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.
We present an unfolding of the codimension-two scenario of the simultaneous occurrence of a discontinuous bifurcation and an Andronov-Hopf bifurcation in a piecewise-smooth, continuous system of autonomous ordinary differential equations in the plane. We find the Hopf cycle undergoes a grazing bifurcation that may be very shortly followed by a saddle-node bifurcation of the orbit. We derive scaling laws for the bifurcation curves that emanate from the codimension-two bifurcation.
Codimension-two bifurcations are fundamental and interesting phenomena in dynamical systems. Fold-Hopf and double-Hopf bifurcations are the most important among them. We study the unfoldings of these two codimension-two bifurcations, and obtain sufficient conditions for their nonintegrability in the meaning of Bogoyavlenskij. We reduce the problems of the unfoldings to those of planar polynomial vector fields and analyze the nonintegrability of the planar vector fields, based on Ayoul and Zung's version of the Morales-Ramis theory. New useful criteria for nonintegrability of planar polynomial …
We investigate the breakdown of normal hyperbolicity of a manifold of equilibria of a flow. In contrast to classical bifurcation theory we assume the absence of any flow-invariant foliation at the singularity transverse to the manifold of equilibria. We call this setting bifurcation without parameters. In the present paper we provide a description of general systems with a manifold of equilibria of codimension one as a first step towards a classification of bifurcations without parameters. This is done by relating the problem to singularity theory of maps.
We study bifurcation behavior in periodic perturbations of two-dimensional symmetric systems exhibiting codimension-two bifurcations with a double eigenvalue when the frequencies of the perturbation terms are small. We transform the periodically perturbed system to a simpler one which is a periodic perturbation of the normal form for codimension-two bifurcations with a double zero eigenvalue and symmetry, and apply the subharmonic and homoclinic Melnikov methods to analyze bifurcations occurring in the system. In particular, we show that there exist transverse homoclinic or heteroclinic orbits…
In this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit…
The Functionalized Cahn-Hilliard free energy supports phase separated morphologies of distinct codimension, including codimension-one bilayer and codimension-two filament morphologies. We characterize the linear stability of bilayer and filament morphologies associated to hypersurfaces within the strong functionalization scaling. In particular we show that the onset of the pearling instability, which triggers fast in-plane oscillations associated to bifurcation to higher codimensional morphology, is controlled by the functionalization parameters and the spatially constant value of the far-fiel…
Tip growth is a growth stage which occurs in fungal cells. During tip growth, the cell exhibits continuous extreme lengthwise growth while its shape remains qualitatively the same. A model for single celled fungal tip growth is given by the Ballistic Ageing Thin viscous Sheet (BATS) model, which consists of a 5-dimensional system of first order differential equations. The solutions of the BATS model that correspond to fungal tip growth arise through a codimension-1 global bifurcation in a 2-parameter family of solutions. In this paper we derive a toy model from the BATS model. The toy model is…
This paper continues the work contained in two previous papers, devoted to the study of the dynamical system generated by a delay differential equation that models leukemia. Here our aim is to identify degenerate Hopf bifurcation points. By using an approximation of the center manifold, we compute the first Lyapunov coefficient for Hopf bifurcation points. We find by direct computation, in some zones of the parameter space (of biological significance), points where the first Lyapunov coefficient equals zero. For these we compute the second Lyapunov coefficient, that determines the type of the …
We unfold the codimension-two simultaneous occurrence of a border-collision bifurcation and a period-doubling bifurcation for a general piecewise-smooth, continuous map. We find that, with sufficient non-degeneracy conditions, a locus of period-doubling bifurcations emanates non-tangentially from a locus of border-collision bifurcations. The corresponding period-doubled solution undergoes a border-collision bifurcation along a curve emanating from the codimension-two point and tangent to the period-doubling locus here. In the case that the map is one-dimensional local dynamics are completely c…
