Hopf-Hopf Bifurcation Research Articles

Effect of Hopf-Hopf bifurcation on the post-flutter behavior of a three-degree-of-freedom airfoil

The post-flutter dynamics of a three-degree-of-freedom nonlinear airfoil with unsteady aerodynamics are investigated based on the Hopf-Hopf bifurcation theory. Many prior works have relied on Hopf bifurcation theory to predict flutter behavior in airfoil systems. Although this approach facilitates flutter prediction and characterization of post-flutter behavior in numerous scenarios, it may be invalid in specific instances, such as when the Hopf bifurcation of the system degenerates. Therefore, this study focuses on a classical degenerate case of Hopf bifurcation in airfoil systems, specifically the Hopf-Hopf bifurcation. We show that the system undergoes various Hopf-Hopf bifurcations under specific parameter conditions as the center of gravity shifts. The local dynamics near the Hopf-Hopf bifurcation points are represented, including quasiperiodic oscillations on a three-dimensional torus. The airfoil begins to oscillate quasiperiodically after the airflow speed crosses specific Hopf-Hopf bifurcation points. The study also uncovers complex quasiperiodic crises and quasiperiodic hysteresis loops, which have not been reported in previous studies of aeroelastic systems. Then, many singularities and bifurcation curves are obtained near the Hopf-Hopf bifurcation point by semiglobal unfolding. Furthermore, the influence of stall effects on the bifurcation structure of the system is represented. It is shown that the types of Hopf-Hopf bifurcations may vary with the changes of stall effects, influencing the system's semiglobal bifurcation structures consequently. For all Hopf-Hopf bifurcation scenarios, stall effects affect one of the Neimark-Sacker bifurcation curve structures unfolded from the Hopf-Hopf bifurcation point significantly, while the other Neimark-Sacker bifurcation curve experiences minimal impact from stall effects. Moreover, a large nonlinear stall coefficient will postpone the onset of quasiperiodic/chaotic oscillations.

Hopf-Hopf bifurcation, period [formula omitted] solutions, slow-fast phenomena, and chimera of an optoelectronic reservoir computing system with single delayed feedback loop

In this paper, we investigate the co-dimension two bifurcations and complicated dynamics of an optoelectronic reservoir computing (RC) system with single delayed feedback loop. We focuses primarily on its underlying system’s Hopf-Hopf bifurcation. Firstly, we apply DDE-BIFTOOL built in Matlab to sketch the bifurcation diagrams with respect to two bifurcation parameters, namely feedback strength β and time delay τ, and find the existence of the Hopf-Hopf bifurcation points. Then, using the multiple scales method, we obtain their normal forms, and using the normal form method, we unfold and classify their local dynamics. Then numerical simulations are conducted to verify these results. We discover rich dynamical behaviors of the system in specific regions. Besides, other complicated dynamics, such as fast-slow phenomena, Period n solutions, and chimera, are found in the system. All these rich dynamical phenomena can provide excellent performance potentially for this optoelectronic reservoir computing system with single delayed feedback loop.

Effect of motor suspension parameters on bifurcations for a nonlinear bogie system

This paper aims to investigate the effect of motor suspension parameters, specifically damping and stiffness, on the bifurcations of a bogie system. A motor bogie model with a nonlinear smooth equivalent conicity function is established. The study includes qualitative analyses of the stability and the Hopf bifurcation of the equilibrium, with the running speed as the single parameter. Furthermore, the generalised Hopf bifurcation and Hopf-Hopf bifurcation of the equilibrium, based on two motor suspension parameters, are also analysed. The paper delves into the codimension-1 and codimension-2 bifurcations of the limit cycles generated by the Hopf bifurcation, including Neimark–Sacker bifurcation, fold bifurcation, cusp bifurcation, 1:3 resonance, and 1:4 resonance. Analytical investigations reveal that the cusp bifurcation alters the type of fold bifurcation (subcritical or supercritical), and the fold bifurcation influences the stability and bifurcation direction of the limit cycle. Additionally, resonance occurs between the motor and the bogie frame. Since the subcritical (supercritical) Neimark–Sacker bifurcation produces an unstable (a stable) torus, the resonance points associated with the subcritical Neimark–Sacker bifurcation will lead to the instability of the motor bogie. The bifurcation analysis on the motor suspension parameters in this paper offers a theoretical reference for enhancing the stability of the motor bogie.

