zero-Hopf Bifurcation Research Articles

Bifurcations and multistability in a virotherapy model with two time delays

In this paper, we establish a delayed virotherapy model including infected tumor cells, uninfected tumor cells and free virus. In this model, both infected and uninfected tumor cells have special growth patterns, and there are at most two positive equilibria. We mainly analyze the stability and Hopf bifurcation of the model under different time delays. For the model without delay, we study the Hopf and Bogdanov–Takens bifurcations. For the delayed model, by center manifold theorem and normal form theory of functional differential equation, we study the direction of Hopf bifurcation and stability of the bifurcated periodic solution. Moreover, we prove the existence of Zero-Hopf bifurcation. Finally, some numerical simulations show the results of our theoretical calculations, and the dynamic behaviors near Zero-Hopf and Bogdanov–Takens point of the system are also observed in the simulations, such as bistability, periodic coexistence and chaotic behavior.

Zero-Hopf Bifurcation of a memristive synaptic Hopfield neural network with time delay

This paper proposes a novel memristive synaptic Hopfield neural network (MHNN) with time delay by using a memristor synapse to simulate the electromagnetic induced current caused by the membrane potential difference between two adjacent neurons. First, some sufficient conditions of zero bifurcation and zero-Hopf bifurcation are obtained by choosing time delay and coupling strength of memristor as bifurcation parameters. Then, the third-order normal form of zero-Hopf bifurcation is obtained. By analyzing the obtained normal form, six dynamic regions are found on the plane with coupling strength of memristor and time delay as abscissa and ordinate. There are some interesting dynamics in these areas, i.e., the coupling strength of memristor can affect the number and dynamics of system equilibrium, time delay can contribute to both trivial equilibrium and non-trivial equilibrium losing stability and generating periodic solutions.

A 4D hyperchaotic Lorenz-type system: zero-Hopf bifurcation, ultimate bound estimation, and its variable-order fractional network

A 4D hyperchaotic Lorenz-type system: zero-Hopf bifurcation, ultimate bound estimation, and its variable-order fractional network

Integrability and bifurcation of a three-dimensional circuit differential system

<p style='text-indent:20px;'>We study integrability and bifurcations of a three-dimensional circuit differential system. The emerging of periodic solutions under Hopf bifurcation and zero-Hopf bifurcation is investigated using the center manifolds and the averaging theory. The zero-Hopf equilibrium is non-isolated and lies on a line filled in with equilibria. A Lyapunov function is found and the global stability of the origin is proven in the case when it is a simple and locally asymptotically stable equilibrium. We also study the integrability of the model and the foliations of the phase space by invariant surfaces. It is shown that in an invariant foliation at most two limit cycles can bifurcate from a weak focus.</p>

Local-Activity and Simultaneous Zero-Hopf Bifurcations Leading to Multistability in a Memristive Circuit

In this paper, we consider a memristive circuit consisting of three elements: a passive linear inductor, a passive linear capacitor and an active memristive device. The circuit is described by a four-parameter system of ordinary differential equations. We study in detail the role of parameters in the dynamics of the system. Using the existence of first integrals, we show that the circuit may present a continuum of stable periodic orbits, which arise due to the occurrence of infinitely many simultaneous zero-Hopf bifurcations on a line of equilibria located in the region where the memristance is negative and, consequently, the memristive device is locally-active. These bifurcations lead to multistability, which is a difficult and interesting problem in applied models, since the final state of a solution depends crucially on its initial condition. We also study the control of multistability by varying a parameter related to the state variable of the memristive device. All analytical results obtained were corroborated by numerical simulations.

Zero–Hopf bifurcations in Yu–Wang type systems

Zero–Hopf bifurcations in Yu–Wang type systems

Co-dimension two bifurcations analysis of a delayed tumor model with Allee effect

The mathematical model has become an important means to study tumor treatment and has developed with the discovery of medical phenomena. In this paper, we establish a delayed tumor model, in which the Allee effect is considered. Different from the previous similar tumor models, this model is mainly studied from the point of view of stability and co-dimension two bifurcations, and some nontrivial phenomena and conclusions are obtained. By calculation, there are at most two positive equilibria in the system, and their stability is investigated. Based on these, we find that the system undergoes Bautin bifurcation, zero-Hopf bifurcation, and Hopf–Hopf bifurcation with time delay and tumor growth rate as bifurcation parameters. The interesting thing is that there is a Zero-Hopf bifurcation, which is not common in tumor models, making abundant dynamic phenomena appear in the system. By using the bifurcation theory of functional differential equations, we calculate the normal form of these Co-dimension two bifurcations. Finally, with the aid of MATLAB package DDE-BIFTOOL, some numerical simulations have been performed to support our theoretical results. In particular, we obtain the bifurcation diagram of the system in the two parameter plane and divide its regions according to the bifurcation curves. Meanwhile, the phenomena of multistability and periodic coexistence of some regions can be also demonstrated. Combined with the simulation results, we can know that when the tumor growth rate and the delay of immune cell apoptosis are small, the tumor may tend to be stable, and vice versa.

