Hopf Bifurcation Problem Research Articles

Hopf Bifurcation Analysis of a Nose Landing Gear System of Aircraft

The Hopf bifurcation problem of a nose landing gear system is investigated. Taking the nearly smooth landing gear system with high-order nonlinear terms as the research object, the nose landing gear system is simplified to a plane dynamic system considering nonlinear factors by using the center manifold simplification method. The first Lyapunov coefficient of the landing gear system is derived by using the classical bifurcation theory, and the Hopf bifurcation type of the system is determined according to its sign. The bifurcation diagram is obtained by solving the differential equations of motion numerically. It is revealed that there is a stable limit cycle in the neighborhood of the supercritical Hopf bifurcation point. The influence of velocity and other factors on the shimmy oscillations of the landing gear system is analyzed. The dynamical behavior in the neighborhood of the Hopf bifurcation point is explored theoretically and numerically. The research results can provide a theoretical basis for the optimal design of aircraft landing gear systems.

Multiple Time Scales and Normal Form of Hopf Bifurcation in a Delayed Reaction–Diffusion–Advection System

In the reaction–diffusion systems, the Laplacian is usually a self-adjoint operator. Incorporating advection terms generally destroys this and yields a quite complicated decomposition of phase space in the Center Manifold Reduction (CMR) method. Considering that the Multiple Time Scales (MTS) method can be directly applied to bifurcation analysis and normal form derivation based on algebraic operations, and the computational process is relatively universal, we apply the MTS method to analyze the Hopf bifurcation problem of reaction–diffusion–advection systems with time delay. First, we introduce the MTS method for the system with a specific boundary condition, which is used to derive the normal form of Hopf bifurcation. Calculating the normal form using the MTS method can be summarized as determining the bifurcation parameters, conducting Taylor expansion based on the MTS idea, and eliminating secular terms. Then, the key coefficients in the normal form are explicitly given, which are also compared with the results obtained by the CMR method, and both methods lead to the same bifurcation results. Finally, we use the MTS method to calculate the normal form of Hopf bifurcation for a predator–prey model with mixed boundary conditions. We find that the spatially nonhomogeneous periodic solutions appear near the equilibrium in the system when the time delay exceeds the critical value, and the theoretical results are illustrated by numerical simulations.

Normal forms of a class of partial functional differential equations

In this paper, we investigate the normal form a class of partial functional differential equations, which takes account of delays in the diffusion terms as well as a wider scope of nonlinear terms. We first study the associated linear theory, mainly including the spectral properties of infinitesimal generator, formal adjoint and decomposition of phase space. Based on these results, the normal form theory for nonlinear equation is established, which can be used to study the local dynamics near the steady state for such equations. As an application, we consider the Hopf bifurcation problem of a scalar diffusive equation with delay not only involved in the diffusive term but also in reaction terms. The normal form, depending on the original coefficients, up to the third order term is calculated, which allows us to determine the direction of Hopf bifurcation and stability of bifurcated periodic solutions. The results are then applied to study the scalar population model with memory-based diffusion for modeling the movement of highly-developed animals.

Global Hopf Bifurcation of State-Dependent Delay Differential Equations

We apply the [Formula: see text]-equivariant degree method to a Hopf bifurcation problem for functional differential equations with a state-dependent delay. The formal linearization of the system at a stationary state is extracted and translated into a bifurcation invariant by using the homotopy invariance of [Formula: see text]-equivariant degree. As a result, the local Hopf bifurcation is detected and the global continuation of periodic solutions is described.

