BIFURCATION ANALYSIS OF EQUILIBRIUM POINT IN TWO NODE POWER SYSTEM

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

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.

Bifurcation analysis of density wave oscillations in natural circulation loop

The autonomous prey-predator system with logistic growth and Holling type II functional response, which describes the population dynamics of two species. In this study, the equilibrium points of the system (1.1) were identified. Two saddle-node points and one non-trivial P3 equilibrium point were found. For this latter point, the conditions were determined for the Jacobian matrix of (1.1), evaluated at P3, to have a pair of purely complex eigenvalues (necessary condition for the Hopf bifurcation). Through this analysis, values c0, δ0, k0 were found that satisfied these conditions. Subsequently, each of these values is considered as a bifurcation parameter value, and the remaining two are considered as control parameters, under the assumptions of the normal form theorem for the Hopf bifurcation, it’s concluded that by varying these values slightly, the system undergoes the Hopf bifurcation. Finally, the first Lyapunov coefficient was calculated to determine the conditions under which the system exhibits supercritical, subcritical, and degenerate Hopf bifurcation. The analysis was supported by using MAPLE and MATLAB software, which enabled graphical visualization of the obtained results.

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

This work reviews and elaborates a frequency domain approximate method of estimating amplitude, frequency, and stability of the limit cycle near Hopf bifurcations. A new iterative method and new criterion for ascertaining stability are presented. The method and criterion are applied to a 3-bus dynamic power system model to give the amplitude, frequency and stability at its Hopf bifurcations. New analytic formulas for bifurcation curves are derived by using analytic method, some new saddle-node bifurcation at low voltage levels are found on the formulas. The influence of rotor inertia on the Hopf bifurcation is also studied.

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

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

In many situations of power system restoration or emergency control load flow, often dynamic load flow, results are needed at a time scale which makes regular load flow computations impractical. In this paper a low order dynamic model is introduced which retains fully detailed and precise representation of the critical components (such as strong backbone lines like EHV or weak tie lines) where the interest is focused and reduces the detail of representation of other parts of the system gradually with their distance from the spotlighted sections where details are needed. The result is a low order and hence very fast computation with not only detailed but also quite precise results for those parts of the system which are critical. This approach is very flexible, it can be used to compute tie line load and frequency swings on an interconnection of areas or static loads on the tie lines of such a system affecting third parties in a power trading situation. It can be used to compute precisely and fast the EHV system loads or any combinations of such elements. The operator is quite free to choose what should be represented in precise detail.

In this paper, the predator–prey model with Michaelis–Menten functional response subject to the Neumann boundary conditions is considered. The stability of the positive equilibrium and the direction of the Hopf bifurcations for the corresponding ordinary differential equation and partial differential equation are studied, respectively. To this end, the normal form method and the center manifold theorem are applied to determine the direction of the Hopf bifurcation and the stability of the bifurcated limit cycle when the diffusion terms are present. It is shown that the corresponding system can undergo either the supercritical or subcritical Hopf bifurcation at the equilibrium point within certain parameter range. Meanwhile, the Turing instability conditions for the equilibrium and the limit cycle bifurcated from the Hopf bifurcation are derived. As a result, some patterns occur in the model. Numerical simulations are carried out to verify the theoretical analysis.

Cascading outages are a series of complex dynamic mechanisms, and could lead to major system disturbances and large blackouts. Some of the key mechanisms of cascading outages can only be studied with a dynamic power system model, which emphasizes the significance of dynamic cascading outage simulations. However, most test systems from related literature do not include sufficient data for dynamic simulations and none of them is thus able to capture all possible cascading mechanisms. The aim of this paper is to propose a small set of test systems with minimum required data, including dynamic information and protection elements, as a standard for dynamic cascading outage simulations. It presents the detailed information about each test system, and describes the algorithms that are used to generate the missing data if the baseline test cases are not complete. All proposed test systems are available through the web link: http://web.engr.oregonstate.edu/~cotillaj/ecs/Cases.html.

In this paper, we present an algorithm for deriving the normal forms of Bautin bifurcations in reaction-diffusion systems with time delays and Neumann boundary conditions. On the center manifold near a Bautin bifurcation, the first and second Lyapunov coefficients are calculated explicitly, which completely determine the dynamical behavior near the bifurcation point. As an example, the Segel-Jackson predator-prey model is studied. Near the Bautin bifurcation we find the existence of fold bifurcation of periodic orbits, as well as subcritical and supercritical Hopf bifurcations. Both theoretical and numerical results indicate that solutions with small (large) initial conditions converge to stable periodic orbits (diverge to infinity).

In this paper, we study a predator–prey mite model of Leslie type with generalized Holling IV functional response. The model is shown to have very rich bifurcation dynamics, including subcritical and supercritical Hopf bifurcations, degenerate Hopf bifurcation, focus‐type and cusp‐type degenerate Bogdanov–Takens bifurcations of codimension 3, originating from a nilpotent focus or cusp of codimension 3 that acts as the organizing center for the bifurcation set. Coexistence of multiple steady states, multiple limit cycles, and homoclinic cycles is also found. Interestingly, the coexistence of two limit cycles is guaranteed by investigating generalized Hopf bifurcation and degenerate homoclinic bifurcation, and we also find that two generalized Hopf bifurcation points are connected by a saddle‐node bifurcation curve of limit cycles, which indicates the existence of global regime for two limit cycles. Our work extends some results in the literature.

Mathematical modeling is essential for understanding infectious disease dynamics and guiding public health strategies. We study the global dynamics of a susceptible-infectious-recovered-susceptible (SIRS) model with a generalized nonlinear incidence function, revealing a rich array of bifurcation phenomena, including saddle-node, cusp, forward and backward bifurcations, Bogdanov-Takens bifurcations, saddle-node bifurcation of limit cycles, subcritical and supercritical Hopf bifurcations, generalized Hopf bifurcations, homoclinic and degenerate homoclinic bifurcations, as well as isola bifurcation. Using normal form theory, we show that the Hopf bifurcation reaches codimension three, resulting in up to three small-amplitude limit cycles. The involvement of the recovered population enables coexistence of these limit cycles, leading to bistable and tristable dynamics. We employ a one-step transformation method to analyze codimension two and three Bogdanov-Takens bifurcations, confirming a maximum codimension of three. In particular, we identify isolas of limit cycles in an SIRS model involving double exposure, introducing a mechanism for generating limit cycles centered on the isola. The findings may have important public health implications, highlighting how nonlinearities in transmission and immunity can produce recurrent outbreaks or persistent infection despite interventions. The existence of multiple limit cycles suggests that small changes in transmission rates or immune response could cause abrupt shifts in outbreak patterns, emphasizing the need for adaptive and flexible intervention strategies.