Dynamical Analysis and Periodic Solution of a Chaotic System with Coexisting Attractors

Chaotic attractors with no equilibria, with an unstable node, and with stable node-focus are presented in this paper. The conservative solutions are investigated by the semianalytical and seminumerical method. Furthermore, multiple coexisting attractors are investigated, and circuit is carried out. To study the system’s global structure, dynamics at infinity for this new chaotic system are studied using Poincaré compactification of polynomial vector fields in R 3 . Meanwhile, the dynamics near the infinity of the singularities are obtained by reducing the system’s dimensions on a Poincaré ball. The averaging theory analyzes the periodic solution’s stability or instability that bifurcates from Hopf-zero bifurcation.

Retardational Effect and Hopf Bifurcations in a New Attitude System of Quad-Rotor Unmanned Aerial Vehicle

The effect of retardation along second vector of angular velocity on a new attitude system of quad-rotor unmanned aerial vehicle (QUAV) is examined in this paper. Catastrophic and hovering conditions of rotor dynamics are verified through bifurcation analysis. It is analyzed that disturbed rotor dynamics during yaw maneuvering lead towards the existence of oscillations and hamper the efficiency of attitude system. We have categorically signified the situation where remote controller of attitude system cannot prevent flipping and consequently, shortens the life of QUAV. This type of situation arises in a strong wind friction zone where positive drag force of rotor dynamics is impeded by the magnitude of rotational moment of inertia. Moreover, subcritical and zero-Hopf bifurcations aroused due to retardational effect in attitude system are analyzed with the aid of normal form and averaging theory. Numerical simulations are also provided for validation of analytical results.

Four-dimensional Zero-Hopf Bifurcation of Quadratic Polynomial Differential System, via Averaging Theory of Third Order

Four-dimensional Zero-Hopf Bifurcation of Quadratic Polynomial Differential System, via Averaging Theory of Third Order

Hopf-zero bifurcation of the ring unidirectionally coupled Toda oscillators with delay

In this paper, the Hopf-zero bifurcation of the ring unidirectionally coupled Toda oscillators with delay was explored. First, the conditions of the occurrence of Hopf-zero bifurcation were obtained by analyzing the distribution of eigenvalues in correspondence to linearization. Second, the stability of Hopf-zero bifurcation periodic solutions was determined based on the discussion of the normal form of the system, and some numerical simulations were employed to illustrate the results of this study. Lastly, the normal form of the system on the center manifold was derived by using the center manifold theorem and normal form method.

The zero-Hopf bifurcations of a four-dimensional hyperchaotic system

We consider the four-dimensional hyperchaotic system ẋ=a(y−x), ẏ=bx+u−y−xz, ż=xy−cz, and u̇=−du−jx+exz, where a, b, c, d, j, and e are real parameters. This system extends the famous Lorenz system to four dimensions and was introduced in Zhou et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 27, 1750021 (2017). We characterize the values of the parameters for which their equilibrium points are zero-Hopf points. Using the averaging theory, we obtain sufficient conditions for the existence of periodic orbits bifurcating from these zero-Hopf equilibria and give some examples to illustrate the conclusions. Moreover, the stability conditions of these periodic orbits are given using the Routh–Hurwitz criterion.

Limit cycles bifurcating from a zero-Hopf equilibrium of a 3-dimensional continuous differential system

A zero-Hopf equilibrium of a differential system in \({\mathbb {R}}^3\) is an equilibrium point whose linear part has eigenvalues 0 and \(\pm \omega i\) with \(\omega > 0\). We provide necessary and sufficient conditions for the existence of two or one limit cycles bifurcating from a zero-Hopf equilibrium of the following 3-dimensional Lypschizian differential systems $$\begin{aligned} \begin{array}{l} {\dot{x}}= y, \\ {\dot{y}}= z,\\ {\dot{z}}= -a|x|-y+3y^{2}-xz-b, \end{array} \end{aligned}$$when \(a=b=0\). Note that due to the existence of an absolute value the vector field associated to this system is only Lypschitz. We shall prove that these limit cycles persist for \(ab>0\) with a and b sufficiently small, i.e. when the differential system has no equilibria. Additionally we provide an estimation of the size of the bifurcating small limit cycles and also characterize their kind of stability or instability. We remark that there are no works which study the zero-Hopf bifurcation of nonsmooth differential systems as we do in this paper, and that the algorithm that we use here can be applied for studying the zero-Hopf bifurcation of an arbitrary nonsmooth differential system.

