Singular Bifurcation Research Articles

Singular bifurcations and regularization theory.

Nonlinear sciences are present today in almost all disciplines, ranging from physics to social sciences. A major task in nonlinear science is the classification of different types of bifurcations (e.g., pitchfork and saddle-node) from a given state to another. Bifurcation analysis is traditionally based on the assumption of a regular perturbative expansion, close to the bifurcation point, in terms of a variable describing the passage of a system from one state to another. However, it is shown that a regular expansion is not the rule due to the existence of hidden singularities in many models, paving the way to a new paradigm in nonlinear science, that of singular bifurcations. The theory is first illustrated on an example borrowed from the field of active matter (phoretic microswimers), showing a singular bifurcation. We then present a universal theory on how to handle and regularize these bifurcations, bringing to light a novel facet of nonlinear sciences that has long been overlooked.

Classification of Codimension-1 Singular Bifurcations in Low-Dimensional DAEs

The study of bifurcations of differential-algebraic equations (DAEs) is the topic of interest for many applied sciences, such as electrical engineering, robotics, etc. While some of them were investigated already, the full classification of such bifurcations has not been done yet. In this paper, we consider bifurcations of quasilinear DAEs with a singularity and provide a full list of all codimension-one bifurcations in lower-dimensional cases. Among others, it includes singularity-induced bifurcations (SIBs), which occur when an equilibrium branch intersects a singular manifold causing certain eigenvalues of the linearized problem to diverge to infinity. For these and other bifurcations, we construct the normal forms, establish the non-degeneracy conditions and give a qualitative description of the dynamics. Also, we study singular homoclinic and heteroclinic bifurcations, which were not considered before.

The nilpotent bifurcations in a model for generalist predatory mite and pest leafhopper with stage structure

We propose a generalist predator-prey model with stage structure for prey to explore the bifurcations and complex dynamics of tea green leafhopper pest and predatory mite. Using qualitative and bifurcation theory of dynamical systems, we carry out bifurcation analysis of the three-dimensional nonlinear system, including saddle-node bifurcation of codimension 1 and 2, Hopf bifurcation, Bogdanov-Takens bifurcation, and bifurcations of nilpotent singularities of elliptic and focus type of codimension 3. We find that the nilpotent focus of codimension 4 serves as an organizing center to connect all the codimension 3 bifurcations in the two-dimensional center manifold of the system, and the bifurcations are also associated with a third order cubic Liénard system. Moreover, we present a classification of the nilpotent singularities of the system using the hatching rate of pest eggs as the parameter. Our results suggest that extending the incubation time of pest eggs can be beneficial for pest control.

The study of the singular bifurcation of a 3D dynamical system of interest in ecology

The classical bifurcations that occur in the SIR system were already studied in many papers. Our attention is focused on the singular bifurcations, when the system has in infinite number of periodic points at the bifurcation limit. The dynamics of the system is analyzed and the results are interpreted from a practical point of view.

Three Crossing Limit Cycles in a 3D-Filippov System Having a T-Singularity

Our starting point is a discontinuous piecewise linear dynamical system in [Formula: see text] with two zones that present a single invisible two-fold straight line and invariant cylinders. We broke the dynamics of this vector field to obtain a new four-parameter piecewise linear vector field with a [Formula: see text]-singularity. For this vector field, we prove that the upper bound of simple crossing limit cycles (simple CLCs) is three, and provide conditions for the existence of 1, 2, or 3 simple CLCs. We also show the occurrence of three distinct bifurcations involving such simple CLCs that are derived from a bifurcation set with two parameters: (i) a Teixeira Singularity bifurcation (TS-bifurcation); (ii) a Fold bifurcation; and (iii) a Cusp bifurcation.

Bifurcation of singularities of fluctuational paths for a noise-driven overdamped two-well system.

