fact that Chua’s system is a classic model system of electronic circuits, we first present modified Chua’s system with a smooth nonlinearity, described by a cubic polynomial in this paper. Then, we explore the distribution of the equilibrium points of the modified Chua circuit system. By using the averaging theory, we consider zero-Hopf bifurcation of the modified Chua system. Moreover, the existence of periodic solutions in the modified Chua system is derived from the classical Hopf bifurcation theorem.

Zero-Hopf bifurcations in 3-dimensional differential systems with no equilibria

Recently sixteen 3-dimensional differential systems exhibiting chaotic motion and having no equilibria have been studied, and it has been graphically observed that these systems have a period-doubling cascade of periodic orbits providing a route to chaos. Here using new results on the averaging theory we prove that these systems exhibit, for some values of their parameters different to the ones having chaotic motion, either a zero-Hopf or a Hopf bifurcation, and graphically we observed that the periodic orbit starting in those bifurcations is at the beginning of the mentioned period-doubling cascade.

Complex dynamics in a generalized Langford system

This paper analyzes complex dynamics of the generalized Langford system (GLS) with five parameters. First, some important local dynamics such as the Hopf bifurcations and the stabilities of hyperbolic and zero/double-zero equilibrium are investigated using normal form theory, center manifold theory and bifurcation theory. Besides, an accurate expression of a periodic orbit and some approximate expressions of bifurcating limit cycles by Hopf bifurcation are obtained. Second, by using averaging theory, the zero-Hopf bifurcation at the origin is analyzed and also the stability of the bifurcating limit cycle is obtained. Of particular interest is that one numerically finds an annulus in three-dimensional space which appears nearby the bifurcating limit cycle under proper conditions. If an amplitude system of GLS has a center, its suspension may lead the GLS to exhibit such annulus. Third, it is proved rigorously that there exist two heteroclinic cycles and the coexistence of such two heteroclinic cycles and a periodic orbit under some conditions. This implies system has no chaos in the sense Shil’nikov heteroclinic criterion. Finally, by further numerical observation, it is shown that three different types of attractors exist simultaneously, such as two kinds of periodic orbits, periodic orbit and invariant torus.

Zero-Hopf Bifurcation in the Rössler’s Second System

This paper is devoted to study the zero-Hopf bifurcation of the Rossler's second system. We characterize the parameters for which a zero-Hopf equilibrium point takes place at each point. We prove that there are three one-parameter families exhibiting such equilibria. The averaging theory of the first order is also applied to prove the existence of one periodic orbit bifurcating from the zero-Hopf equilibrium at the origin. Here, to visualize this, FireFlies software is used.

Zero-Hopf Periodic Orbit of a Quadratic System of Differential Equations Obtained from a Third-Order Differential Equation

We study the zero-Hopf bifurcation of the third-order differential equations $$\begin{aligned} x^{\prime \prime \prime }+ (a_{1}x+a_{0})x^{\prime \prime }+ (b_{1}x+b_{0})x^{\prime }+x^{2} =0, \end{aligned}$$ where $$a_{0}$$ , $$a_{1}$$ , $$b_{0}$$ and $$b_{1}$$ are real parameters. The prime denotes derivative with respect to an independent variable t. We also provide an estimate of the zero-Hopf periodic solution and their kind of stability. The Hopf bifurcations of these differential systems were studied in [5], here we complete these studies adding their zero-Hopf bifurcations.

Zero-Hopf bifurcation for an infection-age structured epidemic model with a nonlinear incidence rate

An infection-age structured epidemic model with a nonlinear incidence rate is investigated. We formulate the model as an abstract non-densely defined Cauchy problem and derive the condition which guarantees the existence and uniqueness for positive age-dependent equilibrium of the model. By analyzing the associated characteristic transcendental equation and applying the normal form theory presented recently for non-densely defined semilinear equations, we show that the SIR (susceptible-infected-recovered) epidemic model undergoes Zero-Hopf bifurcation at the positive equilibrium which is the main result of this paper.

Zero-Hopf bifurcation in a Chua system

A zero-Hopf equilibrium is an isolated equilibrium point whose eigenvalues are ±ωi≠0 and 0. In general for a such equilibrium there is no theory for knowing when it bifurcates some small-amplitude limit cycles moving the parameters of the system. Here we study the zero-Hopf bifurcation using the averaging theory. We apply this theory to a Chua system depending on 6 parameters, but the way followed for studying the zero-Hopf bifurcation can be applied to any other differential system in dimension 3 or higher.In this paper first we show that there are three 4-parameter families of Chua systems exhibiting a zero-Hopf equilibrium. After, by using the averaging theory, we provide sufficient conditions for the bifurcation of limit cycles from these families of zero-Hopf equilibria. From one family we can prove that 1 limit cycle bifurcates, and from the other two families we can prove that 1, 2 or 3 limit cycles bifurcate simultaneously.

