Hopf Points Research Articles

Bifurcation Analysis of a Neutrophil Periodic Oscillation Model with State Feedback Control

The mathematical model of stem cells is discussed with its motivation to describe the tissue relationship by technically introducing a two compartments model. The clear link between the proliferation phase of stem cells and the circulating neutrophil phase is set forth after delay feedback control of the state variable of stem cells. Hopf bifurcation is discussed with varying free parameters and time delays. Based on the center manifold theory, the normal form near the critical point is computed and the stability of bifurcating periodical solution is rigorously discussed. With the aids of the artificial tool on-hand which implies how much tedious work doing by DDE-Biftool software, the bifurcating periodic solution after Hopf point is continued by varying time delay.

Effects of diffusion and delayed immune response on dynamic behavior in a viral model

This paper studies a diffusive viral infection system with delayed immune response in the one-domain system. A system of DDE equations was explored, both analytically and numerically, using the Galerkin method. A condition that helps to find Hopf bifurcation points is determined. Full maps of the Hopf bifurcation points as well regions of stability are constructed and considered in detail. It is shown that the time delay of cytotoxic T lymphocyte (CTL) response and the diffusion parameter can significantly impact upon the stability regions. Furthermore, the influences of the other free values have been examined for their effects on stability. It is found that, as diffusion increases, the CTL response delay increases, and also as the CTL response delay is increased, the Hopf points for both generation rate and activate rate are decreased, whereas the Hopf points for the infection and death rates increased. Moreover, an increase diffusion results in an increase in the Hopf points for growth rate and activation rate, while the Hopf bifurcations are decreased for the death rate of infected cells. Bifurcation diagrams are plotted to show selected examples of limit cycle behavior (periodic oscillation), and 3-D solutions for the three concentrations in the model have been plotted to corroborate all analytical results from the theoretical section.

Hopf Bifurcation Analysis of the Combustion Instability in a Liquid Rocket Engine

The bifurcation process of self-sustained combustion instability pressure perturbations in a liquid rocket combustor is investigated based on the Helmholtz equations and a pressure dependent flame describing function. The modal frequency and growth rates are numerically resolved by the commercial software COMSOL multiphysics. Validation of the numerical approach is firstly conducted on a Rijke tube combustor, and a supercritical bifurcation for the first longitudinal mode is observed. The bifurcation diagrams for the first transverse mode for different time delays and gain index of the flame describing function are analyzed. Only the supercritical bifurcation presents for this configuration. The trajectory of Hopf points and the bifurcation diagram feature period motions with increasing the time delay. The effect of flame length distributions on the bifurcation diagrams is analyzed by considering a non-uniform flame length distribution model. Results show that the distribution has a large impact on the bifurcation process, e.g., the first transverse mode is more unstable for the non-uniform distribution. Finally, a subcritical bifurcation is found when a more complicated flame describing function is considered; the bistable region presents and the condition for this is discussed.

Experimental investigation on the synchronization characteristics of a pitch-plunge aeroelastic system exhibiting stall flutter.

This study focuses on characterizing the bifurcation scenario and the underlying synchrony behavior in a nonlinear aeroelastic system under deterministic as well as stochastic inflow conditions. Wind tunnel experiments are carried out for a canonical pitch-plunge aeroelastic system subjected to dynamic stall conditions. The system is observed to undergo a subcritical Hopf bifurcation, giving way to large-amplitude limit cycle oscillations (LCOs) in the stall flutter regime under the deterministic flow conditions. At this condition, we observe intermittent phase synchronization between pitch and plunge modes near the fold point, whereas synchronization via phase trapping is observed near the Hopf point. Repeating the experiments under stochastic inflow conditions, we observe two different aeroelastic responses: low amplitude noise-induced random oscillations (NIROs) and high-amplitude random LCOs (RLCOs) during stall flutter. The present study shows asynchrony between pitch and plunge modes in the NIRO regime. At the onset of RLCOs, asynchrony persists even though the relative phase distribution changes. With further increase in the flow velocity, we observe intermittent phase synchronization in the flutter regime. To the best of the authors' knowledge, this is the first study reporting the experimental evidence of phase synchronization between pitch and plunge modes of an aeroelastic system, which is of great interest to the nonlinear dynamics community. Furthermore, given the ubiquitous presence of stall behavior and stochasticity in a variety of engineering systems, such as wind turbine blades, helicopter blades, and unmanned aerial vehicles, the present findings will be directly beneficial for the efficient design of futuristic aeroelastic systems.

The emergence of polyglot entrainment responses to periodic inputs in vicinities of Hopf bifurcations in slow-fast systems.