Hopf-Hopf bifurcation in a predator-prey model with nonlocal competition and refuge in prey

Hopf-Hopf bifurcation in a predator-prey model with nonlocal competition and refuge in prey

Spatiotemporal patterns of a delayed diffusive prey-predator model with prey-taxis

<p>This paper explored a delayed diffusive prey-predator model with prey-taxis involving the volume-filling mechanism subject to homogeneous Neumann boundary condition. To figure out the impact on the dynamic of the prey-predator model due to prey-taxis and time delay, we treated the prey-tactic coefficient $ \chi $ and time delay $ \tau $ as the bifurcating parameters and did stability and bifurcation analysis. It showed that the time delay will induce Hopf bifurcations in the absence of prey-taxis, and the bifurcation periodic solution at the first critical value of $ \tau $ was spatially homogeneous. Hopf bifurcations occurred in the model when the prey-taxis and time delay coexisted, and at the first critical value of $ \tau $, spatially homogeneous or nonhomogeneous periodic solutions might emerge. It was also discovered that the bifurcation curves will intersect, which implied that Hopf-Hopf bifurcations can occur. Finally, we did numerical simulations to validate our outcomes.</p>

Multi-stable and spatiotemporal staggered patterns in a predator-prey model with predator-taxis and delay.

The effects of predator-taxis and conversion time delay on formations of spatiotemporal patterns in a predator-prey model are explored. First, the well-posedness, which implies global existence of classical solutions, is proved. Then, we establish critical conditions for the destabilization of the coexistence equilibrium via Turing/Turing-Turing bifurcations by describing the first Turing bifurcation curve; we also theoretically predict possible bistable/multi-stable spatially heterogeneous patterns. Next, we demonstrate that the coexistence equilibrium can also be destabilized via Hopf, Hopf-Hopf and Turing-Hopf bifurcations; also possible stable/bistable spatially inhomogeneous staggered periodic patterns and bistable spatially inhomogeneous synchronous periodic patterns are theoretically predicted. Finally, numerical experiments also support theoretical predictions and partially extend them. In a word, theoretical analyses indicate that, on the one hand, strong predator-taxis can eliminate spatial patterns caused by self-diffusion; on the other hand, the joint effects of predator-taxis and conversion time delay can induce complex survival patterns, e.g., bistable spatially heterogeneous staggered/synchronous periodic patterns, thus diversifying populations' survival patterns.

Stability switches, periodic oscillations and global stability in an infectious disease model with multiple time delays.

A delay differential equation model of an infectious disease is considered and analyzed. In this model, the impact of information due to the presence of infection is considered explicitly. As information propagation is dependent on the prevalence of the disease, the delay in reporting the prevalence is an important factor. Further, the time lag in waning immunity related to protective measures (such as vaccination, self-protection, responsive behaviour etc.) is also accounted. Qualitative analysis of the equilibrium points of the model is executed and it is observed that when the basic reproduction number is less unity, the local stability of the disease free equilibrium (DFE) depends on the rate of immunity loss as well as on the time delay for the waning of immunity. If the delay in immunity loss is less than a threshold quantity, the DFE is stable, whereas, it loses its stability when the delay parameter crosses the threshold value. When, the basic reproduction number is greater than unity, the unique endemic equilibrium point is found locally stable irrespective of the delay effect under certain parametric conditions. Further, we have analyzed the model system for different scenarios of both delays (i.e., no delay, only one delay, and both delay present). Due to these delays, oscillatory nature of the population is obtained with the help of Hopf bifurcation analysis in each scenario. Moreover, at two different time delays (delay in information's propagation), the emergence of multiple stability switches is investigated for the model system which is termed as Hopf-Hopf (double) bifurcation. Also, the global stability of the endemic equilibrium point is established under some parametric conditions by constructing a suitable Lyapunov function irrespective of time lags. In order to support and explore qualitative results, exhaustive numerical experimentations are carried out which lead to important biological insights and also, these results are compared with existing results.

