Multi-parameter Bifurcation Analysis Research Articles

Probabilistic reachability for multi-parameter bifurcation analysis of cardiac alternans

Using a probabilistic reachability-based approach, we present a multi-parameter bifurcation analysis of electrical alternans in the two-current Mitchell–Schaeffer (MS) cardiac-cell model. Electrical alternans is a phenomenon characterized by a variation in successive Action Potential Durations generated by a cardiac cell or tissue. Alternans are known to initiate re-entrant waves and are an important physiological indicator of an impending life-threatening arrhythmia such as ventricular fibrillation. The multi-parameter bifurcation analysis we perform identifies a bifurcation hypersurface in the MS model parameter space, such that a small perturbation to this region results in a transition from highly likely alternans to highly likely non-alternans behavior.Our approach to this problem rests on encoding alternans-like behavior in the MS model as a five-mode, multinomial hybrid automaton. To perform multi-parameter bifurcation analysis of cardiac alternans, we first treat the parameters in question as bounded random variables. We then apply a sophisticated guided-search-based probabilistic reachability analysis to compute a bounded bifurcation region (possibly very tight) that contains the bifurcation hypersurface (BH). Our probabilistic reachability analysis uses a technique that combines a δ-decision procedure with statistical tests. In the process of computing the bifurcation region, we further partition the parameter space into two more regions such that any valuation chosen from one of the regions will either produce alternans or non-alternans behavior with a probability greater than a user-defined threshold.

Multi-parameter bifurcation analysis of subsynchronous interactions in DFIG-based wind farms

This work analyzes the dynamics involved in subsynchronous interactions between wind farms and series-compensated transmission lines. This adverse phenomenon may arise when doubly-fed induction generators are radially connected to an ac transmission line with series capacitive compensation. The resulting subsynchronous oscillations can produce the system instability and cause a severe damage to the wind turbines. The study is carried out combining tools from bifurcation theory, eigenvalue analysis and nonlinear time-domain simulations. The impact on the subsynchronous mode due to the variation of a set of key parameters is considered. These parameters include the series compensation of the transmission line, the wind speed (generated power), the wind energy penetration level, and a gain of the internal control loop of the wind turbine. The bifurcation study determines that the underlying mechanism associated with the appearance of subsynchronous oscillations can be explained by means of Hopf bifurcations of the equilibrium point and their corresponding limit cycles. In addition, the saturation of the wind turbine converters plays an important role in the dynamics, introducing non-smooth bifurcations that are also analyzed.

A numerical study of the effect of degenerate Hopf bifurcations on the voltage stability in power systems

Bifurcation theory has been employed to qualitatively explain the mechanisms of voltage stability problems occurring in electric power systems by relating the voltage collapse and voltage oscillatory problems to the appearance of saddle node and Hopf bifurcations, respectively. The results obtained by bifurcation analysis have revealed the existence of degenerate Hopf bifurcations when two parameters of the system are varied simultaneously; however, the mechanisms of voltage stability problems when the system is operating near these kinds of bifurcations have not yet been investigated. Bearing this in mind, the main scope of this paper is to assess how the voltage stability of a power system is affected by the presence of codimension-2 degenerate Hopf bifurcations. From the multiparameter bifurcation analysis, the existence of degenerate Hopf bifurcation points are identified and a trajectory stability diagram is then computed for classifying the type of degenerate Hopf bifurcations. This classification is used for establishing the voltage dynamics’ behavior when the power system is operating near these bifurcations. Finally, the application of a StatCom is proposed to eliminate sustained voltage oscillations and the appearance of degenerate Hopf bifurcations.

