Bifurcation Boundaries Research Articles

ROBUST BIFURCATION ANALYSIS OF SYSTEMS WITH DYNAMIC UNCERTAINTIES

In this paper, we propose in uncertain dynamical systems a novel method for identifying the region that includes all possible bifurcation boundaries. First, we formulate a robust bifurcation analysis problem for parameter-dependent differential equations with dynamic uncertainties. Next, to solve this problem, we give stability and instability conditions for the uncertain system. Finally, on the basis of these conditions, we propose a method for solving the robust bifurcation analysis problem.

A UNIFIED "SCALAR" DISCRETE-TIME MODEL FOR ENHANCING BIFURCATION PREDICTION IN DIGITALLY CONTROLLED H-BRIDGE GRID-CONNECTED INVERTER

In this paper, a unified "scalar" discrete-time model is proposed for enhanced bifurcation prediction in digitally controlled H-bridge grid-connected inverter system. The model is established by analytically expressing the elements of the matrix exponential involved in the exact discrete-time model. In high-order system, different from the discrete-time model using the state-transition matrix directly, obtaining the discrete-time model of the state-variable vector only, in this system, the discrete-time model of each state variable can be obtained directly and separately. In addition, the stability boundary of Hopf bifurcation, period-doubling bifurcation and saddle-node bifurcation can be obtained by analytical expressions. The modeling method is validated by an LCL-filter system as an example using practical circuit parameters. The results of the analysis can simplify the modeling process and enhance the bifurcation prediction of the digitally controlled inverter system.

A novel automatic algorithm for the segmentation of the lumen of the carotid artery in ultrasound B-mode images

A novel algorithm is proposed for the segmentation of the lumen and bifurcation boundaries of the carotid artery in B-mode ultrasound images. It uses the image contrast characteristics of the lumen and bifurcation of the carotid artery in relation to other tissues and structures for their identification. The relevant ultrasound data regarding the artery presented in the input image is identified using morphologic operators and processed by an anisotropic diffusion filter for speckle noise removal. The information obtained is then used to define two initial contours, one corresponding to the lumen and the other one regarding the bifurcation boundaries, for the application of the Chan-Vese level set segmentation model. A set of longitudinal ultrasound B-mode grayscale images of the common carotid artery was acquired using a GE Healthcare Vivid-e ultrasound system. The results reveal that the new algorithm is effective and robust, and that its main advantage relies on the automatic identification of the carotid lumen, which overcomes the known limitations of the traditional algorithms.

Bifurcation analysis of an endogenous growth model

This paper analyzes the dynamics of a variant of Jones (2002) semi-endogenous growth model within the feasible parameter space. We derive the long-run growth rate of the economy and do a detailed bifurcation analysis of the equilibrium. We show the existence of codimension-1 bifurcations (Hopf, Branch Point, Limit Point of Cycles, and Period Doubling) and codimension-2 (Bogdanov–Takens and Generalized Hopf) bifurcations within the feasible parameter range of the model.It is important to recognize that bifurcation boundaries do not necessarily separate stable from unstable solution domains. Bifurcation boundaries can separate one kind of unstable dynamics domain from another kind of unstable dynamics domain, or one kind of stable dynamics domain from another kind (called soft bifurcation), such as bifurcation from monotonic stability to damped periodic stability or from damped periodic to damped multiperiodic stability. There are not only an infinite number of kinds of unstable dynamics, some very close to stability in appearance, but also an infinite number of kinds of stable dynamics. Hence subjective prior views on whether the economy is or is not stable provide little guidance without mathematical analysis of model dynamics.When a bifurcation boundary crosses the parameter estimatesʼ confidence region, robustness of dynamical inferences from policy simulations are compromised, when conducted, in the usual manner, only at the parametersʼ point estimates.

Catastrophic Bifurcation in Three-Phase Voltage-Source Converters

Active pulse-width-modulated (PWM) voltage rectifiers are commonly used to convert ac power from a three-phase grid to a regulated dc voltage with unity input power factor. The output voltage regulation is normally achieved by an outer voltage feedback loop and a sinusoidal pulse-width-modulated (SPWM) inner current loop. Due to output voltage disturbances, such as those produced by a capacitive load, the inner current loop can be easily driven into over-modulation and possible six-step operating regions. Current research efforts have focused on extending the linearity of the modulated voltage in the over-modulation range to reduce the input current harmonics and on studying the closed-loop stability via conventional small-signal linearization around a fixed operating point. Such linear techniques obviously fall short of predicting large-signal transient stability. In this paper, catastrophic bifurcation of the three-phase voltage-source converter is reported. The physical origin of the phenomenon is identified. Boundaries of catastrophic bifurcation in the parameter space are derived. The phenomenon has been verified experimentally.

