Double Hopf Point Research Articles

Bifurcation in an modified model of neutrophil cells with time delay

The hematological stem cells model is introduced with neutrophil dynamics of two department model setting forth. During the cells differentiation and proliferation process, the neutrophils are functioned with negative feedback with delay history, which contains delayed amplification coefficient. In more general view, the new introduction rate is given to replace the familiar Hill function which is helpful to understand the complex dynamics of neutrophils. The double Hopf bifurcation is calculated with the artificial handtools named DDE-Biftool, which is observed as the self-intersection of Hopf lines. The continuation of periodical solutions arising from Hopf points are done and the longer period solutions are manifested with multi-rhythm and bursting oscillation. The near dynamics of double Hopf points is simulated by DDE-Biftool with different route design, the multi-period attractors, quasi-periodical solutions and chaos are observed.

Double Hopf bifurcation induced by spatial memory in a diffusive predator–prey model with Allee effect and maturation delay of predator

In this paper, we delve into double Hopf bifurcation induced by memory-driven directed movement in a spatial predator–prey model with Allee effect and maturation delay of predators. We first adopt a novel technique to handle the associated characteristic equation and thus obtain the crossing curves as well as the double Hopf points. We then calculate explicit formulae of normal form regarding non-resonant double Hopf bifurcation. We thus divide the dynamics of the developed model into several categories near the double Hopf bifurcation points. Our numerical and theoretical results both demonstrate that the model can exhibit various complex phenomena when the parameters are near the double Hopf bifurcation points. For example, the transition from one stable spatially inhomogeneous periodic orbit with mode-5 to another with mode-4 and the coexistence of them can be observed.

Double Hopf Bifurcation in Microbubble Oscillators with Delay Coupling

Using center manifold reduction methodswe investigate the double Hopf bifurcation in the dynamics of microbubble with delay couplingwith main attention focused on nonresonant double Hopf bifurcation. We obtain the normal form of the system in the vicinity of the double Hopf point and classify the bifurcations in a two-dimensional parameter space near the critical point. Some numerical simulations support the applicability of the theoretical results. In particularwe give the explanation for some physical phenomena of the system using the obtained mathematical results.

Hopf Bifurcation of a Type of Neuron Model with Multiple Time Delays

By applying a geometrical scheme developed to tackle the eigenvalue problem of delay differential equations with multiple time delays, Hopf bifurcation of Hopfield neuron model is analyzed in two-parameter space. By the introduction of two new angles, the calculation of imaginary roots is carried out analytically and effectively. By increasing the parameter to cross over the Hopf bifurcation lines, the stability switching direction is confirmed. The method is a useful tool to show the partition of stable and unstable regions in two-parameter space and detect double Hopf bifurcation further. The typified dynamical behaviors based on nearby double Hopf points are analyzed by applying the normal form technique and center manifold method.

Double Hopf Bifurcation of a Hematopoietic System with State Feedback Control

The dynamics of a system composed of hematopoietic stem cells and its relationship with neutrophils is ubiquitous due to periodic oscillating behavior induce cyclical neutropenia. Underlying the methodology of state feedback control with two time delays, double Hopf bifurcation occurs as varying time delay to reach its threshold value. By applying center manifold theory, the analytical analysis of system exposed the different dynamical feature in the classified regimes near double Hopf point. The novel dynamics as periodical solution and quasi-periodical attractor coexistence phenomena are explored and verified by numerical simulation.

Dynamics of Cutting Near Double Hopf Bifurcation

Bifurcation analysis of the orthogonal cutting model with cutting force nonlinearity is presented with special attention to double Hopf bifurcations. The normal form of the system in the vicinity of the double Hopf point is derived analytically by means of center manifold reduction. The dynamics is restricted to a four-dimensional center manifold, and the long-term behavior is illustrated on simplified phase portraits in two dimensions. The topology of the phase portraits reveal the coexistence of periodic and quasi-periodic solutions, which are computed by approximate analytical formulas.

Codimension-two bifurcations induce hysteresis behavior and multistabilities in delay-coupled Kuramoto oscillators

Hysteresis phenomena and multistability play crucial roles in the dynamics of coupled oscillators, which are now interpreted from the point of view of codimension-two bifurcations. On the Ott-Antonsen's manifold, complete bifurcation sets of delay-coupled Kuramoto model are derived regarding coupling strength and delay as bifurcation parameters. It is rigorously proved that the system must undergo Bautin bifurcations for some critical values, thus there always exists saddle-node bifurcation of periodic solutions inducing hysteresis loop. With the aid of center manifold reduction method and the Matlab Package DDE-Biftool, the location of Bautin and double Hopf points and detailed dynamics are theoretically determined. We find that, near these critical points, at most four coherent states (two of which are stable) and a stable incoherent state may coexist, and that the system undergoes Neimark-Sacker bifurcation of periodic solutions. Finally, the clear scenarios about the synchronous transition in delayed Kuramoto model are depicted.

