CHAOTIC REGIMES, POST-BIFURCATION DYNAMICS, AND COMPETITIVE MODES FOR A GENERALIZED DOUBLE HOPF NORMAL FORM

A dinmica da equao diferencial com retardo + 2a + bx = f (x ), para a funo no linear f (x ) = tanh(x ), foi analisada como funo dos parmetros a, b, e do retardo , onde x = x(t ). Esse modelo descreve um oscilador harmnico amortecido sujeito a retroalimentao

Bifurcation analyses of the impact of the void distribution parameter C0 and the axial power profile on the stability of boiling water reactors (BWRs) are reported. Bifurcation characteristics of heated channels (without nuclear feedback) appear to be very sensitive to the axial power profile. A turning point bifurcation was detected for a (symmetrically) peaked axial power profile. This kind of bifurcation does not occur for a uniformly heated channel.Both supercritical and subcritical Hopf bifurcations were encountered in a (nuclear-coupled) reactor system, depending on the strength of the void reactivity feedback. Subcritical bifurcations become less likely to occur as C0 is significantly larger than unity. In BWRs with a strong nuclear feedback, the oscillation amplitude of limit cycles caused by a supercritical bifurcation is very sensitive to both C0 and the axial power profile.

In this paper, the Hopf bifurcations and limit cycle oscillations (LCOs) of an airfoil with cubic nonlinearity in supersonic\hypersonic flow are investigated. The harmonic balance method and multivariable Floquet theory are applied to analyze the LCOs of the airfoil. Four distinct cases of the LCOs response are detected in this system: (I) supercritical Hopf bifurcation, (II) a single subcritical Hopf bifurcation, (III) two subcritical Hopf bifurcations, and (IV) no Hopf bifurcation. Furthermore, the parameter variations domains separating the supercritical and subcritical Hopf bifurcations are presented using singularity theory.

Characterization of stability limits of Ledinegg instability and density wave oscillations for two-phase flow in natural circulation loops

A bifurcation analysis of boiling water reactor on large domain of parametric spaces

Bifurcation analysis of density wave oscillations in natural circulation loop

On the spectral instability and bifurcation of the 2D-quasi-geostrophic potential vorticity equation with a generalized Kolmogorov forcing

Food uptake ability of higher trophic level species are more complicated and interesting due to their choice and availability of food and consequently their growth and also the effect of top predator interference on the dynamics of a tritrophic food chain model. In this paper, we consider the general framework for calculating the stability of equilibria, Hopf and double Hopf-bifurcation of a prey–predator system with Holling type IV and Beddington–DeAngelis type functional responses of intermediate and top predator respectively. The top predator is of generalist type and its growth is considered due to sexual reproduction. Firstly, we have shown feasibility and boundedness of the solutions of the considered model system, behavior of equilibria and the existence of Hopf-bifurcation. Conditions are determined under which the coexistence equilibrium point remains globally asymptotically stable. We identify the critical values of delay parameter for which stability switches and nature of the Hopf-bifurcation by using normal form theory and center manifold theorem. We investigate the occurence of double Hopf bifurcation at positive equilibrium point when we choose appropriate measure of the predator’s immunity or tolerance of the prey. Furthermore, some dynamic behaviors, such as stability switches, chaos, bifurcation and double Hopf-bifurcation scenarios are observed using numerical simulations. The chaotic behavior of the system is clarified by standard numerical tests.

Abstract. In this paper we study the spatiotemporal properties of waves in the Lorenz-96 model and their dependence on the dimension parameter n and the forcing parameter F. For F > 0 the first bifurcation is either a supercritical Hopf or a double-Hopf bifurcation and the periodic attractor born at these bifurcations represents a traveling wave. Its spatial wave number increases linearly with n, but its period tends to a finite limit as n → ∞. For F < 0 and odd n, the first bifurcation is again a supercritical Hopf bifurcation, but in this case the period of the traveling wave also grows linearly with n. For F < 0 and even n, however, a Hopf bifurcation is preceded by either one or two pitchfork bifurcations, where the number of the latter bifurcations depends on whether n has remainder 2 or 0 upon division by 4. This bifurcation sequence leads to stationary waves and their spatiotemporal properties also depend on the remainder after dividing n by 4. Finally, we explain how the double-Hopf bifurcation can generate two or more stable waves with different spatiotemporal properties that coexist for the same parameter values n and F.