Analysis and experimental verification of Hopf bifurcation in a solar photovoltaic powered hysteresis current‐controlled cascaded‐boost converter

In this study, the non-linear dynamics of a two-stage cascaded-boost converter, which is a popular choice, for solar energy systems is investigated. The stability of the system is studied with the aid of non-linear state equations derived from an averaged continuous model. Analysis of the describing autonomous equations reveals that the system loses stability via supercritical Hopf bifurcation. Extensive computer simulations are performed to capture the system's dynamic behaviour and demarcate the bifurcation boundaries. Furthermore, the possibility of subcritical Hopf bifurcation for variation in input voltage is also examined. Trajectory and Poincare section before and after the bifurcation are shown. Experimental results are also provided to confirm the observed bifurcation scenario.

Stability-margin adjustment on the basis of trials for PWM converters

For a pulsed dc drive, the influence of hysteresis in adjustment of the stability margin to compensate for uncontrollable drift of the bifurcation boundary is considered. Search adaptation is employed, in combination with the symbolic prediction of PWM converter dynamics.

Nondestructive, Epi‐Illumination Surface Microscopic Characterization of Surface Discontinuity in Bone: A New Approach Offers a Descriptive Vocabulary and New Insights

Perspectives are in the eye of the beholder and are expanded with increased scrutiny (e.g., magnification). The term "porosity' for macroscopically or radiologically identifiable discontinuity in bone is so variably applied as to be incomprehensible in its meanings. A closer examination of surface discontinuity seems appropriate. Histological and atomic level analysis of bone alterations misses the forest for the trees and the former destroys the very subject it attempts to characterize. Surface discontinuity was therefore evaluated at the three-dimensional submacroscopic level. An epifluorescent/illumination microscope was used to characterize surface discontinuity in intact specimens of known pathophysiology and/or derivation. The derived perspectives were then utilized to characterize other surface discontinuity. The smooth boundaries of vascular foramen and internal bifurcation, continuity of the expanded base of nerve structures and surrounding elevated rings of Sharpey fibers are all clearly distinguishable from fronts of resorption in erosive arthritis, serpentine discontinuity associated with infectious processes, specific zones of resorption noted in tuberculosis, and structural bone exposure by taphonomic abrasion. These perspectives allow the bone pathology associated with hematoma to be characterized as accentuated Sharpey fiber insertions; maxillary and sphenoid surface discontinuity, as vascular in origin; and periosteal reaction, as manifest by superficial bifurcating channels and in-growth, similar to healing trephanation. Epi-illumination microscopy bridges macroscopic and histological examination, providing greater understanding of the pathology, at no cost to the integrity of the material studied. It explains a number of observed results, while providing discriminatory descriptions of phenomena for which pathophysiology has not yet been clarified.

Hopf bifurcation in the Clarida, Gali, and Gertler model

We explore bifurcation phenomena in the open-economy New Keynesian CGG Model based on Clarida et al. (2001, 2002). We find that the open economy framework can bring about more complex dynamics, along with a wider variety of qualitative behaviors and policy responses. Introducing parameters related to the open economy structure affects the values of bifurcation parameters and changes the location of bifurcation boundaries. As a result, the stratification of the confidence region, as previously seen in closed-economy New Keynesian models, remains an important research and policy risk to be considered in the context of the open-economy New Keynesian functional structures. In fact, econometrics and optimal policy design become more complex within an open economy. Dynamical inferences need to be qualified by the risk of bifurcation boundaries crossing the confidence regions. Without adequate prior econometric research, policy design needs to take into consideration that a change in monetary policy can produce an unanticipated bifurcation.