This paper pursues the study carried out by the authors in {\it Stability and Hopf bifurcation in the Watt governor system} \cite{smb}, focusing on the codimension one Hopf bifurcations in the centrifugal Watt governor differential system, as presented in Pontryagin's book {\it Ordinary Differential Equations}, \cite{pon}. Here are studied the codimension two and three Hopf bifurcations and the pertinent Lyapunov stability coefficients and bifurcation diagrams, illustrating the number, types and positions of bifurcating small amplitude periodic orbits, are determined. As a consequence it is fo…
Hysteresis phenomena and multistability play crucial roles in the dynamics of coupled oscillators, which are now interpreted from the point of view of codimension-two bifurcations. On the Ott-Antonsen's manifold, complete bifurcation sets of delay-coupled Kuramoto model are derived regarding coupling strength and delay as bifurcation parameters. It is rigorously proved that the system must undergo Bautin bifurcations for some critical values, thus there always exists saddle-node bifurcation of periodic solutions inducing hysteresis loop. With the aid of center manifold reduction method and the…
If we are ever to move beyond the study of isolated special cases in theoretical neuroscience, we need to develop more general theories of neural circuits over a given neural model. The present paper considers this challenge in the context of continuous-time recurrent neural networks (CTRNNs), a simple but dynamically-universal model that has been widely utilized in both computational neuroscience and neural networks. Here we extend previous work on the parameter space structure of codimension-1 local bifurcations in CTRNNs to include codimension-2 local bifurcation manifolds. Specifically, we…
The well-studied Hindmarsh-Rose model of neural action potential is revisited from the point of view of global bifurcation analysis. This slow-fast system of three paremeterised differential equations is arguably the simplest reduction of Hodgkin-Huxley models capable of exhibiting all qualitatively important distinct kinds of spiking and bursting behaviour. First, keeping the singular perturbation parameter fixed, a comprehensive two-parameter bifurcation diagram is computed by brute force. Of particular concern is the parameter regime where lobe-shaped regions of irregular bursting undergo a…
Generic bifurcation theory was classically well developed for smooth differential systems, establishing results for $k$-parameter families of planar vector fields. In the present study we focus on a qualitative analysis of $2$-parameter families, $Z_{α,β}$, of planar Filippov systems assuming that $Z_{0,0}$ presents a codimension-two minimal set. Such object, named elementary simple two-fold cycle, is characterized by a regular trajectory connecting a visible two-fold singularity to itself, for which the second derivative of the first return map is nonvanishing. We analyzed the codimension-two…
In this paper are studied the codimensions one, two, three and four Hopf bifurcations and the pertinent Lyapunov stability coefficients. Algebraic expressions obtained with computer assisted calculations are displayed.
Temporal lobe epilepsy is one of the most common chronic neurological disorder characterized by the occurrence of spontaneous recurrent seizures which can be observed at the level of populations through electroencephalogram (EEG) recordings. This paper summarizes some preliminary works aimed to understand from a theoretical viewpoint the occurrence of this type of seizures and the origin of the oscillatory activity in some classical cortical column models. We relate these rhythmic activities to the structure of the set of periodic orbits in the models, and therefore to their bifurcations. We w…
We examine the interaction of transcritical and saddle-node bifurcations in a predator-prey-nutrient system that is stressed by the presence of a toxicant affecting the prey. This model, formulated by Kooi et al. ({\sl Ecol. Model.} {\bf 212}(2008), 304--318), has a two-dimensional invariant sub system with zero predator density. In the sub system, a pair of prey-nutrient equilibria is created in a saddle-node bifurcation, while predator invasion in modelled by a transcritical bifurcation of one of this pair. Interactions of these bifurcations at codimension-two points give rise to bistable, p…
This chapter presents a dynamical systems point of view of the study of systems with delays. The focus is on how advanced tools from bifurcation theory, as implemented for example in the package DDE-BIFTOOL, can be applied to the study of delay differential equations (DDEs) arising in applications, including those that feature state-dependent delays. We discuss the present capabilities of the most recent release of DDE-BIFTOOL. They include the numerical continuation of steady states, periodic orbits and their bifurcations of codimension one, as well as the detection of certain bifurcations of…