Noise-induced escape in a 2D generalized Maier-Stein model with two parameters μ and α is investigated in the weak noise limit. With the WKB approximation, the patterns of extreme paths and singularities are displayed. By employing the Freidlin-Wentzell action functional and the asymptotic series, critical parameters α inducing singularity bifurcation are determined analytically for μ=1. The switching line will appear with singularities and is equivalent to the sliding set in the Filippov system. The pseudo-saddle-node bifurcation on the switching line is found. Then, when -1<μ<1, it is found that all bifurcation values α will decrease as μ decreases and the second-order bifurcation values are bigger than all first-order ones. In addition, the variation of the switching line is also analyzed and a new switching line will emerge when the location of the minimum quasi-potential on the boundary changes. At last, when the noise is anisotropic, only the noise intensity ratio will affect the bifurcation value α.

On the Teixeira singularity bifurcation in a DC–DC power electronic converter

The electronic circuit of a DC–DC boost power converter under a specific sliding mode control strategy is analyzed. This circuit is modeled as a discontinuous piecewise linear three-dimensional dynamical system, whose state space is divided into two open regions by a plane acting as the switching manifold. This system displays sliding motion confined to the switching manifold and limited by two straight lines where lie the points of tangency, one for each involved linear vector field. Such tangency lines can intersect transversally on a invisible two-fold point known as Teixeira singularity. Our goal is to investigate an interesting bifurcation that occurs on the Teixeira singularity, involving both a pseudo-equilibrium point and a crossing periodic orbit. This bifurcation, named TS-bifurcation, occurs when a pseudo-equilibrium point of the discontinuous piecewise linear system, capable of moving between the attractive and repulsive sliding regions as a result of the change in a parameter, collides with the Teixeira singularity. Simultaneously, a periodic orbit arises from the Teixeira singularity crossing the switching manifold in two points. Apart from the TS-bifurcation characterization in the DC–DC converter, we have numerically detected other non-local phenomena like a saddle-node bifurcation of crossing limit cycles. Experimental results to illustrate the effects of the TS-bifurcation on the dynamics of a power electronic converter are also shown.

An Interview with William A. Barnett

An Interview with William A. Barnett

Singularities of fluctuational paths for an overdamped two-well system driven by white noise

Abstract Noise-induced escape from a metastable state in an overdamped double-well system is investigated. With the WKB approximation employed, the pattern of fluctuational paths is given. By using the action plot, which gives rise to MPEPs, the pattern is divided into various regions where fluctuational paths inside show qualitatively distinguished characteristics. The global quasi-potential is calculated and its value at the saddle on the boundary agrees with the one obtained by action plot. We show a contour and 3D plot of the quasi-potential to illustrate the relative stabilities of metastable states. The bifurcations of singularities as the system parameter varies are analyzed by the second-order term of the WKB expansion in the vicinity of x axis. Its asymptotic behavior determines the positions of cusps and furthermore, its evolution turns out to delineate the twisting structure of the Lagrangian manifold.

Bifurcation Analysis of a DC–DC Bidirectional Power Converter Operating with Constant Power Loads

Direct current (DC) microgrids (MGs) are an emergent option to satisfy new demands for power quality and integration of renewable resources in electrical distribution systems. This work addresses the large-signal stability analysis of a DC–DC bidirectional converter (DBC) connected to a storage device in an islanding MG. This converter is responsible for controlling the balance of power (load demand and generation) under constant power loads (CPLs). In order to control the DC bus voltage through a DBC, we propose a robust sliding mode control (SMC) based on a washout filter. Dynamical systems techniques are exploited to assess the quality of this switching control strategy. In this sense, a bifurcation analysis is performed to study the nonlinear stability of a reduced model of this system. The appearance of different bifurcations when load parameters and control gains are changed is studied in detail. In the specific case of Teixeira Singularity (TS) bifurcation, some experimental results are provided, confirming the mathematical predictions. Both a deeper insight in the dynamic behavior of the controlled system and valuable design criteria are obtained.