A Reaction Diffusion Model for Inter-Species Competition and Intra-Species Cooperation

This thesis deals with a two component reaction-diffusion system (RDS) for competing and cooperating species. We have analyse in detail the stability and bifurcation structure of equilibrium solutions of this system, a natural extension of the Lotka-Volterra system. We find seven topologically different regions separated by bifurcation boundaries depending on the number and stability of equilibrium solutions, with four regions in which the solutions are similar to those in the Lotka-Volterra system. We study RDS in the small parameter of the range $0< \lambda \ll 1 $ (fast diffusion and slow reaction), and in a few cases we assume $\lambda=O(1)$. We consider three types of initial conditions, and we find three types of travelling wave solutions using numerical and asymptotic methods. However, neither numerical nor asymptotic methods were able to find a particular travelling wave solution which connects a coexistence state say, $(u_0,w_0)$ to an extinction state $(0,0)$ when $0< \lambda \ll 1 $. This type can be found when the reaction-diffusion system satisfy the symmetry property and $\lambda=1$.

Bistable regimes in an optically injected mode-locked laser

We study experimentally the dynamics of quantum-dot (QD) passively mode-locked semiconductor lasers under external optical injection. The lasers demonstrated multiple dynamical states, with bifurcation boundaries that depended upon the sign of detuning variation. The area of the hysteresis loops grew monotonically at small powers of optical injection and saturated at moderate powers. At high injection levels the hysteresis decreased and eventually disappeared.

Bifurcation boundary conditions for current programmed PWM DC–DC converters at light loading

Three types of bifurcations (instabilities) in the PWM DC–DC converter at light loading under current mode control in continuous-conduction mode (CCM) or discontinuous-conduction mode (DCM) are analysed: saddle-node bifurcation (SNB) in CCM or DCM, border-collision bifurcation during the CCM–DCM transition, and period-doubling bifurcation in CCM. Different bifurcations occur in some particular loading ranges. Bifurcation boundary conditions separating stable regions from unstable regions in the parametric space are derived. A new methodology to analyse the SNB in the buck converter based on the peak inductor current is proposed. The same methodology is applied to analyse the other types of bifurcations and converters. In the buck converter, multiple stable/unstable CCM/DCM steady-state solutions may coexist. Possibility of multiple solutions deserves careful study, because an ignored solution may merge with a desired stable solution and make both disappear. Understanding of SNB can explain some sudden disappearances or jumps of steady-state solutions observed in switching converters.

Symmetry Breaking in a Model of Antigenic Variation with Immune Delay

Effects of immune delay on symmetric dynamics are investigated within a model of antigenic variation in malaria. Using isotypic decomposition of the phase space, stability problem is reduced to the analysis of a cubic transcendental equation for the eigenvalues. This allows one to identify periodic solutions with different symmetries arising at a Hopf bifurcation. In the case of small immune delay, the boundary of the Hopf bifurcation is found in a closed form in terms of system parameters. For arbitrary values of the time delay, general expressions for the critical time delay are found, which indicate bifurcation to an odd or even periodic solution. Numerical simulations of the full system are performed to illustrate different types of dynamical behaviour. The results of this analysis are quite generic and can be used to study within-host dynamics of many infectious diseases.

Bifurcation on synchronous full annular rub of rigid-rotor elastic-support system

An aero-engine rotor system is simplified as an unsymmetrical-rigid-rotor with nonlinear-elastic-support based on its characteristics. Governing equations of the rubbing system, obtained from the Lagrange equation, are solved by the averaging method to find the bifurcation equations. Then, according to the two-dimensional constraint bifurcation theory, transition sets and bifurcation diagrams of the system with and without rubbing are given to study the influence of system eccentricity and damping on the bifurcation behaviors, respectively. Finally, according to the Lyapunov stability theory, the stability region of the steady-state rubbing solution, the boundary of static bifurcation, and the Hopf bifurcation are determined to discuss the influence of system parameters on the evolution of system motion. The results may provide some references for the designer in aero rotor systems.

Bayesian sensitivity analysis of bifurcating nonlinear models

Sensitivity analysis allows one to investigate how changes in input parameters to a system affect the output. When computational expense is a concern, metamodels such as Gaussian processes can offer considerable computational savings over Monte Carlo methods, albeit at the expense of introducing a data modelling problem. In particular, Gaussian processes assume a smooth, non-bifurcating response surface. This work highlights a recent extension to Gaussian processes which uses a decision tree to partition the input space into homogeneous regions, and then fits separate Gaussian processes to each region. In this way, bifurcations can be modelled at region boundaries and different regions can have different covariance properties. To test this method, both the treed and standard methods were applied to the bifurcating response of a Duffing oscillator and a bifurcating FE model of a heart valve. It was found that the treed Gaussian process provides a practical way of performing uncertainty and sensitivity analysis on large, potentially-bifurcating models, which cannot be dealt with by using a single GP, although an open problem remains how to manage bifurcation boundaries that are not parallel to coordinate axes.