A Generalist Predator and the Planar Zero-Hopf Bifurcation

A typical approach for searching periodic orbits of planar dynamical systems is through the Hopf bifurcation. In this work we present a family of predator–prey models with a generalist predator which does not exhibit a Hopf bifurcation, but a planar zero-Hopf bifurcation; that means, in the whole bifurcation process the eigenvalues of the linear approximation around the equilibrium points remain as pure imaginary. Similar models with a nongeneralist predator always possess a Hopf bifurcation.

HOPF-ZERO BIFURCATION OF A DELAYED PREDATOR-PREY MODEL WITH DORMANCY OF PREDATORS

In this paper, We investigate Hopf-zero bifurcation with codimension 2 in a delayed predator-prey model with dormancy of predators. First we prove the specific existence condition of the coexistence equilibrium. Then we take the mortality rate and time delay as two bifurcation parameters to find the occurrence condition of Hopf-zero bifurcation in this model. Furthermore, using the Faria and Magalhases normal form method and the center manifold theory, we obtain the third order degenerate normal form with two original parameters. Finally, through theoretical analysis and numerical simulations, we give a bifurcation set and a phase diagram to show the specific relations between the normal form and the original system, and explain the coexistence phenomena of several locally stable states, such as the coexistence of multiperiodic orbits, as well as the coexistence of a locally stable equilibrium and a locally stable periodic orbit.

Mathematical modeling about nonlinear delayed hydraulic cylinder system and its analysis on dynamical behaviors

In this paper, we study dynamics in delayed nonlinear hydraulic cylinder equation, with particular attention focused on several types of bifurcations. Firstly, basing on a series of original equations, we model a nonlinear delayed differential equations associated with hydraulic cylinder in glue dosing processes for particleboard. Secondly, we identify the critical values for fixed point, Hopf, Hopf-zero, double Hopf and tri-Hopf bifurcations using the method of bifurcation analysis. Thirdly, by applying the multiple time scales method, the normal form near the Hopf-zero bifurcation critical points is derived. Finally, two examples are presented to demonstrate the application of the theoretical results.

Transcritical and zero-Hopf bifurcations in the Genesio system

In this paper we study the existence of transcritical and zero-Hopf bifurcations of the third-order ordinary differential equation $${\dddot{x}} + a {\ddot{x}} + b {\dot{x}} + c x - x^2 = 0$$ , called the Genesio equation, which has a unique quadratic nonlinear term and three real parameters. More precisely, writing this differential equation as a first-order differential system in $$\mathbb {R}^3$$ we prove: first that the system exhibits a transcritical bifurcation at the equilibrium point located at the origin of coordinates when $$c=0$$ and the parameters (a, b) are in the set $$\{(a,b) \in \mathbb {R}^2 : b \ne 0\} {\setminus } \{(0,b) \in \mathbb {R}^2 : b > 0\}$$ , and second that the system has a zero-Hopf bifurcation also at the equilibrium point located at the origin when $$a=c=0$$ and $$b>0$$ .

Zero-Hopf bifurcation analysis in delayed differential equations with two delays

In this manuscript, we provide a framework for zero-Hopf singularity for a general two dimensional system with two delays. The distribution of the eigenvalues for the linearized system at an equilibrium point is studied in detail. Explicit conditions for the system to undergo a zero-Hopf bifurcation are established and the corresponding normal form up to the third order terms is derived. Our theoretical results are applied to a Kaldor–Kalecki model of business cycles.

Zero-Hopf bifurcation and multistability coexistence on a four-neuron network model with multiple delays

In this paper, a four-neuron neural system with four delays is investigated to exhibit the effects of multiple delays and coupled weights on system dynamics. Zero-Hopf bifurcation is obtained, where the system characteristic equation has a simple zero and a simple pair of pure imaginary eigenvalues. The coupled weight and time delay are considered as bifurcation parameters to study dynamical behaviors derived from zero-Hopf bifurcation. Various dynamical behaviors are analyzed near the bifurcation singularity qualitatively and quantitatively in detail by using perturbation- incremental method, and bifurcation diagrams are obtained. Numerical simulations and theoretical results are given to display a stable resting state, multistability coexistence of two resting states and a pair of periodic activities in the neighbor of the zero-Hopf bifurcation point.

