Double Hopf Bifurcation Research Articles

Spatially inhomogeneous periodic patterns induced by distributed memory in the memory-based single population model

In this paper, we investigate the influence of the distributed delay on the stability and spatiotemporal dynamics in the memory-based diffusion single population model. Under the assumption that the positive equilibrium of the corresponding ordinary differential equation is locally asymptotically stable, it has been shown that the weak kernel does not affect the stability of this positive constant steady state, but for the strong kernel, the combined effect of the diffusion rate of the memory diffusion and the distributed delay can affect the stability and lead to the rich dynamics. For the strong kernel, the theoretical conditions for the stability, Hopf bifurcation and double Hopf bifurcation are explicitly determined.

Spatiotemporal pattern formation in a pollen tube model with nonlocal effect and time delay

The dynamics of a pollen tube model with nonlocal effect and time delay is investigated in this paper. Compared with the model without delay, a double Hopf bifurcation due to the interaction of homogeneous and nonhomogeneous Hopf bifurcations can occur and quasi periodic patterns can be observed. Besides, the interaction of Turing bifurcation and spatially nonhomogeneous Hopf bifurcation will lead to some new spatiotemporal patterns.

Extremely rich dynamics of coupled heterogeneous neurons through a Josephson junction synapse

Artificial neural networks are generally used to emulate some biological activities of the brain. Those neurons in the network are connected to others through synapses. As a result, several works have been devoted to the design of various artificial synapses, including the Josephson Junction Synapse (JJS). In this contribution, a model of the Hindmarsh–Rose neuron coupled with a FitzHugh–Nagumo neuron through a JJS is considered. That JJS enables us to simulate the effects of the magnetic field by providing additional phase error between the junctions. The Hamilton function related to the energy released by the coupled neurons during the transition between electrical activities is determined using the Helmholtz theorem. The equilibrium points of the proposed model are investigated, and their stability justifies the self-excited dynamics of the model. During studies, sets of phenomena such as Hopf bifurcation, double Hopf bifurcation, periodic spikings, periodic and chaotic burstings, (resp. firings) are found. More importantly, several coexisting activities are found in the coupled neurons as well as hyperchaotic firing activities rarely reported in such classes of neurons. Finally, PSpice simulations are used to further support results obtained from the mathematical model of coupled neurons through a JJS.

Stability Switches and Chaos in a Diffusive Toxic Phytoplankton–Zooplankton Model with Delay

To explore the effect of toxins produced by phytoplankton in algal blooms, a diffusive toxic phytoplankton–zooplankton system with delay is studied through detailed bifurcation analysis. The local stability of the positive equilibrium is studied by analyzing the characteristic equation. The critical values of delay at which Hopf bifurcations occur are obtained and ordered, and it is proved that the delay can induce stability switches, which means that the delay has a significant impact on both the formation and the termination of algal blooms. During the process of stability switching, double Hopf bifurcation arises. Taking the rate of toxin liberation and time delay as the bifurcation parameters, the normal form for double Hopf bifurcation is derived, from which the classification for dynamics and bifurcation sets near double Hopf bifurcation point is obtained. It is proved that the system has complex dynamics, such as periodic oscillations, quasi-periodic oscillations on two- or three-torus and even chaos. Numerical simulations are presented to support the theoretical results.

Complex dynamics in the improved Koren–Feingold cloud–rain system

In this paper, we study the improved aerosol–cloud–precipitation system. Since in our previous article, we guessed that the delayed term y(t−τ) in the denominator tends to cause singularity. So we remove the denominator by time transformation, the original cloud–rain model is transformed into a simpler polynomial system under certain conditions, namely, the improved Koren–Feingold cloud–rain system. Fortunately, the new model meets our expectations. The main work of the present paper is to study the mechanism of cloud and rain interaction by double Hopf bifurcation analysis via the methods of multiple scales. We find some complicated dynamical behaviors of the system via analytical method, such as stable quasi-periodic solutions and chaos. The numerical simulations agree well with those of theoretical analysis, which shows that we are improving the model in the right direction, meanwhile perhaps it suggests that the MMS employed in this paper provides us an effective and accurate method of analyzing double Hopf bifurcation. The above complex dynamical phenomena are of great significance for the researchers to understand the mechanism of aerosol–cloud–precipitation system. And it is also significant to ecosystem and climate control.