This paper is devoted to study the generic fold-fold singularity of Filippov systems on the plane, its unfoldings and its Sotomayor-Teixeira regularization. We work with general Filippov systems and provide the bifurcation diagrams of the fold-fold singularity and their unfoldings, proving that, under some generic conditions, is a codimension one embedded submanifold of the set of all Filippov systems. The regularization of this singularity is studied and its bifurcation diagram is shown. In the visible-invisible case, the use of geometric singular perturbation theory has been useful to give t…
We study a homoclinic flip bifurcation of case~\textbf{C}, where a homoclinic orbit to a saddle equilibrium with real eigenvalues changes from being orientable to nonorientable. This bifurcation is of codimension two, and it is the lowest codimension for a homoclinic bifurcation of a real saddle to generate chaotic behavior in the form of (suspended) Smale horseshoes and strange attractors. We present a detailed numerical case study of how global stable and unstable manifolds of the saddle equilibrium and of bifurcating periodic orbits interact close to such bifurcation. This is a step forward…
This paper pursues the study carried out by the authors in "Stability and Hopf bifurcation in a hexagonal governor system", focusing on the codimension one Hopf bifurcations in the hexagonal Watt governor differential system. Here are studied the codimension two, three and four Hopf bifurcations and the pertinent Lyapunov stability coefficients and bifurcation diagrams, ilustrating the number, types and positions of bifurcating small amplitude periodic orbits, are determined. As a consequence it is found an open region in the parameter space where two attracting periodic orbits coexist with an…
Boundary equilibria bifurcation (BEB) arises in piecewise-smooth systems when an equilibrium collides with a discontinuity set under parameter variation. Singularly perturbed BEB refers to a bifurcation arising in singular perturbation problems which limit as some $ε\to 0$ to piecewise-smooth (PWS) systems which undergo a BEB. This work completes a classification for codimension-1 singularly perturbed BEB in the plane initiated by the present authors in [19], using a combination of tools from PWS theory, geometric singular perturbation theory (GSPT) and a method of geometric desingularization …
When a real saddle equilibrium in a three-dimensional vector field undergoes a homoclinic bifurcation, the associated two-dimensional invariant manifold of the equilibrium closes on itself in an orientable or non-orientable way. We are interested in the interaction between global invariant manifolds of saddle equilibria and saddle periodic orbits for a vector field close to a codimension-two homoclinic flip bifurcation, that is, the point of transition between having an orientable or non-orientable two-dimensional surface. Here, we focus on homoclinic flip bifurcations of case $\textbf{B}$, wh…
For piecewise-linear maps, the phenomenon that a branch of a one-dimensional unstable manifold of a periodic solution is completely contained in its stable manifold is codimension-two. Unlike codimension-one homoclinic corners, such `subsumed' homoclinic connections can be associated with stable periodic solutions. The purpose of this paper is to determine the dynamics near a generic subsumed homoclinic connection in two dimensions. Assuming the eigenvalues associated with the periodic solution satisfy $0 < |λ| < 1 < σ< \frac{1}{|λ|}$, in a two-parameter unfolding there exists an infinite sequ…
In this paper we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we prove that any nilpotent singularity of codimension four in $\mathbb{R}^4$ unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in $\mathbb{R}^3$ unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes. Under generic a…
We analyze the competitive evolution of codimension one and two morphologies within the $H^{-1}$ gradient flow of the strong Functionalized Cahn-Hilliard equation. On a slow time scale a sharp hypersurface reduction yields a degenerate Mullins-Sekerka evolution for both codimension one and two hypersurfaces, leading to a geometric flow that depends locally on curvatures couples to the dynamic value of the spatially constant far-field chemical potential. Both codimension one and two morphologies admit two classes of bifurcations, one leads to pearling, a short-wavelength in-plane modulation of …
From two-dimensional continuous maps to one-dimensional discontinuous maps: a novel reduction explaining complex bifurcation structures in piecewise-linear families of maps
This paper presents results concerning bifurcations of 2D piecewise-smooth vector fields. In particular, the generic unfoldings of codimension three fold-addle singularities of Filippov systems, where a boundary-saddle and a fold coincide, are considered and the bifurcation diagrams exhibited.
