Narrow Results

In this paper, the effect of time delay is investigated on the system dynamics of a glucose-insulin model incorporating obesity. Treating the time delay as a bifurcation parameter, the stability switching on the positive equilibrium with global bifurcation is obtained. With the method of normal forms and central manifold theory, the direction and stability of limit cycles arising from Hopf bifurcation are analyzed. Using the method of multiple time scales, the normal form associated with non-resonant double Hopf bifurcation is derived. Moreover, the bifurcations are classified in the two-dimensional parameter plane near the critical point, and numerical simulations are presented to demonstrate the applicability of the theoretical results. Our results indicate that time delay in the glucose-insulin model can not only induce Hopf bifurcation and double Hopf bifurcation but also generate multiple stable periodic solutions. These results may help to understand the dynamical mechanism of glucose-insulin metabolic regulation systems, and to design control strategies for regulating and mitigating the occurrence of related diseases.

In this paper, a four-neuron delayed bidirectional associative memory (BAM) model with inertia is considered. Weak resonant double Hopf bifurcations are completely analyzed in the parameter space of the coupling weight and the coupling delay by the perturbation-incremental scheme (PIS). Numerical simulations are given for justifying the theoretical results. To the best of our knowledge, the paper is the first one to introduce inertia to a four-neuron delayed system and clarify the relationship between system parameters and dynamical behaviors.

This note presents closed-form formulas for determining the critical points of general n-dimensional differential equations. The formulas do not require commutating the eigenvalues of the Jacobian of a system. Based on the Hurwitz criterion, explicit necessary and sufficient conditions are obtained. Particular attention is focused on Hopf and double Hopf bifurcations. A model of induction machine is presented to show the application of the results.

In this work, the dynamics of a coupled electric circuit is studied. Several bifurcation diagrams associated with the truncated normal form of the Hopf-Hopf bifurcation are presented. The bifurcation curves are obtained by numerical continuation methods. The existence of quasi-periodic solutions with two (2D torus) and three (3D torus) frequency components is shown. These, in certain way, are close (or have a tendency to end up) to chaotic motion. Furthermore, two fold-flip bifurcations are detected in the vicinity of the Hopf-Hopf bifurcation, and are classified correspondingly. The analysis is completed with time simulations, the continuation of several limit cycle bifurcations and the indication of resonance points.

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.

In this paper, we study dynamics in delayed van der Pol–Duffing equation, with particular attention focused on nonresonant double Hopf bifurcation. Both multiple time scales and center manifold reduction methods are applied to obtain the normal forms near a double Hopf critical point. A comparison between these two methods is given to show their equivalence. Bifurcations are classified in a two-dimensional parameter space near the critical point. Numerical simulations are presented to demonstrate the applicability of the theoretical results.

In this paper, a general and systematic scheme is provided to research strong and weak resonances derived from delay-induced various double Hopf bifurcations in delayed differential systems. The method of multiple scales is extended to obtain a common complex amplitude equation when the double Hopf bifurcation with frequency ratio k₁:k₂ occurs in the systems under consideration. By analyzing the complex amplitude equation, we give the conditions of the strong and weak resonances respectively in the analytical expressions. The weak resonances correspond to the codimension-two double Hopf bifurcations since the amplitudes and the phases may be decoupled, but the strong resonances to the codimension-three double Hopf bifurcations in the system. It is seen that the weak resonances happen in the system even for a lower-order ratio, i.e. k₁ + k₂ ≤ 4. As applications, two examples are displayed. Three cases of the delay-induced resonance with 1:2, 1:3, 1:5 and are discussed in detail and the corresponding normals are represented. Thus, the relative dynamical behaviors can be easily classified in the physical parameter space in terms of nonlinear dynamics. The results show the provided conditions may be used to determine that a resonance is strong or weak.

A predator–prey system with neutral delay is investigated from the viewpoint of bifurcation analysis on neutral delay differential equations. Stability and Hopf bifurcation of the inner equilibrium are given, by which we show how the neutral terms affect the dynamical behavior of the prey and the predator. To give more detailed information on the periodic oscillations, the direction and stability of Hopf bifurcation are studied by using the normal form theory of neutral equation. We find neutral delay makes the predator–prey system more complicated and usually induces stability switches or double Hopf bifurcations. Near the double Hopf bifurcation we give the detailed bifurcation set by calculating the universal unfoldings. It is shown that the population of prey or predator may exhibit transient quasiperiodic oscillations driven by the neutral delay. Finally, we carry out several groups of illustrations.

In this paper, we investigate the codimension-two double Hopf bifurcation in delay-coupled van der Pol–Duffing oscillators. By using normal form theory of delay differential equations, the normal form associated with the codimension-two double Hopf bifurcation is calculated. Choosing appropriate values of the coupling strength and the delay can result in nonresonance and weak resonance double Hopf bifurcations. The dynamical classification near these bifurcation points can be explicitly determined by the corresponding normal form. Periodic, quasi-periodic solutions and torus are found near the bifurcation point. The numerical simulations are employed to support the theoretical results.