Computation of double Hopf points for delay differential equations

Abstract Relating to the crucial problem of branch switching, the calculation of codimension 2 bifurcation points is one of the major issues in numerical bifurcation analysis. In this paper, we focus on the double Hopf points for delay differential equations and analyze in detail the corresponding eigenspace, which enable us to obtain the finite dimensional defining system of equations of such points, instead of an infinite dimensional one that happens naturally for delay systems. We show that the double Hopf point, together with the corresponding eigenvalues, eigenvectors and the critical values of the bifurcation parameters, is a regular solution of the finite dimensional defining system of equations, and thus can be obtained numerically through applying the classical iterative methods. We show our theoretical findings by a numerical example.

On local bifurcations in neural field models with transmission delays

Neural field models with transmission delays may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results.

THE HOMOTOPY ANALYSIS METHOD IN BIFURCATION ANALYSIS OF DELAY DIFFERENTIAL EQUATIONS

In this paper we apply the homotopy analysis method (HAM) to study the van der Pol equation with a linear delayed feedback. The paper focuses on the calculation of periodic solutions and associated bifurcations, Hopf, double Hopf, Neimark–Sacker, etc. In particular, we discuss the behavior of the systems in the neighborhoods of double Hopf points. We demonstrate the applicability of HAM to the analysis of bifurcation and stability.

Hopf bifurcation for a class of fractional differential equations with delay

The main purpose of this manuscript is to prove the existence of solutions for delay fractional order differential equations (FDE) at the neighborhood of its equilibrium point. After we convert the delay FDE into linear delay FDE by using its equilibrium point, we define the 1:2 resonant double Hopf point set with its characteristic equation. We find the members of this set in different cases. The bifurcation curves for a class of delay FDE are obtained within a differential operator of Caputo type with the lower terminal at −∞.

Hopf and resonant double Hopf bifurcation in congestion control algorithm with heterogeneous delays

The congestion control algorithm, which has dynamic adaptations at both user ends and link ends, with heterogeneous delays is considered and analyzed. Some general stability criteria involving the delays and the system parameters are derived by generalized Nyquist criteria. Furthermore, by choosing one of the delays as the bifurcation parameter, and when the delay exceeds a critical value, a limit cycle emerges via a Hopf bifurcation. Resonant double Hopf bifurcation is also found to occur in this model. An efficient perturbation-incremental method is presented to study the delay-induced resonant double Hopf bifurcation. For the bifurcation parameter close to a double Hopf point, the approximate expressions of the periodic solutions are updated iteratively by use of the perturbation-incremental method. Simulation results have verified and demonstrated the correctness of the theoretical results.

DYNAMICS AND DOUBLE HOPF BIFURCATIONS OF THE ROSE–HINDMARSH MODEL WITH TIME DELAY

In this paper, we are concerned with the Rose–Hindmarsh model with time delay. By applying the generalized Sturm criterion, a number of imaginary roots of the characteristic equation are classified. The absolutely stable regions for any value of time delay are detected. By the continuous software DDE-Biftool, both the Hopf bifurcation curves and double Hopf bifurcation points are illustrated in parametric spaces. The normal form and universal unfolding at double Hopf bifurcation points are considered by the center manifold method. Some examples also indicate that the corresponding unique attractor near each double Hopf point is asymptotically stable.

An Efficient Method for Studying Weak Resonant Double Hopf Bifurcation in Nonlinear Systems with Delayed Feedbacks

An efficient method, called the perturbation‐incremental scheme (PIS), is proposed to study, both qualitatively and quantitatively, the delay‐induced weak or high‐order resonant double Hopf bifurcation and the dynamics arising from the bifurcation of nonlinear systems with delayed feedback. The scheme is described in two steps, namely, the perturbation and the incremental steps, when the time delay and the feedback gain are taken as the bifurcation parameters. As for applications, the method is employed to investigate the delay‐induced weak resonant double Hopf bifurcation and dynamics in the van der Pol–Duffing and the Stuart–Landau systems with delayed feedback. For bifurcation parameters close to a double Hopf point, all solutions arising from the resonant bifurcation are classified qualitatively and expressed approximately in a closed form by the perturbation step of the PIS. Although the analytical expression may not be accurate enough for bifurcation parameters away from the double Hopf point, it is...

An Efficient Method for Studying Weak Resonant Double Hopf Bifurcation in Nonlinear Systems with Delayed Feedbacks

An efficient method, called the perturbation‐incremental scheme (PIS), is proposed to study, both qualitatively and quantitatively, the delay‐induced weak or high‐order resonant double Hopf bifurcation and the dynamics arising from the bifurcation of nonlinear systems with delayed feedback. The scheme is described in two steps, namely, the perturbation and the incremental steps, when the time delay and the feedback gain are taken as the bifurcation parameters. As for applications, the method is employed to investigate the delay‐induced weak resonant double Hopf bifurcation and dynamics in the van der Pol–Duffing and the Stuart–Landau systems with delayed feedback. For bifurcation parameters close to a double Hopf point, all solutions arising from the resonant bifurcation are classified qualitatively and expressed approximately in a closed form by the perturbation step of the PIS. Although the analytical expression may not be accurate enough for bifurcation parameters away from the double Hopf point, it is used as an initial guess for the incremental step which updates the approximate expression iteratively and performs parametric continuation. The analytical predictions on the two systems show that the delayed feedback can, on the one hand, drive a periodic solution into an amplitude death island where the motion vanishes and, on the other hand, create complex dynamics such as quasi‐periodic and coexisting motions. The approximate expression of periodic solutions with parameter varying far away from the double Hopf point can be calculated to any desired accuracy by the incremental step. The validity of the results is shown by their consistency with numerical simulations. We show that as an analytical tool the PIS is simple but efficient.