Double Hopf Bifurcation Analysis in the Memory-based Diffusion System

In this paper, we derive the algorithm for calculating the normal form of the double Hopf bifurcation that appears in a memory-based diffusion system via taking memory-based diffusion coefficient and the memory delay as the perturbation parameters. Using the obtained theoretical results, we study the dynamical classification near the double Hopf bifurcation point in a predator-prey system with Holling type II functional response. We show the existence of different kinds of stable spatially inhomogeneous periodic solutions, the transition from one kind to the other as well as the coexistence of two types of periodic solutions with different spatial profiles by varying the memory-based diffusion coefficient and the memory delay.

Complicated Dynamics of a Delayed Photonic Reservoir Computing System

In this paper, we consider the complicated dynamics of a delay-based photonic reservoir computing system. Since conventional computer architectures are approaching their limit, it is imperative to find new, efficient and fast ways of data processing. Photonic reservoir computing (RC) is a promising way which combines the computational capabilities of recurrent neural networks with high processing speed and energy efficiency of photonics. As the RC system is very promising, we analyze its dynamics so that we can make better use of it. In this paper, we mainly focus on its double Hopf bifurcation. We first analyze the existence of double Hopf bifurcation points. Then we use DDE-BIFTOOL to draw the bifurcation diagrams with respect to two bifurcation parameters, i.e. feedback strength [Formula: see text] and delay [Formula: see text], and give a clear picture of the double Hopf bifurcation points of the system. These figures show stability switches and the existence of double Hopf bifurcation points. Finally, we employ the method of multiple scales to obtain their normal forms, use the method of normal form to unfold and classify their local dynamics. The classification and unfolding of these double Hopf bifurcation points are obtained. Three types of double Hopf bifurcations are found. We verify the results by numerical simulations and find its complicated behavioral dynamics. For example, there exist stable equilibrium, stable periodic and quasi-periodic solutions in distinct regions. The discovered rich dynamical phenomena can help us to choose suitable values of parameters to achieve excellent performance of RC.

Normal forms of double Hopf bifurcation for a reaction-diffusion system with delay and nonlocal spatial average and applications

In this paper, we are concerned with a reaction-diffusion model incorporating delay and nonlocal effects. The normal form of double Hopf bifurcation is derived. The diffusive model of pollen tube tip growth is discussed and numerical simulations show that spatially homogeneous and inhomogeneous periodic solutions can be both stable or connected by a heteroclinic orbit under certain conditions. In addition, the diffusive Lotka-Volterra model with delay and nonlocality is considered and spatially inhomogeneous quasi-periodic solution is obtained.

Transitions of zonal flows in a two-layer quasi-geostrophic ocean model

We consider a 2-layer quasi-geostrophic ocean model where the upper layer is forced by a steady Kolmogorov wind stress in a periodic channel domain, which allows to mathematically study the nonlinear development of the resulting flow. The model supports a steady parallel shear flow as a response to the wind stress. As the maximal velocity of the shear flow (equivalently the maximal amplitude of the wind forcing) exceeds a critical threshold, the zonal jet destabilizes due to baroclinic instability and we numerically demonstrate that a first transition occurs. We obtain reduced equations of the system using the formalism of dynamic transition theory and establish two scenarios which completely describe this first transition. The generic scenario is that a conjugate pair of modes loses stability and a Hopf bifurcation occurs as a result. Under an appropriate set of parameters describing related midlatitude oceanic flows, we show that this first transition is continuous: a supercritical Hopf bifurcation occurs and a stable time periodic solution bifurcates. We also investigate the case of double Hopf bifurcations which occur when four modes of the linear stability problem simultaneously destabilize the zonal jet. In this case, we prove that, in the relevant parameter regime, the flow exhibits a continuous transition accompanied by a bifurcated attractor homeomorphic to \(S^3\). The topological structure of this attractor is analyzed in detail and is shown to depend on the system parameters. In particular, this attractor contains (stable or unstable) time-periodic solutions and a quasi-periodic solution.

Spatially nonhomogeneous periodic patterns in a delayed predator–prey model with predator-taxis diffusion

In this paper, we study the effect of time delay on the dynamics of a diffusive predator–prey model with predator-taxis under Neumann boundary condition. The joint effect of predator-taxis and delay can lead to spatially nonhomogeneous periodic patterns via spatially nonhomogeneous Hopf bifurcations. It is also shown that there exist double Hopf bifurcations due to the interaction either between homogeneous and nonhomogeneous or between nonhomogeneous Hopf bifurcations with different modes, which cannot occur for the system with only either delay or predator-taxis diffusion.

