Determination Of Bifurcation Points Research Articles

Influence of metrics on determination of bifurcation points in time series of meteorological values

The influence of a metric kind and a method of setting the "dividing" polynomial on the results of finding bifurcation points in time series of meteorological values has been studied. The bifurcation point reflects the moment of change of the established system mode, caused by a significant change in the factors that generate the process under study. This article searches for a moment when the direction of the process changes in the time series of measurement data as the boundary between local trends. Zero to third degree polynomials have served as an indicator for determining the position of the bifurcation point. As the metrics allowing to determine the optimal position of these polynomials within the time series under investigation the following ones have been considered: standard deviation, Euclidean metric, Manhattan and relative distance. The statistical significance of the split is checked using the Fischer test. The proposed algorithm has been tested on time series of instantaneous and averaged air temperature and atmospheric pressure. The degree of the initial variation unexplained by the model is used as an estimate of the approximation accuracy. The use of Manhattan distance as a metric has been found to be the best for the time series considered. The approach proposed in this paper can be applied in the future, in particular, when determining the start time of climate change in different regions of our planet

Studying the performance of critical slowing down indicators in a biological system with a period-doubling route to chaos

This paper aims to investigate critical slowing down indicators in different situations where the system’s parameters change. Variation of the bifurcation parameter is important since it allows finding bifurcation points. A system’s parameters can vary through different functions. In this paper, five cases of bifurcation parameter variation are considered in a biological model with a period-doubling route to chaos. The first case is a slow and small stepwise variation of the bifurcation parameter. The second case is a cyclic, state-dependent variation of the bifurcation parameter. In the third case, a small cyclic variation is combined with a sizeable stochastic resonance. The fourth case involves variations by a large noise, and finally, in the fifth case, significant stepwise changes in the parameter are studied. To identify the conditions under which critical slowing down occurs, an improved version of four well-known critical slowing down indicators (autocorrelation at lag-1, variance, kurtosis, and skewness) are used. The results show that when bifurcations are caused by a sudden change in a parameter or state, critical slowing down cannot be observed before the bifurcation points. However, in cases with slowly varying parameters, critical slowing down can be detected before the bifurcation points. Thus critical slowing down indicators can predict these bifurcation points. In other words, in three cases, the system approaches bifurcation points slowly. In other cases, the bifurcations occur suddenly because of a significant shift in the parameter or state. Thus critical slowing down indicators cannot predict those bifurcation points. However, critical slowing down indicators can predict the bifurcation points in other cases.

Numerical Optimization Method for Determination of Bifurcation Points and its Application in Stability Analysis of Power System

Nonlinear bifurcation theory is very powerful and efficient to deal with multifarious nonlinear phenomena emerging from physics, biology, engineering, and economics. However, the execution of augmented system for bifurcation points will be very complicated for a system with too many variables. The purpose of this paper is to give a new optimization based method for computation of fold and Hopf bifurcation points and discuss their applications in stability analysis of power system. The validity of the proposed method has been testified by a WSCC9-bus power system in which the load increment at any bus was taken as a parameter.

Detecting bifurcation points in a memristive neuron model

In this paper, bifurcations of a memristive neuron model are analyzed. The system shows different limit cycles and chaotic attractors by varying external current. The focus of this paper is finding bifurcation points of the system and predicting them using critical slowing down indicators. The system has different tipping points such as transition from a period-2 limit cycle to period-3 limit cycle, period-3 limit cycle to period-6 limit cycle and limit cycle to chaos. Two critical slowing down indicators have been used to predict tipping points of the system. The first critical slowing down indicator is autocorrelation at lag-1 which cannot indicate bifurcation points of the system. The second one is Lyapunov exponent which shows acceptable results in prediction of bifurcation points of the memristive neuron model.

On the stability of reinforced arches

This paper deals with the problems of stability of circular arches supported by inextensible threads that cannot withstand compressive forces. Both ends of the thread are attached to the axis of the arch so that the distance between the attachment points cannot increase during deformation. The problems of stability and supercritical behavior of elastic systems in the presence of unilateral restrictions on the movement lead to the need to study the bifurcation points of the equations or to find the parameters for which some variational problem with restrictions on the desired functions in the form of inequalities has a non-unique solution. In the numerical study, this problem is reduced to finding and studying the bifurcation points of solutions to a nonlinear programming problem. The problem of finding bifurcation points for solutions of nonlinear programming problems is reduced to the problem of identification of conditional positive definiteness of quadratic forms on cones. There are criteria of conditional positive definiteness of quadratic forms on cones in the important special case, when the cone is a positive orthant in Euclidean space. Their application poses the necessity to calculate a large number of determinants, which is computationally extremely inefficient, although in the case of a small number of variables, they offer promise for solving Lyapunov stability problems. The paper proposes and justifies the method of searching for options to solve the problem of nonconvex quadratic programming that arises during the study of the stability of elastic unilateral restrictions on the movement. The results can be used in the design of arches and arch systems to improve their bearing capacity.

Bifurcation points of the generalized solution of the Hamilton-Jacobi-Bellman equation

Properties of a minimax piecewise smooth solution of the Hamilton-Jacobi-Bellman equation are considered in the article. We study necessary and sufficient conditions for finding bifurcation points. Such points are points of “nucleation” of the set where the solution is not differentiable.