An Investigation of the Bifurcation Behavior of an F-18 Aircraft Model

Bifurcations of equilibria and limit cycles of an F-18 aircraft model have been investigated in this paper based on one or two continuation parameters. First, it is shown that the aircraft can be in a stable straight-and-level flight state by coordinating the elevator deflection and the engine thrust and fixing the other parameters. Second, the aircraft may be disturbed by perturbations out of the longitudinal plane and get into the lateral-directional motion mode by a branch point bifurcation. Third, when the straight flight state loses its stability, body axis angular rates will present periodic or quasi-periodic motion pattern by the Hopf or Neimark–Sacker bifurcation. Finally, bifurcation structure of limit cycles has been exhibited, including the generalized Hopf bifurcation, the Neimark–Sacker bifurcation, the Hopf-Hopf bifurcation and the fold-Neimark–Sacker bifurcation. Meanwhile corresponding bifurcation curves in two-parameter plane have been depicted with the help of numerical continuation techniques.

Bifurcation analysis in a diffusion mussel-algae interaction system with delays considering the half-saturation constant

In this paper, the kinetics of a class of delayed reaction-diffusion mussel-algae system under Neumann boundary conditions with the half-saturation constant is studied. The global existence and priori bounds as well as the existence conditions of positive equilibrium are obtained. The half-saturation constant affects the stability of the system and may result in Turing instability. When the half-saturation constant exceeds a certain critical value, the boundary equilibrium is globally asymptotically stable which means that the larger half-saturation constant forces the mussel population toward extinction. By analyzing the distribution of the roots of the characteristic equation with two delays, the stability conditions of the positive equilibrium in the parameter space are obtained. The stability of the positive equilibrium can be changed by steady-state bifurcation, Hopf bifurcation, Hopf-Hopf bifurcation or Hopf-steady state bifurcation, which can be verified by some numerical simulations. Among the parameters, the half-saturation constant and two delays drive the complexity of system dynamics.

Bogdanov-Takens bifurcation with $ Z_2 $ symmetry and spatiotemporal dynamics in diffusive Rosenzweig-MacArthur model involving nonlocal prey competition

<p style='text-indent:20px;'>A diffusive Rosenzweig-MacArthur model involving nonlocal prey competition is studied. Via considering joint effects of prey's carrying capacity and predator's diffusion rate, the first Turing (Hopf) bifurcation curve is precisely described, which can help to determine the parameter region where coexistence equilibrium is stable. Particularly, coexistence equilibrium can lose its stability through not only codimension one Turing (Hopf) bifurcation, but also codimension two Bogdanov-Takens, Turing-Hopf and Hopf-Hopf bifurcations, even codimension three Bogdanov-Takens-Hopf bifurcation, etc., thus the concept of Turing (Hopf) instability is extended to high codimension bifurcation instability, such as Bogdanov-Takens instability. To meticulously describe spatiotemporal patterns resulting from <inline-formula><tex-math id="M2">\begin{document}$ Z_2 $\end{document}</tex-math></inline-formula> symmetric Bogdanov-Takens bifurcation, the corresponding third-order normal form for partial functional differential equations (PFDEs) involving nonlocal interactions is derived, which is expressed concisely by original PFDEs' parameters, making it convenient to analyze effects of original parameters on dynamics and also to calculate normal form on computer. With the aid of these formulas, complex spatiotemporal patterns are theoretically predicted and numerically shown, including tri-stable nonuniform patterns with the shape of <inline-formula><tex-math id="M3">\begin{document}$ \cos \omega t\cos \frac{x}{l}- $\end{document}</tex-math></inline-formula>like or <inline-formula><tex-math id="M4">\begin{document}$ \cos \frac{x}{l}- $\end{document}</tex-math></inline-formula>like, which reflects the effects of nonlocal interactions, such as stabilizing spatiotemporal nonuniform patterns.</p>

Coexistence in a four-species food web model with general functional responses

The dynamics of an intraguild predation system is analyzed. Two shared resources, a middle predator and a top predator are considered in the model. Parameter conditions for coexistence in the intraguild system are given. The coexistence of species, through the existence of an equilibrium point or a stable limit cycle or an invariant torus is shown. Conditions to have Andronov-Hopf and Hopf-Hopf bifurcation are given. Then, limit sets and complex dynamical behaviors, can exhibit in the system. For a wide family of functional responses, these results are valid. Numerical simulations with varied functional responses are given and different limit sets are shown.