ABSTRACTMost researches on railway vehicle stability analysis are focused on the codimension 1 (for short, codim 1) bifurcations like subcritical and supercritical Hopf bifurcation. The analysis of codim 1 bifurcation can be completed based on one bifurcation parameter. However, two bifurcation parameters should be considered to give a general view of the motion of the system when it undergoes a degenerate Hopf bifurcation. This kind of bifurcation named the generalised Hopf bifurcation belongs to the codimension 2 (for short, codim 2) bifurcations where two bifurcation parameters need to be taken into consideration. In this paper, we give a numerical analysis of the codim 2 bifurcations of a nonlinear railway wheel-set with the QR algorithm to calculate the eigenvalues of the linearised system incorporating the Golden Cut method and the shooting method to calculate the limit cycles around the Hopf bifurcation points. Here, we found the existence of a generalised Hopf bifurcation where a subcritical Hopf bifurcation turns into a supercritical one with the increase of the bifurcation parameters, which belong to the codim 2 bifurcations, in a nonlinear railway wheel-set model. Only the nonlinear wheel/rail interactive relationship has been taken into consideration in the lateral model that is formulated in this paper. The motion of the wheel-set has been investigated when the bifurcation parameters are perturbed in the neighbourhood of their critical parameters, and the influences of different parameters on critical values of the bifurcation parameters are also given. From the results, it can be seen that the bifurcation types of the wheel-set will change with a variation of the bifurcation parameters in the neighbourhood of their critical values.

Limit cycle oscillation suppression of 2-DOF airfoil using nonlinear energy sink

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.

The space-independent xenon oscillation problem relevant to power nuclear reactors is studied. Xenon oscillations – both, without temperature feedback and in the presence of temperature feedback – have been analyzed semi-analytically and numerically. The effects of various parameters on the nature of bifurcation are studied. Bifurcation analysis shows that Hopf bifurcation occurs, and it is found that both sub- critical and super-critical Hopf bifurcation can occur in different regions of parameter space. Numerical experiments show that 'outsidey the unstable periodic solutions that exist for the sub-critical Hopf bifurcation case, there exist large amplitude, stable periodic solutions to which the initial conditions, outside the basin of attraction of the stable fixed point, evolve. Though the existence of sub-critical Hopf bifurcation indicates that large amplitude perturbations even in the stable region may lead to initially diverging oscillations, it is reassuring that the oscillation amplitude remains bounded

Non-resonant double Hopf bifurcation of a class-B laser system

This study examines the Hopf (double Hopf) bifurcations and transitions of two dimensional quasi-geostrophic (QG) flows that model various large-scale oceanic and atmospheric circulations. Using the Kolmogorov function to represent an external forcing in the tropical region, it is shown that the equilibrium of the QG model loses its stability if the combination of the Rossby number, the Ekman number, and the eddy viscosity satisfies a specific condition. Further use of the center manifold technique reveals two different types of the dynamical transition from either a pair of simple complex eigenvalues or a double pair of complex conjugate eigenvalues. These dynamical transitions are confirmed in the numerical analyses of the QG dynamics at the equilibrium, which capture Hopf (double Hopf) bifurcations due to the existence of a nonzero imaginary part of the first eigenvalue. The transition from a pair of simple complex eigenvalues is of particular interest, because it indicates the existence of a stable periodic pattern that is similar to atmospheric easterly waves and related tropical cyclone formation in the tropical atmosphere.