Several distinct entrainment patterns can occur in the FitzHugh-Nagumo (FHN) model under external periodic forcing. Investigating the FHN model under different types of periodic forcing reveals the existence of multiple disconnected 1:1 entrainment segments for constant, low enough values of the input amplitude when the unforced system is in the vicinity of a Hopf bifurcation. This entrainment structure is termed polyglot to distinguish it from the single 1:1 entrainment region (monoglot) structure typically observed in Arnold tongue diagrams. The emergence of polyglot entrainment is then explained using phase-plane analysis and other dynamical system tools. Entrainment results are investigated for other slow-fast systems of neuronal, circadian, and glycolytic oscillations. Exploring these models, we found that polyglot entrainment structure (multiple 1:1 regions) is observed when the unforced system is in the vicinity of a Hopf bifurcation and the Hopf point is located near a knee of a cubic-like nullcline.

Abelian Integrals and Non-generic Turning Points

In this paper we initiate the study of the Chebyshev property of Abelian integrals generated by a non-generic turning point in planar slow-fast systems. Such Abelian integrals generalize the Abelian integrals produced by a slow-fast Hopf point (or generic turning point), introduced in Dumortier et al. (Discrete Contin Dyn Syst Ser S 2(4):723–781, 2009), and play an important role in studying the number of limit cycles born from the non-generic turning point.

Dynamics and bifurcations of a ratio-dependent predator-prey model

In this paper, we study a ratio-dependent predator-prey model with modied Holling-Tanner formalism, by using dynamical techniques and numerical continuation algorithms implemented in Matcont. We determine codim-1 and 2 bifurcation points and their corresponding normal form coecients. We also compute a curve of limit cycles of the system emanating from a Hopf point.

Bifurcation analysis of a predator-prey system with density dependent disease recovery

The center manifold is an invariant manifold that plays a crucial role in the bifurcation analysis of dynamical systems. The center manifold existence theorem assures the local existence of an invariant submanifold of the state space of a dynamical system around a non-hyperbolic equilibrium point. Center manifold theory is essential in the reduction of different bifurcation scenarios to their normal forms. Our study focuses on a predator-prey interactive system with density-dependent growth in predators subject to a contagious disease. The disease is assumed to be horizontally transmitted, and the rate of recovery of the infected predator is assumed to be density-dependent. At the trivial (zero) equilibrium, the center manifold is calculated whose dynamical behaviour is similar to that of the original system. Further, using the center manifolds, the normal form of a Hopf bifurcation point is determined fromwhich the criticality of the system can be deduced. Finally, numerical simulations are performed with biologically plausible parameters to substantiate the analytical findings. Using numerical continuation methods we detect Generalized Hopf and Zero-Hopf bifurcation points. We discuss their normal form coefficients, compute their two-parameter unfoldings and relate these results to the mathematical theory of codimension two bifurcations.

Dynamical analysis of two-pathogen coinfection in influenza and other respiratory diseases

Influenza is a contagious pathogen caused by the ribonucleic acid (RNA) virus and attacks the respiratory tract. This disease is still one of the leading causes of mortality and morbidity due to the continuous emergence of new virus variants. One of the causes of the emergence of this new virus is the coinfection of two different strains. Apart from the emergence of new variants, several coinfection cases with other pathogens are reported with severe conditions. We constructed an SI1I2I3R1R2R3 model to describe the coinfection of influenza and other respiratory pathogen transmissions by considering aspects of waning immunity, reinfected with the different pathogen and coinfected individuals. Analysis of this model reveals four equilibria: disease-free, two boundary endemics, and coexistence equilibria. The first three equilibria are expressed explicitly, along with the basic reproductive ratio. Stability conditions for boundary equilibria are shown analytically for the case with no coinfection and the case of symmetry. These conditions provide possible bifurcation points for numerical coexistence exploration and the nature of their stability. Numerical simulations show that no coexistence equilibrium exists in the case of no coinfection. As with coinfection, stable coexistence occurs for a relatively large coinfection rate, and Hopf bifurcations appear at two different values of infection rate of each pathogen. Stable limit cycles exist for pathogen infection rates between the two Hopf points. We also identify the periods of limit cycle increases as the infection rate of pathogen increases and decreases after reaching the maximum value.

Rigorous validation of a Hopf bifurcation in the Kuramoto–Sivashinsky PDE

We use computer-assisted proof techniques to prove that a branch of non-trivial equilibrium solutions in the Kuramoto–Sivashinsky partial differential equation undergoes a Hopf bifurcation. Furthermore, we obtain an essentially constructive proof of the family of time-periodic solutions near the Hopf bifurcation. To this end, near the Hopf point we rewrite the time periodic problem for the Kuramoto–Sivashinsky equation in a desingularized formulation. We then apply a parametrized Newton–Kantorovich approach to validate a solution branch of time-periodic orbits. By construction, this solution branch includes the Hopf bifurcation point.