Resonant double Hopf bifurcation in a diffusive Ginzburg–Landau model with delayed feedback

Resonant double Hopf bifurcation in a diffusive Ginzburg–Landau model with delayed feedback

Synchronization, symmetry and rotating periodic solutions in oscillators with Huygens’ coupling

Synchronization is widespread in complex systems made by multiple individuals. And rotating waves are usually used to describe the synchronization in coupled oscillators systems. Many systems are observed to produce patterns of rotating waves, but it is difficult to predict the type of them or understand the conditions under which they form. Here we present the concept of the rotating-periodic solution and develop new bifurcation theory about rotating-periodic solutions to show the mechanisms for the existence of various rotating waves.We use a Huygens’ coupling model to study this topic. All kinds of rotating waves like in-phase, anti-phase, periodic, cluster synchronous are different types of rotating-periodic solutions with different rotating matrices. In the symmetric Huygens model, the rotating matrices Q satisfying the system rotating invariance just form a special symmetry group. By using the conjugate classes of symmetry groups and the diagonalization method, we can obtain all kinds of rotating waves of finite identical oscillator systems. We calculate all possible rotating waves in three and four identical oscillator systems, and get a general result that the phase difference of the oscillators in a rotating wave can only be kπn (n is the number of the oscillators, 0≤k≤n, k is even). Furthermore, in order to obtain the existence of rotating waves, we establish a new rotating periodic solution Hopf bifurcation theory. This particular bifurcation theory can be used to replace Hopf bifurcation, double Hopf bifurcation and other more complex periodic solution bifurcation in a unified way to study the rotating waves.

Bubbling, Bistable Limit Cycles and Quasi-Periodic Oscillations in Queues with Delayed Information

We consider a model describing the length of two queues that incorporates customer choice behavior based on delayed queue length information. The symmetric case, where the values of the time-delay parameter in each queue are the same, was recently studied. It was shown that under some conditions, the stable equilibrium solution becomes unstable as the common time delay passes a threshold value. This one-time stability switch occurs only at a symmetry-breaking Hopf bifurcation where a family of stable asynchronous limit-cycle solutions arise. In this paper, we examine the non-symmetric case, wherein the values of the time-delay parameter in each queue are different. We show that, in contrast to the symmetric case, the non-symmetric case allows bubbling, multiple stability switches and coexistence of distinct families of stable limit cycles. An investigation of the dynamical behavior of the non-symmetric system in a neighborhood of a double-Hopf bifurcation using numerical continuation explains the occurrence of the bistable limit cycles. Quasi-periodic oscillations were also observed due to the presence of torus bifurcations near the double-Hopf bifurcation. These identifications of the underlying mechanisms that cause unwanted oscillations in the system give a better understanding of the effects of providing delayed information and consequently help in better management of queues.

Persistence and Bifurcation Analysis of a Plankton Ecosystem with Cross-Diffusion and Double Delays

To investigate the joint effects of diffusion and multiple delays, a toxin producing phytoplankton–zooplankton system with cross-diffusion is studied and two-parameter bifurcation analysis is carried out. Firstly, for the delay-free system, the stability of equilibria, existence of solutions and priori bound are given. The globally asymptotic stability of equilibrium is investigated for the two situations with or without the diffusion terms. The linear stability analyses show that the cross-diffusion is a key mechanism for Turing instability to occur and Turing instability interval becomes larger with increasing cross diffusion coefficient. Secondly, the focus is on the effects of delays on the system including persistence, stability and Hopf bifurcation. The sufficient conditions for the diffusion system with delays to remain persistent are given. Under certain conditions, the boundary equilibrium is globally asymptotically stable. By analyzing the distribution of roots of the characteristic equation, using the crossing curve methods, the stable changes of positive equilibrium in two-delay plane are obtained. Explicit algorithm is derived for the properties of the bifurcating periodic solutions by normal form theory and the center manifold theorem for partial functional differential equations. Finally, numerical simulations confirm the correctness of theoretical analyses and find that double Hopf bifurcation may occur.

Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcations

<p style='text-indent:20px;'>We investigate spatiotemporal patterns near the Turing-Hopf and double Hopf bifurcations in a diffusive Holling-Tanner model on a one- dimensional spatial domain. Local and global stability of the positive constant steady state for the non-delayed system is studied. Introducing the generation time delay in prey growth, we discuss the existence of Turing-Hopf and double Hopf bifurcations and give the explicit dynamical classification near these bifurcation points. Finally, we obtain the complicated dynamics, including periodic oscillations, quasi-periodic oscillations on a three-dimensional torus, the coexistence of two stable nonconstant steady states, the coexistence of two spatially inhomogeneous periodic solutions, and strange attractors.</p>

An analysis of two degenerate double-Hopf bifurcations

<abstract><p>The generic double-Hopf bifurcation is presented in detail in literature in textbooks like references. In this paper we complete the study of the double-Hopf bifurcation with two degenerate (or nongeneric) cases. In each case one of the generic conditions is not satisfied. The normal form and the corresponding bifurcation diagrams in each case are obtained. New possibilities of behavior which do not appear in the generic case were found.</p></abstract>

Nonlinear dynamics of a SIRI model incorporating the impact of information and saturated treatment with optimal control.