Sustaining of the quality index of controlling pulse converters based on nonlinear dynamics forecasting

The problem of uncontrolled variations in characteristics of a pulsed energy-conversion system conditioned by a “drift” of the actual bifurcation boundary due to external and internal perturbations is discussed. To preserve the control-quality indices at the specified level, it is proposed to use search adaptation based on forecasting of the directions of conversion of transient processes by interpreting geometrical regularities in the profiles of the time series of the state variables.

UNCERTAINTY OF PULSE ENERGY CONVERSION SYSTEM DYNAMICS NEAR BIFURCATION BOUNDARY

The paper is devoted to the problem of uncertainty in the behavior of a pulse energy conversion system (PECS) near the bifurcation boundary. As a result of the investigations, the magnitude of the zone within which a heightened sensitivity of system dynamics varying to external and internal disturbances is revealed. Also the phenomenon of intermittency is confirmed. The validity of the results is based on an analysis of the synchronized data presented by oscillograms, bifurcation diagrams and time series, acquired on the experimental setup of DC motor drive (2 kW) with pulse-width modulation.

An Investigation on the Sensitivity of Limit Cycle Oscillations for Detecting Damage in an Aeroelastic Panel

There are many nonlinear systems used in engineering which require damage detection. For such systems, nonlinear damage detection methods may be useful. In this paper, a novel method is introduced that uses limit cycle oscillations that arise once the bifurcation (flutter) boundary is exceeded and shows greater sensitivity for damage detection versus linear damage detection techniques. Another advantage of this method is that it can be used for health monitoring of linear or nonlinear systems. Here a nonlinear aeroelastic panel is considered as a model to show the capabilities of the proposed damage detection technique. Also Proper Orthogonal Decomposition is used to find the number of independent damage locations in the panel.

Impact of the exciter voltage limit to small signal stability region of a three-bus power system

We investigate the influence of non-differential components to the power system small signal stability region in this paper. A method named unified expression function (UEF) is introduced to describe the piecewise, continuous and non-differential function in the small signal stability study. It converts non-differential points of the original function to poles of the UEF’s derivative function. Then the derivative at those poles can be approximated by UEF’s left-hand or right-hand derivative limits according to the requirement. Based on this method, impact of the exciter voltage limit to power system small signal stability region is then deeply discussed using a simple three-bus power system. We find that the exciter voltage limit can change elements of the system Jacobian matrix so as to cause jump of the system critical eigenvalue. As a result, two new Hopf bifurcation boundaries and a new instability hole emerge in the small signal stability region. When the exciter voltage limit varies, the new instability hole and the new Hopf bifurcation boundaries will change significantly. It makes the topological characteristics of the small signal stability region much more complicated. Since there are many non-differential components in power systems, they should be correctly considered in power system stability studies.

Stress and strain behaviour modelling of the carotid bifurcation

The aim of this study is to investigate the biomechanical stress and strain behaviour within the wall of the artery and its influence on plaque formation and rupture using computational fluid dynamics (CFD). A three-dimensional finite-element model of the carotid bifurcation was generated to analyse the wall stress and strain behaviour. Both single-layer and multilayer models were created and structural analysis was compared between these two types of models. Systolic pressure of 180 mm Hg (~24 kPa) was applied in the inner boundary of the carotid bifurcation, and CFD analysis was performed to show the wall shear stress and pressure. The highest wall stress was found at the carotid bifurcation. When a high blood pressure (280 mm Hg) was applied to the carotid CFD model, the results showed that the stress at the carotid bifurcation may reach the rupture value. The multilayer carotid bifurcation model behaved differently from the equivalent single-layer model, with peak stress (Von-Mises) being higher in the multilayer model. The peak stress and strain was located at the origins of the internal and external carotid arteries. Significant shearing occurred between the layers in the wall of the artery at the bifurcation. Intramural shear stress in the CFD multilayer model has potential for intramural vascular injury. This may be responsible for plaque formation, plaque rupture and an injury/healing cycle.