Neural Excitability and Singular Bifurcations.

We discuss the notion of excitability in 2D slow/fast neural models from a geometric singular perturbation theory point of view. We focus on the inherent singular nature of slow/fast neural models and define excitability via singular bifurcations. In particular, we show that type I excitability is associated with a novel singular Bogdanov–Takens/SNIC bifurcation while type II excitability is associated with a singular Andronov–Hopf bifurcation. In both cases, canards play an important role in the understanding of the unfolding of these singular bifurcation structures. We also explain the transition between the two excitability types and highlight all bifurcations involved, thus providing a complete analysis of excitability based on geometric singular perturbation theory.

Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II

Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II

Symbolic Quest into Homoclinic Chaos

We explore the multifractal, self-similar organization of heteroclinic and homoclinic bifurcations of saddle singularities in the parameter space of the Shimizu–Morioka model that exhibits the Lorenz chaotic attractor. We show that complex transformations that underlie the transitions from the Lorenz attractor to wildly chaotic dynamics are intensified by Shilnikov saddle-foci. These transformations are due to the emergence of Shilnikov flames originating from inclination-switch homoclinic bifurcations of codimension-two. We demonstrate how the original computational technique, based on the symbolic description and kneading invariants, can disclose the complexity and universality of parametric structures and their link with nonlocal bifurcations in this representative model.

Studying Voltage Stability in Power System Considering Load Dynamics

This paper investigates on-load tap changer OLTC effect on voltage stability. Chaos theory used in this analysis is via the calculation of the maximum power transmission. Resulted chaotic behavior is exhibited by singular induced bifurcation which clearly shows the voltage stability of the power system with OLTC from a singular induced bifurcation point of view. The paper is organized as follows: 1) increasing in the maximum reactive power via increasing the ratio of OLTC and the possibility of voltage collapse decrease under the same load command. 2) Noting that the voltage instability is directly affected by the load characteristics, shown via the effect of OLTC on the voltage instability under different static load models. 3) Studying voltage stability in state space and the effects of the generator excitation current limit, the OLTC and load dynamics on voltage collapse. In addition, a small-disturbance voltage stability region of a power system is constructed. The voltage stability of the system is determined by the dynamic characteristics of both the OLTC and the load.

Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part I

In this paper, we study the connection between the bifurcation of diffuse transition layers and that of the underlying limit interfacial problem in a degenerate spatially inhomogeneous medium. In dimension one, we prove the existence of bifurcation of diffuse interfaces in a pitchfork spatial inhomogeneity for a partial differential equation with bistable type nonlinearity. Bifurcation point is characterized quantitatively as well. The main conclusion is that the bifurcation diagram of the diffuse transition layers inherits mostly from that of the zeros of the spatial inhomogeneity. However, explicit examples are given for which the bifurcation of these two are different in terms of (im)perfection. This is a continuation of [8] which makes use of bilinear nonlinearity allowing the use of explicit solution formula. In the current work, we extend the results to a general smooth nonlinear function. We perform detail analysis of the principal eigenvalue and eigenfunction of some singularly perturbed eigenvalue problems and their interaction with the background inhomogeneity. This is the first result that takes into account simultaneously the interaction between singular perturbation, spatial inhomogeneity and bifurcation.

ОПТИМИЗАЦИЯ СТЕРЖНЕВЫХ КОНСТРУКЦИЙ СО СЛУЧАЙНЫМИ НЕСОВЕРШЕНСТВАМИ ПРИ ОГРАНИЧЕНИЯХ ПО УСТОЙЧИВОСТИ

A method for the optimization of geometrically non-linear bar structures containing random global imperfections is presented. The initial imperfections are distributed randomly according to the normal law. Optimization is carried out with constraints on the unfailing operation probability of the structure for the overall loss of stability and constraints on the design variables. Criteria are formulated for the detection of asymmetric, symmetric special bifurcation points of the first or second type and the limit point. The probability density functions of buckling loads for the four types of critical points are derived. The optimization method is based on quadratic approximation of the objective function and linear approximation of the constraints on the probability that the structure does not lose stability. The method was verified on three test structures. Keywords: optimization of geometrically non-linear rod structures, the global random imperfections, unfailing operation probability of the structure for the overall loss of stability, non-multiple singular bifurcation points, non-multiple limit points.