Hopf bifurcation problems near double positive equilibrium points for a class of quartic Kolmogorov model

<abstract><p>The Kolmogorov model is a class of significant ecological models and is initially introduced to describe the interaction between two species occupying the same ecological habitat. Limit cycle bifurcation problem is close to Hilbertis 16th problem. In this paper, we focus on investigating bifurcation of limit cycle for a class of quartic Kolmogorov model with two positive equilibrium points. Using the singular values method, we obtain the Lyapunov constants for each positive equilibrium point and investigate their limit cycle bifurcations behavior. Furthermore, based on the analysis of their Lyapunov constants' structure and Hopf bifurcation, we give the condition that each one positive equilibrium point of studied model can bifurcate 5 limit cycles, which include 3 stable limit cycles.</p></abstract>

Numerical investigation on hopf bifurcation problem for nonlinear dynamics of a towed vessel in calm water and waves

In this study, the Hopf bifurcation problem with regard to the nonlinear dynamics of a towed vessel was numerically investigated. A time-domain simulation method based on three degrees-of-freedom maneuvering equations of motion was applied to study the nonlinear dynamic responses of a towed vessel in calm water, wind and waves. In the towing operation, it is assumed that a vessel is passively towed by a tug via a towline with a constant towing speed following a straight line. In the case of wind or wave conditions, the steady wind or wave drift forces are considered as additional external force components. The hydrodynamic forces acting on the towed vessel are modeled as a modular-type hull force model, and the towline force is simply modeled as a linear spring. First, for the validation, the motion trajectories of the towed vessel were compared between the experimental data and numerical simulation results under the calm water condition. Discussions are provided on the effect of skegs on the dynamics of the towed vessel and related Hopf bifurcation according to the skeg size. Next, the wind speed was changed, and the resulting limit cycles were studied. It can be observed that the Hopf bifurcation can be delayed according to the wind speed and direction. Finally, the dynamic characteristics of the towed vessel in regular waves were investigated via numerical simulations. The stability of the towed vessel with no skegs increases owing to the mean wave drift forces for both the model test and numerical simulation. The effects of the wave height and period on the Hopf bifurcation were studied based on numerical simulations. It was found that the occurrence of Hopf bifurcation for nonlinear dynamics of a towed vessel is sensitive to the conditions of wave height and period.

Spatiotemporal dynamics on a class of [formula omitted]-dimensional reaction–diffusion neural networks with discrete delays and a conical structure

In this paper, the stability and Hopf bifurcation problems on a class of high-order reaction–diffusion neural networks are investigated, where the conical structure and signal transmission delays are introduced to make the model more accurate. By the linear stability analysis, Hopf bifurcation theory and Turing instability theorem, some criteria on the local stability and Hopf bifurcation of this system are established by selecting the time delay as bifurcation parameter. Especially, the Coates formula and the holistic element method are taken to determine the critical value, at which the purely imaginary root appears in the associated high-degree characteristic equation. On the other hand, the effects of diffusion terms on the spatial distribution of periodic solutions are explored. New sufficient conditions are derived to guarantee the occurrence of spatially homogeneous periodic solutions. Finally, some numerical examples are provided to validate the main results. It is found that the critical value can also be affected by the network size and the self-feedback rate.

New exploration on bifurcation for fractional-order quaternion-valued neural networks involving leakage delays.

During the past decades, many works on Hopf bifurcation of fractional-order neural networks are mainly concerned with real-valued and complex-valued cases. However, few publications involve the quaternion-valued neural networks which is a generalization of real-valued and complex-valued neural networks. In this present study, we explorate the Hopf bifurcation problem for fractional-order quaternion-valued neural networks involving leakage delays. Taking advantage of the Hamilton rule of quaternion algebra, we decompose the addressed fractional-order quaternion-valued delayed neural networks into the equivalent eight real valued networks. Then the delay-inspired bifurcation condition of the eight real valued networks are derived by making use of the stability criterion and bifurcation theory of fractional-order differential dynamical systems. The impact of leakage delay on the bifurcation behavior of the involved fractional-order quaternion-valued delayed neural networks has been revealed. Software simulations are implemented to support the effectiveness of the derived fruits of this study. The research supplements the work of Huang et al. (Neural Netw 117:67-93, 2019).