Bifurcation analysis for a novel flight vehicle with pitch-control single moving mass

Moving mass flight vehicle is a strongly nonlinear system under high speed flying conditions. The system attitude dynamics becomes even more complex due to the coupling between the internal moving mass with large mass ratio and the vehicle body. This article investigates the open-loop nonlinear dynamics of a novel flight vehicle with pitch-control single moving mass from the prospective of bifurcation theory and continuation methods. Of particular interest is the influence of moving mass parameters on the number of system equilibrium points, stability of equilibrium curves, bifurcation characteristics, and the longitudinal static stability. Numerical results reveal the bifurcation phenomena existing in the proposed flight vehicle; the generated bifurcation diagrams illustrate that the multiple sets of limit points and Hopf points divide the moving mass parameter space into different regions with different values and types of stability, thus indicating the significant role of the moving mass parameters in the system nonlinear dynamics. Finally, a design strategy for the moving mass parameters is concluded based on the bifurcation analysis results.

Stability analysis of rotating shaft under electromagnetic and bearing oil-film forces

The stability of the fixed pints of a flexible rotor supported by journal bearings is investigated through the Bifurcation diagrams. In this paper the effect of nonlinear electromagnetic harmonic force is added to the simultaneous consideration of the effects of the nonlinearity in curvature and inertia, mass eccentricity and hydrodynamic forces of the journal bearings. Solution of the Reynolds equation renders the pressure distribution in the bearings. The bearing forces are found from integrating the pressure distribution on the surface of bearings. Derivation of the equations is performed using the Hamilton's principle and the nonlinearity terms are related to the eccentricity and the curvature of shaft. Primary and combination resonances are imposed to the rotor-bearing system and steady state responses show the effects of magnetic and bearing forces on the stability of amplitudes. In combination resonance, frequency of the electromagnetic load is tuned as the average of the forward and backward natural frequencies of the rotor. Existence of unstable Hopf points demonstrates that periodic, quasi-periodic and chaotic motions may be arisen in the nonlinear behavior of rotor.

Diffusion‐driven codimension‐2 Turing–Hopf bifurcation in the general Brusselator model

In this paper, spatiotemporal dynamics for a general reaction–diffusion system of Brusselator type under the homogeneous Neumann boundary condition is considered. It is shownthat the reaction–diffusion system has a unique steady state solution. For some suitable parameters, we prove that the steady state solution can be a codimension‐2 Turing–Hopf point. To understand the spatiotemporal dynamics in the vicinity of the Turing–Hopf bifurcation point, we calculate and analyze the normal form on the center manifold by analytical methods. A wealth of complex spatiotemporal dynamics near the degenerate point are obtained. It is proved that the system undergoes a codimension‐2 Turing–Hopf bifurcation. Moreover, several numerical simulations are carried out to illustrate the validity of our theoretical results.

Fast–Slow Variable Dissection with Two Slow Variables: A Case Study on Bifurcations Underlying Bursting for Seizure and Spreading Depression

The identification of nonlinear dynamics of bursting patterns related to multiple time scales and pathology of brain tissues is still an open problem. In the present paper, representative cases of bursting related to seizure (SZ) and spreading depression (SD) simulated in a theoretical model are analyzed. When the fast–slow variable dissection method with only one slow variable (extracellular potassium concentration, [Formula: see text]) taken as the bifurcation parameter of the fast subsystem is used, the mismatch between bifurcation points of the fast subsystem and the beginning and ending phases of burst appears. To overcome this problem, both slow variables [Formula: see text] and [Formula: see text] (intracellular sodium concentration) are regarded as bifurcation parameters of the fast subsystem, which exhibits three codimension-2 bifurcation points and multiple codimension-1 bifurcation curves containing the saddle-node bifurcation on an invariant cycle (SNIC), the supercritical Hopf bifurcation (the border between spiking and the depolarization block), and the saddle homoclinic (HC) bifurcation. The bursting patterns for SD are related to the Hopf bifurcation and the depolarization block while for SZ to SNIC. Furthermore, at the intersection points between the bursting trajectory and the bifurcation curves in plane ([Formula: see text], [Formula: see text]), the initial or termination phases of burst match the SNIC or HC point well or the Hopf point to a certain extent due to the slow passage effect, showing that the fast–slow variable dissection method with suitable process is still effective to analyze bursting activities. The results present the complex bifurcations underlying the bursting patterns and a proper performing process for the fast–slow variable dissection with two slow variables, which are helpful for modulation to bursting patterns related to brain disfunction.

Effect of the frequency detuning on the stability analysis in a ‎semiconductor laser subject to optical injection: Hopf and Routh-Horwitz ‎conditions

In this paper, the rate equations for a laser diode subject to an optical injection are studied both ‎analytically and numerically. By determining the Hopf and Routh- Hurwitz conditions, the stability ‎boundaries of the laser system have been studied. The stability diagrams have been discussed in terms ‎of detuning and the injection rate for different values of the linewidth enhancement factor.‎‏ ‏Also, we ‎obtain critical relations for the Hopf points that lead to instability in the laser system. It has also ‎shown that obtained relations depend on four parameters: detuning, linewidth enhancement factor, ‎optical injection strength and pumping current‎.