Simple bifurcation of coupled advertising oscillators with delay

In this work, the singular bifurcation of a ring of three coupled advertising oscillators with delay, each of them being an advertising model, is considered. The center manifold reduction and normal form method are employed to study the bifurcation from the double-zero singularity which is induced by the coupled strength. Numerical simulation supports the analysis results.

Existence of singularity bifurcation in an Euler-equations model of the United States economy: Grandmont was right

Grandmont (1985) found that the parameter space of the most classical dynamic general-equilibrium macroeconomic models are stratified into an infinite number of subsets supporting an infinite number of different kinds of dynamics, from monotonic stability at one extreme to chaos at the other extreme, and with all forms of multiperiodic dynamics between. But Grandmont provided his result with a model in which all policies are Ricardian equivalent, no frictions exist, employment is always full, competition is perfect, and all solutions are Pareto optimal. Hence he was not able to reach conclusions about the policy relevance of his dramatic discovery. As a result, Barnett and He (1999, 2001, 2002) investigated a Keynesian structural model, and found results supporting Grandmont's conclusions within the parameter space of the Bergstrom–Wymer continuous-time dynamic macroeconometric model of the UK economy. That prototypical Keynesian model was produced from a system of second order differential equations. The model contains frictions through adjustment lags, displays reasonable dynamics fitting the UK economy's data, and is clearly policy relevant. In addition, results by Barnett and Duzhak (2010) demonstrate the existence of Hopf and flip (period doubling) bifurcation within the parameter space of recent New Keynesian models. Lucas-critique criticism of Keynesian structural models has motivated development of Euler equations models having policy-invariant deep parameters, which are invariant to policy rule changes. Hence, we continue the investigation of policy-relevant bifurcation by searching the parameter space of the best known of the Euler equations general-equilibrium macroeconometric models: the path-breaking Leeper and Sims (1994) model. We find the existence of singularity bifurcation boundaries within the parameter space. Although never before found in an economic model, singularity bifurcation may be a common property of Euler equations models, which often do not have closed form solutions. Our results further confirm Grandmont's views. Beginning with Grandmont's findings with a classical model, we continue to follow the path from the Bergstrom–Wymer policy-relevant Keynesian model, to New Keynesian models, and now to Euler equations macroeconomic models having deep parameters.

GRAPH-THEORETIC CHARACTERIZATION OF BIFURCATION PHENOMENA IN ELECTRICAL CIRCUIT DYNAMICS

This paper addresses bifurcation properties of equilibria in lumped electrical circuits. The goal is to tackle these properties in circuit-theoretic terms, characterizing the bifurcation conditions in terms of the underlying network digraph and the electrical features of the circuit devices. The attention is mainly focused on so-called singular bifurcations, resulting from the semistate (differential-algebraic) nature of circuit models, but the scope of our approach seems to extend to other types of bifurcations. The bifurcation analysis combines different tools coming from graph theory (such as proper trees in circuit digraphs, Maxwell's determinantal expansions or the colored branch theorem) with several results from linear algebra (matrix pencils, the Cauchy–Binet formula, Schur complements). Several examples illustrate the results.

Slow relaxations and bifurcations of the limit sets of dynamical systems. I. Bifurcations of limit sets

We consider one-parameter semigroups of homeomorphisms depending continuously on the parameters. We study the phenomenon of slow relaxation that consists in anomalously slow motion to the limit sets. We investigate the connection between slow relaxations and bifurcations of limit sets and other singularities of the dynamics. The statements of some of the problems stem from mathematical chemistry.