Zero-Hopf Bifurcation in a Generalized Genesio Differential Equation

The purpose of the present paper is to study the presence of bifurcations of zero-Hopf type at a generalized Genesio differential equation. More precisely, by transforming such differential equation in a first-order differential system in the three-dimensional space R3, we are able to prove the existence of a zero-Hopf bifurcation from which periodic trajectories appear close to the equilibrium point located at the origin when the parameters a and c are zero and b is positive.

TWO-DIMENSIONAL ATTRACTING TORUS IN AN INTRAGUILD PREDATION MODEL WITH GENERAL FUNCTIONAL RESPONSES AND LOGISTIC GROWTH RATE FOR THE PREY

<abstract> The population coexistence in an intraguild food web model is analyzed. Three populations, the prey, the predator and super predator, are considered, where these last two populations are specialists. The sufficient conditions to guarantee a coexistence point, where the intraguild predation model exhibits a zero-Hopf bifurcation, are given. For a wide family of functional responses, these conditions are valid. The numerical simulations varying the functional responses are given. Different limit sets such as, limit cycles or invariant torus are shown. </abstract>

Analysis of Zero-Hopf Bifurcation in Two Rössler Systems Using Normal Form Theory

In recent publications [Llibre, 2014; Llibre &amp; Makhlouf, 2020], time-averaging method was applied to studying periodic orbits bifurcating from zero-Hopf critical points of two Rössler systems. It was shown that the averaging method is successful for a certain type of zero-Hopf critical points, but fails for some type of such critical points. In this paper, we apply normal form theory to reinvestigate the bifurcation and show that the method of normal forms is applicable for all types of zero-Hopf bifurcations, revealing why the time-averaging method fails for some type of zero-Hopf bifurcation.

The zero-Hopf bifurcations in the Kolmogorov systems of degree 3 in [formula omitted

In this work we study the periodic orbits which bifurcate from all zero-Hopf bifurcations that an arbitrary Kolmogorov system of degree 3 in R3 can exhibit. The main tool used is the averaging theory.

Zero-Hopf bifurcation in a 3D jerk system

Let the three-dimensional differential system defined by the jerk equation x⃛=−aẍ+xẋ2−x3−bx+cẋ, with a,b,c∈R. When a=b=0 and c<0 the equilibrium point localized at the origin of coordinates is a zero-Hopf equilibrium. We analyse the zero-Hopf bifurcation occurring at this singular point after persuading a quadratic perturbation of the coefficients. Particularly, by using averaging theory of second order, we prove that up to three periodic orbits born as the parameter of the perturbation tends to zero.

Zero–Hopf Bifurcations in Three-Dimensional Chaotic Systems with One Stable Equilibrium

In [Molaie et al., 2013] the authors provided the expressions of 23 quadratic differential systems in [Formula: see text] with the unusual feature of having chaotic dynamics coexisting with one stable equilibrium point. In this paper, we consider 23 classes of quadratic differential systems in [Formula: see text] depending on a real parameter [Formula: see text], which, for [Formula: see text], coincide with the differential systems given by [Molaie et al., 2013]. We study the dynamics and bifurcations of these classes of differential systems by varying the parameter value [Formula: see text]. We prove that, for [Formula: see text], all the 23 considered systems have a nonisolated zero–Hopf equilibrium point located at the origin. By using the averaging theory of first order, we prove that a zero–Hopf bifurcation takes place at this point for [Formula: see text], which leads to the creation of three periodic orbits bifurcating from it for [Formula: see text] small enough: an unstable one and a pair of saddle type periodic orbits, that is, periodic orbits with a stable and an unstable manifold. Furthermore, we numerically show that the hidden chaotic attractors which exist for these systems when [Formula: see text] are obtained by period-doubling route to chaos.

Zero-Hopf Periodic Orbits for a Rössler Differential System

We study the zero-Hopf bifurcation of the Rössler differential system [Formula: see text] where the dot denotes the derivative with respect to the independent variable [Formula: see text] and [Formula: see text], [Formula: see text], [Formula: see text] are real parameters.