Extraction of the Retinal Blood Vessels and Detection of the Bifurcation Points

The changes in retinal blood vessels Structure and progression of diseases such as diabetes, hypertension and retinopathy of prematurity (ROP) has been the subject of several large scale clinical studies. Proposed algorithm for the detection and measurement of blood vessels of the retina and finding the bifurcation points of blood vessels is general enough that it can be applied to high resolution fundus photographs. The algorithm proceeds through three main steps 1. Preprocessing operations on high resolution fundus images 2. For retinal vessel extraction, simple vessel segmentation techniques formulated in the language of 2D Median Filter 3. Minutiae techniques for finding bifurcation points of the extracted blood vessels. Performance of this algorithm is tested using the fundus image database ( 240 images) taken from Dr. Manoj Saswade, Dr.Neha Deshpande and online available databases diaretdb0, diaretdb1 and DRIVE. This algorithm achieves accuracy of 96% with 0.92 sensitivity and 0 specificity for Saswade database , for diaretdb0 accuracy 95% with 0.95 sensitivity and 0 specificity, for diaretdb1 accuracy 96% with 0.96 sensitivity and 0 specificity, and for DRIVE database 98% accuracy with 0.98 sensitivity and 0 specificity.

Numerical Path Following as an Analysis Method for Electrostatic MEMS

This paper aims to find all static states, including stable and unstable states, of electrostatically actuated microelectromechanical systems (MEMS) device models. We apply the numerical path-following technique to solve for the curve connecting the static states. We demonstrate that device models with 2 DOF can already exhibit symmetry-breaking bifurcations in the curve of static states and can have multiple disjoint solution paths. These features are also found in a finite-element method (FEM) model for a flexible beam suspended by a torsion spring. We have observed multiple hysteresis loops in measurements of a capacitive RF-MEMS device and have captured the qualitative features of these measurements in a model with 5 DOF. Numerical procedures for determining stability of solutions and finding bifurcation points are provided. Numerical path following is shown to be an efficient technique to find the curve of static states both in low-dimensional models and in FEM models.

Numerical Method for Finding Bifurcation Points of the Linear Two-parameter Eigenvalue Problem

AbstractThe iteration algorithm of finding bifurcation points of simple eigenvalue curves of the linear algebraic two-parameter eigenvalue problem is considered. The algorithm is based on the efficient numerical procedure of calculation of the derivative of matrix determinant. Numerical results are given.

Investigation of Dynamic Behavior of the Bray−Liebhafsky Reaction in the CSTR. Determination of Bifurcation Points

Experimental results obtained by operating the Bray−Liebhafsky (BL) reaction in the CSTR are presented. The dynamic behavior of the BL reaction is examined at several operation points in the concentration phase space, by varying different parameters, the specific flow rate, temperature, and mixed inflow concentrations of the feed species, one at a time. Different types of bifurcation leading to simple periodic orbits, supercritical and subcritical Hopf bifurcations, saddle node infinite period bifurcation (SNIPER), saddle loop infinite period bifurcation, and jug handle bifurcation, are observed. Moreover, complex dynamic behavior, including transition from simple periodic oscillations to complex mixed-mode oscillations and chaos, and bistability are also discovered.

An alternative method of finding bifurcation points in pseudo‐barotropes

AbstractStarting from the equations of hydrodynamics, which include both collisional and collisionless ideal fluids, the special case of pseudo‐arotropes with anisotropic pressure distribution along tangential or vertical direction (in respect to a “vertical” rotation axis) is considered. We find that vertical motions are equivalent to an imaginary rotation, and call “pseudo‐otation” the combination of rotation and real or imaginary peculiar circular motions. By use of a necessary condition for equilibrium, i.e. the coincidence of the boundary with an isopotential (gravitational + rotational centrifugal + real peculiar or imaginary peculiar centrifugal) surface, a new method of finding bifurcation points from axisymmetric to triaxial pseudo‐arotropic sequences – discussed in an earlier paper – is dealt with for the systems under discussion. A simple application to the special case where the isopycnic surfaces are spheroidal discloses that, on one hand, steadily pseudo‐otating, heterogeneous spheroids, cannot be equilibrium configurations and, on the other hand, the same results obtained by use of the virial technique hold when the current method can work. Then the coincidence of the boundary with an isopotential surface and the validity of the virial equations of the second order make two independent (necessary) conditions for equilibrium and both must be satisfied. Under the further assumptions of steady rotation and homeoidally or focaloidally striated distribution of matter, the configurations for which a bifurcation point must necessarily occurr, are determined. In the former alternative a connection is also established, between local (on the boundary) and global anisotropy of pressure distribution; in particular, it is found that local isotropy (on the boundary) involves global isotropy and vice versa. In both cases, the right amount of of anisotropy turns out to yield violation of a treshold for stability, ϵrot = −Erot/Epot ⩽ 0.14, conjectured by Ostriker and Peebles (1973). This result gives additional support to a conclusion established in an earlier paper: it seems more germane to speak about Ostriker‐eebles conjecture for stability in connection with visible bodies of galaxies, instead of Ostriker‐eebles criterion for stability in connection with self‐ravitating fluids.

On the Computation of Multiple Bifurcations with Multiple Parameters and Symmetry

This paper concerns numerical methods for the determination of bifurcation points of certain steady state multiparameter problems in the presence of symmetries. The approach is based on the fact that under general conditions the solution set forms a manifold in the space of all state and parameter variables. The reduced manifold with respect to some subsymmetry is introduced. Methods are presented for the local computation of the submanifold of bifurcation points of the same symmetry.