Bogdanov\u2013Takens bifurcation of a Holling IV prey\u2013predator model with constant-effort harvesting

A prey–predator model with constant-effort harvesting on the prey and predators is investigated in this paper. First, we discuss the number and type of the equilibria by analyzing the equations of equilibria and the distribution of eigenvalues. Second, with the rescaled harvesting efforts as bifurcation parameters, a subcritical Hopf bifurcation is exhibited near the multiple focus and a Bogdanov–Takens bifurcation is also displayed near the BT singularity by analyzing the versal unfolding of the model. With the variation of bifurcation parameters, the system shows multi-stable structure, and the attractive domains for different attractors are constituted by the stable and unstable manifolds of saddles and the limit cycles bifurcated from Hopf and Bogdanov–Takens bifurcations. Finally, a cusp point and two generalized Hopf points are found on the saddle-node bifurcation curve and the Hopf bifurcation curves, respectively. Several phase diagrams for parameters near one of the generalized Hopf points are exhibited through the generalized Hopf bifurcation.

Fractal dimensions and two-dimensional slow-fast systems

In our paper we present a fractal analysis of canard cycles and slow-fast Hopf points in 2-dimensional singular perturbation problems under very general conditions. Our focus is on the orientable case (e.g. R2) and the non-orientable case (e.g. the Möbius band). Given a slow-fast system, we generate a sequence of real numbers using the so-called slow relation function and compute a fractal dimension of that sequence. Then the value of the fractal dimension enables us to determine the cyclicity and bifurcations of canard cycles in the slow-fast system. We compute the fractal dimension of a slow-fast Hopf point depending on its codimension. Our focus is on the box dimension, one-sided dimensions and the fractal zeta-function. We also find explicit fractal formulas of Cahen-type for the computation of the above fractal dimensions and use them to detect numerically the number of canard limit cycles.

Numerical bifurcation analysis for a prey-predator type interactions with a time lag and habitat complexity

In this paper, a two-component generic prey-predator system incorporated with habitat complexity in predator functional response, and with constant time delay in predator gestation is considered. Although the role of time delay on the system dynamics is widely studied in the literature, only a few researchers have addressed the effect of habitat complexity in the prey-predator type interactions. In the first part of the paper the equilibria and stability analysis of the mathematical model is mentioned. In the second part, particular attention is paid on the numerical bifurcation analysis of the prey and predator densities based on two system parameters:(i) the strength of homogeneous habitat complexity and (ii) predator attack rate with and without time delay. It is found that dynamics with time delay in predator gestation are found to be much richer compared to that without time delay. The system stability may change from stable to unstable through a Hopf bifurcation and the solution branches emanating from these Hopf points are usually stable and supercritical. However, delay driven system may lead unstable orbits arising from Hopf bifurcations. It is also found that increasing the strength of habitat complexity may lead the stability change from unstable to stable.

Bifurcation and Stability Analysis of HIV Infectious Model with Two Time Delays

The HIV problem is studied by version of delay mathematical models which consider the apoptosis of uninfected CD4+ T cells which cultured with infected T cells in big volume. The opportunistic infection and the apoptosis of uninfected CD4+ T cells are caused directly or indirectly by a toxic substance produced from HIV genes. Ubiquitously, the nonlinear incidence rate brings forth the increasing number of infected CD4+ T cells with introduction of small time delay, and in addition, there also exists a natural time delay factor during the process of virus replication. With state feedback control of time delay, the bifurcating periodical oscillating phenomena is induced via Hopf bifurcation. Mathematically, with the geometrical criterion applied in the stability analysis of delay model, the critical threshold of Hopf bifurcation in multiple delay differential equations which satisfy the transversal condition is derived. By applying reduction dimensional method combined with the center manifold theory, the stability of the bifurcating periodical solution is analyzed by the perturbation near Hopf point.

Tuned Positive Position Feedback Control of an Active Magnetic Bearings System With 16-Poles and Constant Stiffness

This paper presents a tuned positive position feedback controller (TPPF) to overcome the dual high peaks of the classical PPF. This control signal is merged with a proportional-derivative (PD) control signal to suppress the vibrations of a constant-stiffness 16-poles rotor active magnetic bearings (AMBs) system. Another benefit of the applied TPPF is turning the rotor’s quasiperiodic unstable motion into a periodic stable motion by eliminating both saddle-node and Hopf points. The whole group equations of motion are derived and their approximate solutions are sought with the aid of the multiple scales method. Different response curves are plotted to clarify the difference between the rotor’s vibratory behaviors before and after PPF and also with TPPF.