Lyapunov Quantities for Andronov–Hopf Bifurcation Problem in Reaction–Diffusion Systems

Lyapunov Quantities for Andronov–Hopf Bifurcation Problem in Reaction–Diffusion Systems

Singular Limit in Hopf Bifurcation for Doubly Diffusive Convection Equations I: Linearized Analysis at Criticality

A singularly perturbed system for doubly diffusive convection equations, called the artificial compressible system, is considered on a two-dimensional infinite layer for a parameters range where the Hopf bifurcation occurs in the corresponding incompressible system. The spectrum of the linearized operator in a time periodic function space is investigated in detail near the bifurcation point when the singular perturbation parameter is small. The results of this paper are the basis of the study of the nonlinear Hopf bifurcation problem and the singular limit of the time periodic bifurcating solutions.

Hidden Symmetries, Coupled Networks and Equivariant Degrees

Hidden symmetries have been previously explored in the context of coupled cell networks and coupled cell systems. These include interior symmetry, quotient symmetry and quotient interior symmetry. We introduce here an equivariant degree theory that incorporates these different forms of hidden symmetry based on lattice structures of synchrony subspaces. The result is a unified theory capable of treating synchrony-breaking Hopf bifurcation problems in coupled cell systems with various forms of hidden symmetries, which leads to full topological classifications of bifurcating branches arising from single bifurcation points under mutual influence of these hidden symmetries.

Hopf bifurcation of the fractional-order Hindmarsh–Rose neuron model with time-delay

The Hopf bifurcation problem of the fractional-order Hindmarsh–Rose neuron model with time-delay is investigated. The effects of time-delay to the bifurcation behavior of system is considered by linearizing the system at the equilibria method. With the bifurcation theory, the condition for the existence of Hopf bifurcation is obtained when considering time-delay τ as variable parameter. As illustrated by the numerical example, the proposed theoretical results work effectively and accurately.

Normal Forms for Partial Neutral Functional Differential Equations with Applications to Diffusive Lossless Transmission Line

A class of partial neutral functional differential equations are considered. For the linearized equation, the semigroup properties and formal adjoint theory are established. Based on these results, we develop two algorithms of normal form computation for the nonlinear equation, and then use them to study Hopf bifurcation problems of such equations. In particular, it is shown that the normal forms, derived from these two different approaches, for the Hopf bifurcation are exactly the same. As an illustration, the diffusive lossless transmission line equation where a Hopf singularity occurs is studied.

Hopf bifurcation and the centers on center manifold for a class of three‐dimensional Circuit system

In this paper, Hopf bifurcation and center problem for a generic three‐dimensional Chua's circuit system are studied. Applying the formal series method of computing singular point quantities to investigate the two cases of the generic circuit system, we find necessary conditions for the existence of centers on a local center manifold for the systems, then Darboux method is applied to show the sufficiency. Further, we determine the maximum number of limit cycles that can bifurcate from the corresponding equilibrium via Hopf bifurcation.

Influence of Time Delay on Bifurcation in Fractional Order BAM Neural Networks With Four Delays

Over the past several decades, numerous scholars have studied the stability and Hopf bifurcation problem of integer-order delayed neural networks. However, the fruits about the stability and Hopf bifurcation for fractional-order delayed neural networks are very scarce. In this paper, we will consider the stability and the existence of Hopf bifurcation of fractional-order bidirectional associative memory (BAM) neural networks with four delays. A set of sufficient criteria to ensure the stability and the existence of Hopf bifurcation for the fractional-order BAM neural networks with four delays are established by choosing the sum of two different delays as a bifurcation parameter. This paper manifests that the delay has an important influence on the stability and Hopf bifurcation of involved networks. An example is displayed to test the rationality of the derived theoretical findings. The derived results of this paper are new and play a key role in optimizing networks and improving human life.

On the non-autonomous Hopf bifurcation problem: systems with rapidly varying coefficients.

On the non-autonomous Hopf bifurcation problem: systems with rapidly varying coefficients.

On the nonautonomous Hopf bifurcation problem

Under well-known conditions, a one-parameter family of two-dimensional, autonomous ordinary differential equations admits a supercritical\break Andronov-Hopf bifurcation. Let such a family be perturbed by a non-autonomous term. We analyze the sense in which and some conditions under which the Andronov-Hopf pattern persists under such a perturbation.