In this paper, we study the dynamics in delayed nonlinear financial system, with particular attention focused on Hopf and double Hopf bifurcations. Firstly, we identify the critical values for stability switches, Hopf and double Hopf bifurcations. We show how the parameters affect the dynamical behavior of the system. Secondly, the normal forms near the Hopf and double Hopf bifurcations, as well as the classifications of local dynamics are analyzed. These bifurcations lead a chaotic system to be stable states, such as the coexistence of a pair of stable equilibria or a pair of stable periodic oscillations, and then chaos disappears. Numerical simulations are presented to verify the analytical predictions. Furthermore, detailed numerical analysis using MATLAB extends the local bifurcation analysis to a global picture, namely, a family of stable periodic solutions exist in a large region of delay and “chaos switchover” phenomenon appears. Therefore, in accordance with the above theoretical analysis, reasonable parameters can be designed in order to achieve various applications.

This paper investigates a toxic phytoplankton–zooplankton model with Michaelis–Menten type phytoplankton harvesting. The model has rich dynamical behaviors. It undergoes transcritical, saddle-node, fold, Hopf, fold-Hopf and double Hopf bifurcation, when the parameters change and go through some of the critical values, the dynamical properties of the system will change also, such as the stability, equilibrium points and the periodic orbit. We first study the stability of the equilibria, and analyze the critical conditions for the above bifurcations at each equilibrium. In addition, the stability and direction of local Hopf bifurcations, and the completion bifurcation set by calculating the universal unfoldings near the double Hopf bifurcation point are given by the normal form theory and center manifold theorem. We obtained that the stable coexistent equilibrium point and stable periodic orbit alternate regularly when the digestion time delay is within some finite value. That is, we derived the pattern for the occurrence, and disappearance of a stable periodic orbit. Furthermore, we calculated the approximation expression of the critical bifurcation curve using the digestion time delay and the harvesting rate as parameters, and determined a large range in terms of the harvesting rate for the phytoplankton and zooplankton to coexist in a long term.

In this paper, we discuss the double Hopf bifurcation and the existence of quasi-periodic invariant tori in a generalized Gopalsamy neural network model. Regarding the connection weight and the delay as bifurcation parameters of the double Hopf bifurcation, we derive the normal form up to the fifth order near the critical point by using the center manifold theorem and the normal form method, and obtain sufficient conditions on the existence of invariant 2-tori for the truncated normal form. Moreover, we investigate the effect of higher-order terms on these 2-tori by a KAM theorem. It is proved that in a sufficiently small neighborhood of the bifurcation point, the neural network model has quasi-periodic invariant 2-tori for most of the parameter set where its truncated normal form possesses invariant 2-tori. We give a numerical example to verify the conditions on all results in remarks.

This paper focuses on the influence of two scales in the frequency domain on the behaviors of a typical dynamical system with a double Hopf bifurcation. By introducing an external periodic excitation to the normal form of the vector field with double Hopf bifurcation at the origin and taking the exciting frequency far less than the natural frequency, a theoretical model with two scales in the frequency domain is established. Regarding the whole exciting term as a slow-varying parameter leads to a generalized autonomous system, in which the equilibrium branches and their bifurcations with the variation of the slow-varying parameter can be derived. With the increase of the exciting amplitude, different types of bifurcations may be involved in the generalized autonomous system, resulting in several qualitatively different forms of bursting attractors, the mechanism of which is presented by overlapping the transformed phase portraits and the bifurcations of the equilibrium branches. It is found that the single mode 2D torus may evolve to the bursting attractors with mixed modes, in which the trajectory alternates between the single mode oscillations and the mixed mode oscillations. Furthermore, the transitions between the quiescent states and the spiking states may not occur exactly at the bifurcation points because of the slow passage effect, while Hopf bifurcations may cause different forms of repetitive spiking oscillations.

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 $\eta$ and delay $\tau$, 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.

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.

A Stuart–Landau system under delay feedback control with the nonlinear delay-dependent parameter e^-pτ is investigated. A geometrical demonstration method combined with theoretical analysis is developed so as to effectively solve the characteristic equation. Multi-stable regions are separated from unstable regions by allocations of Hopf bifurcation curves in (p,τ) plane. Some weak resonant and non-resonant oscillation phenomena induced by double Hopf bifurcation are discovered. The normal form for double Hopf bifurcation is deduced. The local dynamical behavior near double Hopf bifurcation points are also clarified in detail by using the center manifold method. Some states of two coexisting stable periodic solutions are verified, and some torus-broken procedures are also traced.

In this paper, we study dynamics in a predator–prey model with delay, in which predator can be infected, with particular attention focused on nonresonant double Hopf bifurcation. By using center manifold reduction methods, we obtain the equivalent normal forms near a double Hopf critical point in this system. Moreover, bifurcations are classified in a two-dimensional parameter space near the critical point. Numerical simulations are presented to demonstrate the applicability of the theoretical results.

A nonlinear infectious disease model with information-influenced vaccination behavior and contact patterns is proposed in this paper, and the impact of information related to disease prevalence on increasing vaccination coverage and reducing disease incidence during the outbreak is considered. First, we perform the analysis for the existence of equilibria and the stability properties of the proposed model. In particular, the geometric approach is used to obtain the sufficient condition which guarantees the global asymptotic stability of the unique endemic equilibrium $E_e$ when the basic reproduction number $R_0>1$. Second, mathematical derivation combined with numerical simulation shows the existence of the double Hopf bifurcation around $E_e$. Third, based on the numerical results, it is shown that the information coverage and the average information delay may lead to more complex dynamical behaviors. Finally, the optimal control problem is established with information-influenced vaccination and treatment as control variables. The corresponding optimal paths are obtained analytically by using Pontryagin’s maximum principle, and the applicability and validity of virous intervention strategies for the proposed controls are presented by numerical experiments.