Nonlinear dynamics in tumor-immune system interaction models with delays

<p style='text-indent:20px;'>In this paper, we review some recent results on the nonlinear dynamics of delayed differential equation models describing the interaction between tumor cells and effector cells of the immune system, in which the delays represent times necessary for molecule production, proliferation, differentiation of cells, transport, etc. First we consider a tumor-immune system interaction model with a single delay and present results on the existence and local stability of equilibria as well as the existence of Hopf bifurcation in the model when the delay varies. Second we investigate a tumor-immune system interaction model with two delays and show that the model undergoes various possible bifurcations including Hopf, Bautin, Fold-Hopf (zero-Hopf), and Hopf-Hopf bifurcations. Finally we discuss a tumor-immune system interaction model with three delays and demonstrate that the model exhibits more complex behaviors including chaos. Numerical simulations are provided to illustrate the nonlinear dynamics of the delayed tumor-immune system interaction models. More interesting issues and questions on modeling and analyzing tumor-immune dynamics are given in the discussion section.

Hopf-Hopf bifurcation in the delayed nutrient-microorganism model

In view of time delay in the transport of nutrients, a delayed reaction-diffusion system with homogeneous Neumann boundary conditions is presented to understand the formation of the heterogeneous distribution of bacteria and nutrients in the sediment. With the effects of time delay and diffusion, the system will experience various dynamical behaviors, such as stability, the Turing instability, successive switches of stability of equilibria, the Hopf and the Hopf-Hopf bifurcations. To further understand the dynamics of the Hopf-Hopf bifurcation, the multiple time scale (MTS) technique is employed to derive the amplitude equations at this co-dimensional bifurcation point, and the dynamical classification near such bifurcation point is also identified by analyzing the obtained amplitude equations. Some numerical simulations are carried out to demonstrate the validity of the theoretical analysis.

Multiple bifurcations and coexistence in an inertial two-neuron system with multiple delays.

In this paper, we construct an inertial two-neuron system with multiple delays, which is described by three first-order delayed differential equations. The neural system presents dynamical coexistence with equilibria, periodic orbits, and even quasi-periodic behavior by employing multiple types of bifurcations. To this end, the pitchfork bifurcation of trivial equilibrium is analyzed firstly by using center manifold reduction and normal form method. The system presents different sequences of supercritical and subcritical pitchfork bifurcations. Further, the nontrivial equilibrium bifurcated from trivial equilibrium presents a secondary pitchfork bifurcation. The system exhibits stable coexistence of multiple equilibria. Using the pitchfork bifurcation curves, we divide the parameter plane into different regions, corresponding to different number of equilibria. To obtain the effect of time delays on system dynamical behaviors, we analyze equilibrium stability employing characteristic equation of the system. By the Hopf bifurcation, the system illustrates a periodic orbit near the trivial equilibrium. We give the stability regions in the delayed plane to illustrate stability switching. The neural system is illustrated to have Hopf-Hopf bifurcation points. The coexistence with two periodic orbits is presented near these bifurcation points. Finally, we present some mixed dynamical coexistence. The system has a stable coexistence with periodic orbit and equilibrium near the pitchfork-Hopf bifurcation point. Moreover, multiple frequencies of the system induce the presentation of quasi-periodic behavior. The system presents stable coexistence with two periodic orbits and one quasi-periodic behavior.

Switching to Nonhyperbolic Cycles from Codimension Two Bifurcations of Equilibria of Delay Differential Equations

In this paper, we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf, and transcritical-Hopf bifurcations in delay differential equat...

Chaotic behavior in the unfolding of Hopf-Bogdanov-Takens singularities

A discussion on local bifurcations of codimension one and two is presented for generic unfoldings of Hopf-Bogdanov-Takens singularities of codimension three. Among all identified bifurcations, we focus on Hopf-Zero and Hopf-Hopf bifurcations, since, in certain cases, they can explain the emergence of chaotic dynamics. Moreover, numerical simulations are provided to illustrate that strange attractors appear at least when the second order normal form of the unfolding is considered.

Hopf-Hopf bifurcation and chaotic attractors in a delayed diffusive predator-prey model with fear effect

We investigate a diffusive predator-prey model by incorporating the fear effect into prey population, since the fear of predators could visibly reduce the reproduction of prey. By introducing the mature delay as bifurcation parameter, we find this makes the predator-prey system more complicated and usually induces Hopf and Hopf-Hopf bifurcations. The formulas determining the properties of Hopf and Hopf-Hopf bifurcations by computing the normal form on the center manifold are given. Near the Hopf-Hopf bifurcation point we give the detailed bifurcation set by investigating the universal unfoldings. Moreover, we show the existence of quasi-periodic orbits on three-torus near a Hopf-Hopf bifurcation point, leading to a strange attractor when further varying the parameter. We also find the existence of Bautin bifurcation numerically, then simulate the coexistence of stable constant stationary solution and periodic solution near this Bautin bifurcation point.