Stability and multi-parametric Hopf bifurcation analyses of viral infection model with time delay

Ever since HIV was first diagnosed in human, a great number of scientific works have been undertaken to explore the biological mechanisms involved in the infection and progression of the disease. This paper deals with stability and bifurcation analyses of mathematical model that represents the dynamics of HIV infection of thymus. The existence and stability of the equilibria are investigated. The model is described by a system of delay differential equations with logistic growth term, cure rate and discrete type of time delay. Choosing the time delay as a bifurcation parameter, the analysis is mainly focused on the Hopf bifurcation problem to predict the existence of a limit cycle bifurcating from the infected steady state. Further, using center manifold theory and normal form method we derive explicit formulae to determine the stability and direction of the limit cycles. Moreover the mitosis rate r also plays a vital role in the model, so we fix it as second bifurcation parameter in the incidence of viral infection. Our analysis shows that, while both the bifurcation parameters can destabilize the equilibrium E* and cause limit cycles. Numerical simulations are performed to investigate the qualitative behaviors of the inherent model.

Nonlinear interactions between the two wakes behind a pair of square cylinders

AbstractNonlinear interactions between the two wakes behind a pair of square cylinders, which are placed side by side in a uniform flow, are investigated by the linear and weakly nonlinear stability analyses and numerical simulations. It is known from the linear stability analysis that the flow past a pair of cylinders becomes unstable to a symmetric or an antisymmetric mode of disturbance, depending on the gap ratio, the ratio of the gap distance between the two cylinders to the cylinder diameter. The antisymmetric mode gives the critical condition for smaller gap ratios than a threshold value, and for larger gap ratios the symmetric mode becomes the most unstable. We focus on the flow pattern arising through the nonlinear interactions of the two modes of disturbance for gap ratios around the threshold value when both modes are growing. We derive a couple of amplitude equations for the two modes to properly describe the nonlinear interaction between them by applying the weakly nonlinear stability theory. The amplitude equations are shown to have three equilibrium solutions except the null solution such as a mixed-mode solution, symmetric and antisymmetric single-mode solutions. Examination of the stability of each equilibrium solution leads to a conclusion that the mixed-mode solution exchange its stability with both the symmetric and the antisymmetric single-mode solutions simultaneously. In the case where the mixed-mode solution is stable, both the symmetric and antisymmetric modes have finite amplitudes, and the resultant flow has an asymmetric flow pattern comprising of finite amplitudes of the two modes of disturbance superposed on the steady symmetric flow. While in the case where both the single-mode solutions are stable, either of the symmetric- and antisymmetric-mode solutions survives, overwhelming the other. Then, if the symmetric mode attains at an equilibrium finite amplitude and the antisymmetric mode vanishes, the resultant flow is symmetric, and if the antisymmetric mode survives and the symmetric mode decays out, the flow becomes asymmetric with the antisymmetric mode of disturbance superposed on the steady symmetric flow. Thus, the flow appearing due to instability differs depending on the initial condition, not uniquely determined, when both single-mode solutions are stable. We numerically delineated the region in the parameter space of the gap ratio and the Reynolds number where the mixed-mode solution is stable. The theoretical results obtained from the weakly nonlinear stability analyses are confirmed by numerical simulations. The conclusion derived from the stability analysis of the equilibrium solutions of the amplitude equations is widely applicable also to other double Hopf bifurcation problems.

Multiple limit cycles and centers on center manifolds for Lorenz system

For Lorenz system we investigate multiple Hopf bifurcation and center-focus problem of its equilibria. By applying the method of symbolic computation, we obtain the first three singular point quantities. It is proven that Lorenz system can generate 3 small limit cycles from each of the two symmetric equilibria. Furthermore, the center conditions are found and as weak foci the highest order is proved to be the third, thus we obtain at most 6 small limit cycles from the symmetric equilibria via Hopf bifurcation. At the same time, we realize also that though the same for the related three-dimensional chaotic systems, Lorenz system differs in Hopf bifurcation greatly from the Chen system and Lü system.