Bursting is a phenomenon found in a variety of physical and biological systems. For example, in neuroscience, bursting is believed to play a key role in the way information is transferred in the nervous system. In this work, we propose a model that, appropriately tuned, can display several types of bursting behaviors. The model contains two subsystems acting at different timescales. For the fast subsystem we use the planar unfolding of a high codimension singularity. In its bifurcation diagram, we locate paths that underly the right sequence of bifurcations necessary for bursting. The slow sub…
We use blowup to study the regularization of codimension one two-fold singularities in planar piecewise smooth (PWS) dynamical systems. We focus on singular canards, pseudo-equlibria and limit cycles that can occur in the PWS system. Using the regularization of Sotomayor and Teixeira \cite{Sotomayor96}, we show rigorously how singular canards can persist and how the bifurcation of pseudo-equilibria is related to bifurcations of equilibria in the regularized system. We also show that PWS limit cycles are connected to Hopf bifurcations of the regularization. In addition, we show how regularizati…
We consider the bifurcation diagram in a suitable parameter plane of a quadratic vector field in $\mathbb{R}^3$ that features a homoclinic flip bifurcation of the most complicated type. This codimension-two bifurcation is characterized by a change of orientability of associated two-dimensional manifolds and generates infinite families of secondary bifurcations. We show that curves of secondary $n$-homoclinic bifurcations accumulate on a curve of a heteroclinic bifurcation involving infinity. We present an adaptation of the technique known as Lin's method that enables us to compute such connect…
Detect the birth of limit cycles in non-smooth vector fields is a very important matter into the recent theory of dynamical systems and applied sciences. The goal of this paper is to study the bifurcation of limit cycles from a continuum of periodic orbits filling up a two-dimensional isochronous cylinder of a vector field in $\mathbb{R}^{3}$. The approach involves the regularization process of non-smooth vector fields and a method based in the Malkin's bifurcation function for $C^{0}$ perturbations. The results provide sufficient conditions in order to obtain limit cycles emerging from the cy…
A transition state for a Hamiltonian system is a closed, invariant, oriented, codimension-2 submanifold of an energy-level that can be spanned by two compact codimension-1 surfaces of unidirectional flux whose union, called a dividing surface, locally separates the energy-level into two components and has no local recrossings. For this to happen robustly to all smooth perturbations, the transition state must be normally hyperbolic. The dividing surface then has locally minimal geometric flux through it, giving an upper bound on the rate of transport in either direction. Transition states diffe…
Dynamics of generic 1-parameter perturbations of a vector field with a singular point of codimension k
Institut de Mathématiques de Toulouse, Université Paul Sabatier（图卢兹数学研究所，图卢兹第三大学）；Département de mathématiques et de statistique Université de Montréal（蒙特利尔大学数学与统计系）
本文通过构造几何模型（星形域）和分类通用族，详细描述了复向量场族在奇点展开时的局部分岔图，并与简单模型$\dot z = z^{k+1}-ε$进行比较。
Two dynamical models that have been proposed to describe transitions between low and high confinement states (L-H transitions) in confined plasmas are analysed using singularity theory and stability theory. It is shown that the stationary-state bifurcation sets have qualitative properties identical to standard normal forms for the pitchfork and transcritical bifurcations. The analysis yields the codimension of the highest-order singularities, from which we find that the unperturbed systems are overdetermined bifurcation problems and derive appropriate universal unfoldings. Questions of mutual …
Three-dimensional convex bodies can be classified in terms of the number and stability types of critical points on which they can balance at rest on a horizontal plane. For typical bodies these are nondegenerate maxima, minima, and saddle-points, the numbers of which provide a primary classification. Secondary and tertiary classifications use graphs to describe orbits connecting these critical points in the gradient vector field associated with each body. In previous work it was shown that these classifications are complete in that no class is empty. Here we construct 1- and 2-parameter famili…
A wide variety of intricate dynamics may be created at border-collision bifurcations of piecewise-smooth maps, where a fixed point collides with a surface at which the map is nonsmooth. For the border-collision normal form in two dimensions, a codimension-three scenario was described in previous work at which the map has a saddle-type periodic solution and an infinite sequence of stable periodic solutions that limit to a homoclinic orbit of the saddle-type solution. This paper introduces an alternate scenario of the same map at which there is an infinite sequence of stable periodic solutions d…