Complex dynamics of a nutrient-plankton system with nonlinear phytoplankton morality and allelopathy

Understanding the plankton dynamics can help us take effective measures to settle the critical issue on how to keep plankton ecosystem balance. In this paper, a nutrient-phytoplankton-zooplankton (NPZ) model is formulated to understand the mechanism of plankton dynamics. To account for the harmful effect of the phytoplankton allelopathy, a prototype for a non-monotone response function is used to model zooplankton grazing, and nonlinear phytoplankton mortality is also included in the NPZ model. Using the model, we will focus on understanding how the phytoplankton allelopathy and nonlinear phytoplankton mortality affect the plankton population dynamics. We first examine the existence of multiple equilibria and provide a detailed classification for the equilibria, then stability and local bifurcation analysis are also studied. Sufficient conditions for Hopf bifurcation and zero-Hopf bifurcation are given respectively. Numerical simulations are finally conducted to confirm and extend the analytic results. Both theoretical and numerical findings imply that the phytoplankton allelopathy and nonlinear phytoplankton mortality may lead to a rich variety of complex dynamics of the nutrient-plankton system. The results of this study suggest that the effects of the phytoplankton allelopathy and nonlinear phytoplankton mortality should receive more attention to understand the plankton dynamics.

Dynamics at Infinity, Degenerate Hopf and Zero-Hopf Bifurcation for Kingni–Jafari System with Hidden Attractors

To understand the complex dynamics of Kingni–Jafari system with hidden attractors, the first objective of this paper is to study the global dynamics, and give a complete description of the dynamics of Kingni–Jafari system at infinity by using the Poincaré compactification of a polynomial vector field in [Formula: see text]. The second objective of this paper is to prove the existence of periodic solutions in the Kingni–Jafari system by classical Hopf bifurcation and degenerate Hopf bifurcation. Moreover, based on averaging theory, a small amplitude periodic solution that bifurcates from a zero-Hopf equilibrium was derived in the Kingni–Jafari system. The theoretical analysis and simulations demonstrate the rich dynamics of the system.

Bifurcation Analysis and Chaos Control in a Modified Finance System with Delayed Feedback

We investigate the effect of delayed feedback on the finance system, which describes the time variation of the interest rate, for establishing the fiscal policy. By local stability analysis, we theoretically prove the existences of Hopf bifurcation and Hopf-zero bifurcation. By using the normal form method and center manifold theory, we determine the stability and direction of a bifurcating periodic solution. Finally, we give some numerical solutions, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable equilibrium or periodic orbit.

Zero-Hopf bifurcation in the generalized Michelson system

We provide sufficient conditions for the existence of two periodic solutions bifurcating from a zero–Hopf equilibrium for the differential systemx˙=y,y˙=z,z˙=a+by+cz−x2/2,where a, b and c are real arbitrary parameters. The regular perturbation of this differential system provides the normal form of the so–called triple–zero bifurcation.

Hopf and zero-Hopf bifurcations in the Hindmarsh–Rose system

We prove the existence of the classical Hopf bifurcation and of the zero-Hopf bifurcation in the Hindmarsh–Rose system. For doing this, some adequate change in parameters must be done in order that the computations become easier.

Bifurcation analysis and chaos control of the modified Chua’s circuit system

From the view of bifurcation and chaos control, the dynamics of modified Chua’s circuit system are investigated by a delayed feedback method. Firstly, the local stability of the equilibria is discussed by analyzing the distribution of the roots of associated characteristic equation. The regions of linear stability of equilibria are given. It is found that there exist Hopf bifurcation and Hopf-zero bifurcation when the delay passes though a sequence of critical values. By using the normal form method and the center manifold theory, we derive the explicit formulas for determining the direction and stability of Hopf bifurcation. Finally, chaotic oscillation is converted into a stable equilibrium or a stable periodic orbit by designing appropriate feedback strength and delay. Some numerical simulations are carried out to support the analytic results.

On the periodic orbit bifurcating from one single non-hyperbolic equilibrium in a chaotic jerk system

This paper proposes a chaotic jerk system coexisting with only one non-hyperbolic equilibrium with one zero eigenvalue and a pair of complex conjugate eigenvalues. The system has no classical Hopf bifurcations and belongs to a newly category of chaotic systems. Based on the averaging theory, an analytic proof of the existence of zero-Hopf bifurcation is exhibited. Moreover, unstable periodic orbits from the zero-Hopf bifurcation are obtained. This approach may be useful to clarify chaotic attractors with non-hyperbolic equilibrium hidden behind complicated phenomena.