Principles for the application of bifurcation theory for the systematic analysis of nuclear reactor stability, Part2: Application

This paper is regarded as a continuation of the paper “Principles for the application of bifurcation theory for the systematic analysis of nuclear reactor stability, Part1” with the intention to provide examples demonstrating the application of the bifurcation analysis method in the framework of reactor stability analysis. Hence, we continue with chapter 5 which is devoted to three examples: (1) two-phase flow stability analysis, (2) occurrence of a generalized Hopf bifurcation (GHB) during a real nuclear reactor stability test and (3) existence of a complex stability behaviour in an environment of a double Hopf bifurcation point (Hopf-Hopf bifurcation, HHB). The efficiency of the RAM-ROM method is demonstrated for an operating point of NPP Leibstadt for which a sufficient experimental and system code database is available. These three examples of system dynamics demonstrate the partly very complex stability behaviour of nonlinear systems which cannot be explained by the application of linear stability analysis methods such as the estimation of the decay ratio (as a linear stability indicator). The consequences of the found bifurcation types in examples 2 and 3 on the particular solution structure of the underlying dynamic system will be discussed by using their respective normal forms in order to provide the reader a more clear access to the complex system behaviour around these bifurcation points. In case of the Hopf-Hopf bifurcation, we only present a selected part of solutions in this paper and refer the reader to a future paper, where more details of this bifurcation type are summarized and consequences to the full system are interpreted.

Delayed information induces oscillations in a dynamical model for infectious disease

During disease outbreak, it has been observed that information about the disease prevalence induces the individual’s behavioral changes. This information is usually assumed to be generated by the density of infective individuals and active mass media. The delay in reporting of these infective individuals may have its impact on generated information. Hence, to study the impact of delay on information generation, and therefore on the disease dynamics, a delay differential equation model is proposed and analyzed. The dynamics of information with delay effect is also modeled by a separate rate equation. Model analysis is performed and a unique infected equilibrium is obtained when the basic reproduction number ([Formula: see text]) is greater than one, whereas the disease free equilibrium always exists. When [Formula: see text], the disease free equilibrium is found to be locally stable independent of delay effect. The unique infected equilibrium is found to be locally stable till delay reaches a threshold value. The global stability of the unique infected equilibrium is also established under some parametric conditions by constructing a suitable Lyapunov function. The occurrence of Hopf bifurcation is observed when the delay in information crosses the threshold value. Analytically, the direction and stability of bifurcating periodic solutions is established. Further, we observed the occurrence of Hopf-Hopf bifurcation at two different delays. At first delay threshold, the endemic equilibrium loses its stability and produces periodic oscillations via Hopf bifurcation. It further regains its stability at second delay threshold via another Hopf bifurcation. Hence, the delay effect on information shows possibility of stability switches. Numerical experiments are carried out to support the obtained analytical results. Our study infers that the disease will show persistent oscillations if there is a significant time lag in reporting of infective after the disease outbreak. Thus, the delay in dissemination of information shows rich and complex dynamics in the model and provides important insights. We also observe numerically that the saturation in information plays a significant role on stability of infected equilibrium in presence of delay.

Hopf Bifurcation, Hopf-Hopf Bifurcation, and Period-Doubling Bifurcation in a Four-Species Food Web

Complex dynamics of a four-species food web with two preys, one middle predator, and one top predator are investigated. Via the method of Jacobian matrix, the stability of coexisting equilibrium for all populations is determined. Based on this equilibrium, three bifurcations, i.e., Hopf bifurcation, Hopf-Hopf bifurcation, and period-doubling bifurcation, are analyzed by center manifold theorem, bifurcation theorem, and numerical simulations. We reveal that, influenced by the three bifurcations, the food web can exhibit very complex dynamical behaviors, including limit cycles, quasiperiodic behaviors, chaotic attractors, route to chaos, period-doubling cascade in orbits of period 2, 4, and 8 and period 3, 6, and 12, periodic windows, intermittent period, and chaos crisis. However, the complex dynamics may disappear with the extinction of one of the four populations, which may also lead to collapse of the food web. It suggests that the dynamical complexity and food web stability are determined by the food web structure and existing populations.