Two-fold Singularity Research Articles

Two-fold singularities in nonsmooth dynamics-Higher dimensional analogs.

When a system of ordinary differential equations is discontinuous along some threshold, its flow may become tangent to that threshold from one side or the other, creating a fold singularity, or from both sides simultaneously, creating a two-fold singularity. The classic two-fold exhibits intricate local dynamics and accumulating sequences of local bifurcations and is by now rather well understood, but it is just the simplest of an infinite hierarchy of two-folds and multi-folds in higher dimensions. These arise when a system is discontinuous along multiple intersecting thresholds, and the induced sliding flows on those thresholds become tangent to their intersections. We show here, surprisingly, that these higher dimensional analogs of the two-fold reduce to the equations of the classic two-fold, providing the first step into their study and a new tool to understand higher dimensional systems with discontinuities.

Study of Periodic Orbits in Periodic Perturbations of Planar Reversible Filippov Systems Having a Twofold Cycle

We study the existence of periodic solutions in a class of planar Filippov systems obtained from non-autonomous periodic perturbations of reversible piecewise smooth differential systems. It is assumed that the unperturbed system presents a simple two-fold cycle, which is characterized by a closed trajectory connecting a visible two-fold singularity to itself. It is shown that under certain generic conditions the perturbed system has sliding and crossing periodic solutions. In order to get our results, Melnikov's ideas were applied together with tools from the geometric singular perturbation theory. Finally, a study of a perturbed piecewise Hamiltonian model is performed.

Bifurcations at a degenerate two-fold singularity and crossing limit cycles

In the 1990s Teixeira studied discontinuous piecewise-smooth (DPWS) dynamical systems in R3, whose state space is divided in two open regions by a plane acting as the switching manifold, and presenting two lines of quadratic tangency, one for each involved vector field. We call T-singularity (Teixeira singularity) the point of transverse intersection of the quadratic tangency lines that are of the invisible type. In this paper, we analyze DPWS systems that present two branches of T-singularities. In our approach we consider the cases where either a pseudo-equilibrium collides to just one of the two T-singularities or the two T-singularities (and also a pseudo-equilibrium between them) collides in a single point. As a consequence of these bifurcations, until now not covered by the literature, we can observe the birth of limit cycles, determine their stability and provide an upper limit for the number of limit cycles that occur in this scenario.

Birth of an arbitrary number of T-singularities in 3D piecewise smooth vector fields

The T-singularity (invisible two-fold singularity) is one of the most intriguing objects in the study of 3D piecewise smooth vector fields. The occurrence of just one T-singularity already arouses the curiosity of experts in the area due to the wealth of behaviors that may arise in its neighborhood. In this work we show the birth of an arbitrary number, including infinite, of such singularities. Moreover, we are able to show the existence of an arbitrary number of limit cycles, hyperbolic or not, surrounding each one of these singularities.

Resolution of the Piecewise Smooth Visible–Invisible Two-Fold Singularity in $$\mathbb {R}^3$$ R 3 Using Regularization and Blowup

Two-fold singularities in a piecewise smooth (PWS) dynamical system in $$\mathbb {R}^3$$ have long been the subject of intensive investigation. The interest stems from the fact that trajectories which enter the two-fold are associated with forward non-uniqueness. The key questions are: how do we continue orbits forward in time? Are there orbits that are distinguished among all the candidates? We address these questions by regularizing the PWS dynamical system for the case of the visible–invisible two-fold. Within this framework, we consider a regularization function outside the class of Sotomayor and Teixeira. We then undertake a rigorous investigation, using geometric singular perturbation theory and blowup. We show that there is indeed a forward orbit U that is distinguished amongst all the possible forward orbits leaving the two-fold. Working with a normal form of the visible–invisible two-fold, we show that attracting limit cycles can be obtained (due to the contraction towards U), upon composition with a global return mechanism. We provide some illustrative examples.

Revisiting the Teixeira Singularity Bifurcation Analysis: Application to the Control of Power Converters

For a discontinuous piecewise smooth (DPWS) dynamical system in [Formula: see text], whose state space is divided into two open regions by a plane acting as the switching manifold, there generically appear two lines of quadratic tangency, one for each involved vector field. When these two tangency lines have a transversal intersection, such a point is called a two-fold singularity. If furthermore, both tangencies are of invisible type, then the two-fold point is known as a Teixeira singularity (TS). The Teixeira singularity can undergo an interesting bifurcation, namely when a pseudo-equilibrium point crosses the two-fold singularity, passing from the attractive sliding region to the repulsive sliding region (or vice versa) and, simultaneously, a crossing limit cycle (CLC) arises. After deriving carefully a local canonical form, we revisit the previous works regarding this bifurcation thus correcting some detected misconceptions. Furthermore, we provide by means of a more direct approach the critical coefficients characterizing the bifurcation, also giving computational procedures for them. The achieved results are applied to some illustrative examples, within the realm of discontinuous piecewise linear (DPWL) systems. This family acts like a normal form for the bifurcation, since DPWL systems are able to reproduce all the unfolded dynamics. The study of TS-points in electronic DC-DC Boost power converters under a sliding mode control (SMC) strategy is addressed. Apart from being a relevant application, it allows to show the real usefulness of the analysis done.

The generic unfolding of a codimension-two connection to a two-fold singularity of planar Filippov systems

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, , of planar Filippov systems assuming that Z0,0 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 scenario through the exhibition of its bifurcation diagram.