In this article, we propose and analyze an infectious disease model with reinfection and investigate disease dynamics by incorporating saturated treatment and information effect. In the model, we consider the case where an individual’s immunity deteriorates and they become infected again after recovering. According to our findings, multiple steady states and backward bifurcation may occur as a result of treatment saturation. Further, if treatment is available for all, the disease will be eradicated provided {mathcal {R}}_0<1; however, because limited medical resources caused saturation in treatment, the disease may persist even if {mathcal {R}}_0<1. The global stability of the unique endemic steady state is established using a geometric approach. We also establish certain conditions on the transmission rate for the occurrence of periodic oscillations in the model system. Among nonlinear dynamics, we show supercritical Hopf bifurcation, bi-stability, backward Hopf bifurcation, and double Hopf bifurcation. To illustrate and validate our theoretical results, we present numerical examples. We found that when disease information coverage is high, infection cases fall considerably, and the disease persists when the reinfection rate is high. We then extend our model by incorporating two time-dependent controls, namely inhibitory interventions and treatment. Using Pontryagin’s maximum principle, we prove the existence of optimal control paths and find the optimal pair of controls. According to our numerical simulations, the second control is less effective than the first. Furthermore, while implementing a single intervention at a time may be effective, combining both interventions is most effective in reducing disease burden and cost.

Double Hopf bifurcation and stability of Koren–Feingold cloud–rain system with rain production delay

Double Hopf bifurcation and stability of Koren–Feingold cloud–rain system with rain production delay

Periodic, Quasi-Periodic and Phase-Locked Oscillations and Stability in the Fiscal Dynamical Model with Tax Collection and Decision-Making Delays

In this paper, we consider the complex dynamics of a fiscal dynamical model, which was improved from Wolfstetter classical growth cycle model by Sportelli et al. The main work of the present paper is to study the impact of fiscal policy delays on the national income adjustment processes using a dynamical method, such as double Hopf bifurcation analysis. We first use DDE-BIFTOOL to find the double Hopf bifurcation points of the system, and draw the bifurcation diagrams with two bifurcation parameters, i.e. the tax collection delay [Formula: see text] and the public expenditure decision-making delay [Formula: see text]. Then we employ the method of multiple scales to obtain two amplitude equations. By analyzing these amplitude equations, we derive the classification and unfolding of these double Hopf bifurcation points. And three types of double Hopf bifurcations are found. Finally, we verify the results by numerical simulations. We find complex dynamic behaviors of the system via the analytical method, such as stable equilibrium, stable periodic, quasi-periodic and phase-locked solutions in respective regions. The dynamical phenomena can help policy makers to choose a proper range of the delays so that they could effectively formulate fiscal policies to stabilize the economy.

Dynamics in a Predator–Prey Model with Cooperative Hunting and Allee Effect

This paper deals with a diffusive predator–prey model with two delays. First, we consider the local bifurcation and global dynamical behavior of the kinetic system, which is a predator–prey model with cooperative hunting and Allee effect. For the model with weak cooperation, we prove the existence of limit cycle, and a loop of heteroclinic orbits connecting two equilibria at a threshold of conversion rate p=p#, by investigating stable and unstable manifolds of saddles. When p&gt;p#, both species go extinct, and when p&lt;p#, there is a separatrix. The species with initial population above the separatrix finally become extinct, and the species with initial population below it can be coexisting, oscillating sustainably, or surviving of the prey only. In the case with strong cooperation, we exhibit the complex dynamics of system, including limit cycle, loop of heteroclinic orbits among three equilibria, and homoclinic cycle with the aid of theoretical analysis or numerical simulation. There may be three stable states coexisting: extinction state, coexistence or sustained oscillation, and the survival of the prey only, and the attraction basin of each state is obtained in the phase plane. Moreover, we find diffusion may induce Turing instability and Turing–Hopf bifurcation, leaving the system with spatially inhomogeneous distribution of the species, coexistence of two different spatial-temporal oscillations. Finally, we consider Hopf and double Hopf bifurcations of the diffusive system induced by two delays: mature delay of the prey and gestation delay of the predator. Normal form analysis indicates that two spatially homogeneous periodic oscillations may coexist by increasing both delays.