Regenerative Tool Chatter Near a Codimension 2 Hopf Point Using Multiple Scales

We study a well-known regenerative machine tool vibration model (a delay differential equation) near a codimension 2 Hopf bifurcation point. The method of multiple scales is used directly, bypassing a center manifold reduction. We use a nonstandard choice of expansion parameter that helps understand practically relevant aspects of the dynamics for not-too-small amplitudes. Analytical expressions are then obtained for the double Hopf points. Both sub- and supercritical bifurcations are predicted to occur near the reference point; and analytical conditions on the parameter variations for each type of bifurcation to occur are obtained as well. Analytical approximations are supported by numerics.

Tangent double Hopf bifurcation in a differentially rotating cylinder flow.

A tangent double Hopf bifurcation has been found in a rotating cylinder flow driven by the counter-rotation of the top endwall. The codimension-3 tangent double Hopf point has been located using linear stability analysis. The nonlinear analysis of the multiple solutions in a neighborhood of this bifurcation point is performed by numerical solutions of the three-dimensional Navier-Stokes equations. At the tangent double Hopf point, two rotating waves and an unstable modulated rotating wave bifurcate simultaneously. A center manifold/normal form analysis is also performed, which is in very good agreement with both the linear and nonlinear computations. By a detailed exploration of the nonlinear flow, we have unraveled the complex dynamics generated by the mode competition, which is organized by the codimension-3 tangent double Hopf bifurcation.

Mode interactions in an enclosed swirling flow: a double Hopf bifurcation between azimuthal wavenumbers 0 and 2

A double Hopf bifurcation has been found of the flow in a cylinder driven by the rotation of an endwall. A detailed analysis of the multiple solutions in a large region of parameter space, computed with an efficient and accurate three-dimensional Navier-Stokes solver, is presented. At the double Hopf point, an axisymmetric limit cycle and a rotating wave bifurcate simultaneously. The corresponding mode interaction generates an unstable two-torus modulated rotating wave solution and gives a wedge-shaped region in parameter space where the two periodic solutions are both stable. By exploring in detail the three-dimensional structure of the flow, we have identified the two mechanisms that compete in the neighbourhood of the double Hopf point. Both are associated with the jet that is formed when the Ekman layer on the rotating endwall is turned by the stationary sidewall.

Nonlinear oscillations in business cycle model with time lags

In this paper we analyse the dynamics of the Kaldor–Kalecki business cycle model. This model is based on the classical Kaldor model in which capital stock changes are caused by past investment decisions. This lag is connected with time delay needed for new capital to be installed. The dynamics of the model is reduced to the form of damped oscillator with negative feedback connected with lag parameter and next it is analysed in terms of bifurcation theory. We find conditions for existence and persistence of oscillatory behaviour which is represented by limit cycle on some central manifold in phase space, i.e., single Hopf bifurcation. We demonstrate that the Hopf cycles may be exhibited for nonzero measure set of the parameter space. The conditions for bifurcation of co-dimension two connected with interaction of bifurcations as well as bifurcation diagrams are also given. Finally, we obtain numerical values describing an amplitude and a period of oscillation for different parameter of the system. It is also proved that while the investment function is not nonlinear a quasi-periodic solution (a 1:2 resonant double Hopf point) can appear. The source of such a behaviour is rather a consequence of time lag than nonlinearity of the investment function. Our results confirm the existence of asymmetric (two periodic) cycles in the Kaldor–Kalecki model with time-to-build.

Defining Functions for Multiple Hopf Bifurcations

Let $A(u,\alpha)=F_u(u,\alpha)$ ($u \in R^n, \alpha \in R^k$) be a family of real $n \times n$ matrices arising as the Jacobian matrices of equilibrium solutions to the dynamical system $ \stackrel{.}{u}=F(u,\alpha)$. An equilibrium point is called a Hopf point if A has a conjugate pair of pure imaginary eigenvalues $\pm i\omega$, $\omega> 0$. It is called a double Hopf point if there are two such pairs $\pm i\omega_1, \pm i\omega_2$ and a 1:1 resonant double Hopf point if, in addition, $\omega_1=\omega_2$. Defining functions are obtained for the numerical detection, computation, and continuation of Hopf, double Hopf, and 1:1 resonant double Hopf points. They are based on a combination of matrix biproduct and bordered matrix methods. Example computations are done in a fairly realistic and complicated neural model problem with $n=13$ and $k=29$. However, to make the methods applicable to large-scale problems (e.g., discretized boundary value problems) we reduce the state space to a subspace that essentially contains the generalized eigenspaces of the eigenvalues with largest real part.