Multi-parameter bifurcations of the Kaldor–Kalecki model of business cycles with delay

In this paper, we study the following Kaldor–Kalecki model of business cycles describing the interaction of the gross product Y and the capital product K: {dY(t)dt=α[I(Y(t))−βK(t)−γY(t)],dK(t)dt=I(Y(t−τ))−(β+q)K(t). Single and multi-parameter bifurcation analysis of the system is carried out to perform center manifold reduction. We show that this system can exhibit codimension-1 singularity (transcritical and pitchfork bifurcations) and codimenion-2 singularity (nontransversal and symmetric BT bifurcations) under certain conditions on the time delay τ and the nonlinearity of I(Y) by using the normal form method for DDEs due to Faria and Magalhães. In addition, we use the normal forms of the system to predict bifurcation diagrams such as pitchfork bifurcations, homoclinic bifurcations and double limit cycle bifurcations, which are confirmed by numerical simulations.

Bifurcation Analysis of a Spatially Extended Laser with Optical Feedback

Vertical cavity surface-emitting lasers (VCSELs) are a new type of semiconductor laser characterized by the spatial extent of their disk-shaped output apertures. As a result, a VCSEL supports several optical modes (patterns of light) transverse to the direction of light propagation. When any laser is coupled to other optical elements, there is unavoidable optical feedback via reflecting surfaces, which influences the stability of the laser output. For a VCSEL, the question is how the transverse optical modes interact dynamically in the presence of optical feedback and how this affects stability of the system. In this paper, we start from a PDE description of the VCSEL. We proceed by using an expansion in suitable eigenfunctions to resolve the spatial dependence. In the presence of optical feedback we obtain a model in the form of a system of delay differential equations (DDEs). As we show with the example of a VCSEL that supports two transverse modes, the spatially expanded DDE model is small enough to allow for a multiparameter bifurcation analysis with numerical continuation tools. Specifically, we present stability regions of steady states and periodic solutions in dependence on the feedback strength and a homotopy parameter that models the amount of self- versus cross-feedback between the two modes. Bifurcations of more complicated spatiotemporal mode dynamics are then discussed.

Hopf bifurcations induced by SVC Controllers: A didactic example

This paper illustrates the effects of a static var compensator (SVC) device on the stability of a simple “single-machine dynamic-load” system. Firstly, the stability of the test system without SVC is investigated. The analysis is based on bifurcation diagrams, small signal stability analysis and time domain simulations. Without the SVC device, the system presents a saddle-node bifurcation. Then, static and dynamic analyses are repeated for the system with a SVC device located at the load bus. The SVC is able to improve system loadability but leads to a Hopf bifurcation. Finally, a multi-parameter bifurcation analysis is presented to study the effect of system parameters on the Hopf bifurcation.

Multiparameter bifurcation analysis of the South Brazilian Power System

This paper presents a detailed bifurcation analysis of real multiparameter power systems. Equilibrium points are used to evaluate the system eigenvalues, obtain different bifurcation diagrams, and evaluate the stability for two systems (i.e., a simple theoretical system to illustrate the basic concepts and highlight the main issues) and the southern section of the Brazilian network to demonstrate how these ideas can be applied to a real power system. The paper studies the influence of soft and hard-limits and automatic voltage regulators (AVR) droop on the systems, using detailed generator models. The stability regions and the effect that various parameters and bifurcations have on them are studied in detail.

Oscillatory coexistence in a food chain model with competing predators

A food chain model with two predators feeding on a single prey in a chemostat is studied. Using a multiparameter bifurcation analysis, we find parameters values for which there is stable oscillatory coexistence of the predators. It is also shown how these coexistent states provide a transition between two possible states of competitive exclusion. It is shown that the competitive exclusion principle need not hold if one or more of the predators has oscillatory behavior in the absence of other predators.

Oscillatory Coexistence in the Chemostat: A Codimension Two Unfolding

A nontrivial branch of stable oscillatory solutions of a two predator–one prey competition model is found in the positive octant by a multiparameter bifurcation analysis. The branch of solutions is “global” in that it is a continuous branch connecting the oscillations in each of the one predator–one prey planes. This result gives analytical verification of the numerical observations of other authors that a branch of nonplanar oscillations crosses the positive octant in certain parameter ranges.

A graphic method for the study of alternation in cardiac action potentials.

A graphic method for the study of alternation in cardiac action potentials.