Slow–fast n-dimensional piecewise linear differential systems

In this article we analyze n-dimensional slow–fast systems in a piecewise linear framework. In particular, we prove a Fenichel's-like Theorem where we give an explicit expression for the invariant slow manifold, that leads to the proof of the existence and location of maximal canards orbits. We show that these orbits perturb from singular orbits through contact points, of order greater than or equal to two, between the reduced flow and the fold manifold. In the particular case n=3, we show that the unique contact point is a visible two-fold singularity.

On the Use of Blowup to Study Regularizations of Singularities of Piecewise Smooth Dynamical Systems in $\mathbb{R}^3$

In this paper we use the blowup method of Dumortier and Roussarie, in the formulation due to Krupa and Szmolyan, to study the regularization of singularities of piecewise smooth dynamical systems in $\mathbb R^3$. Using the regularization method of Sotomayor and Teixeira, we first demonstrate the power of our approach by considering the case of a fold line. We quickly extend a main result of Reves and Seara in a simple manner. Then, for the two-fold singularity, we show that the regularized system only fully retains the features of the singular canards in the piecewise smooth system in the cases when the sliding region does not include a full sector of singular canards. In particular, we show that every locally unique primary singular canard persists the regularizing perturbation. For the case of a sector of primary singular canards, we show that the regularized system contains a canard, provided a certain nonresonance condition holds. Finally, we provide numerical evidence for the existence of secondary ...

On resolving singularities of piecewise-smooth discontinuous vector fields via small perturbations

A two-fold singularity is a point on a discontinuity surface of a piecewise-smooth vector field at which the vector field is tangent to the surface on both sides. Due to the double tangency, forward evolution from a two-fold is typically ambiguous. This is an especially serious issue for two-folds that are reached by the forward orbits of a non-zero measure set of initial points. However, arbitrarily small perturbations of the vector field can make forward evolution well-defined, and from an applied perspective, such perturbations may represent additional model features that enhance the realism of a piecewise-smooth mathematical model. Three physically motivated forms of perturbation: hysteresis, time-delay, and noise, are analysed individually. The purpose of this paper is to characterise the perturbed dynamics in the limit that the size of the perturbation tends to zero. This concept is applied to a two-fold in two dimensions. In each case the limit leads to a novel probabilistic notion of forward evolution from the two-fold.

Birth of limit cycles bifurcating from a nonsmooth center

This paper is concerned with a codimension analysis of a two-fold singularity of piecewise smooth planar vector fields, when it behaves itself like a center of smooth vector fields (also called nondegenerate Σ-center). We prove that any nondegenerate Σ-center is Σ-equivalent to a particular normal form Z0. Given a positive integer number k we explicitly construct families of piecewise smooth vector fields emerging from Z0 that have k hyperbolic limit cycles bifurcating from the nondegenerate Σ-center of Z0 (the same holds for k=∞). Moreover, we also exhibit families of piecewise smooth vector fields of codimension k emerging from Z0. As a consequence we prove that Z0 has infinite codimension.

The two-fold singularity of nonsmooth flows: Leading order dynamics in [formula omitted]-dimensions

A discontinuity in a system of ordinary differential equations can create a flow that slides along the discontinuity locus. Prior to sliding, the flow may have collapsed onto the discontinuity, making the reverse flow non-unique, as happens when dry-friction causes objects to stick. Alternatively, a flow may slide along the discontinuity before escaping it at some indeterminable time, implying non-uniqueness in forward time. At a two-fold singularity these two behaviours are brought together, so that a single point may have multiple possible futures as well as histories. Two-folds are a generic consequence of discontinuities in three or more dimensions, and play an important role in both local and global dynamics. Despite this, until now nothing was known about two-fold singularities in systems of more than 3 dimensions. Here, the normal form of the two-fold is extended to higher dimensions, where we show that much of its lower dimensional dynamics survives.

Stability conditions in piecewise smooth dynamical systems at a two-fold singularity

Some qualitative and geometric aspects of 3-dimensional non-smooth vector fields theory are discussed. Our main aim is to study the dynamics near typical singularities of piecewise smooth dynamical systems, the so-called two-fold singularities. More specifically, we are interested in discussing stability problems of such systems around these singularities.

PIECEWISE SMOOTH REVERSIBLE DYNAMICAL SYSTEMS AT A TWO-FOLD SINGULARITY

This paper focuses on the existence of closed orbits around a two-fold singularity of 3D discontinuous systems of the Filippov type in the presence of symmetries.

Structural Stability of the Two-Fold Singularity

At a two-fold singularity, the velocity vector of a flow switches discontinuously across a codimension one switching manifold, between two directions that both lie tangent to the manifold. Particularly intricate dynamics arises when the local flow curves toward the switching manifold from both sides, a case referred to as the Teixeira singularity. The flow locally performs two different actions: it winds around the singularity by crossing repeatedly through, and passes through the singularity by sliding along, the switching manifold. The case when the number of rotations around the singularity is infinite has been analyzed in detail. Here we study the case when the flow makes a finite, but previously unknown, number of rotations around the singularity between incidents of sliding. We show that the solution is remarkably simple: the maximum and minimum numbers of rotations made anywhere in the flow differ only by one and increase incrementally with a single parameter---the angular jump in the flow directio...

Nondeterministic Chaos, and the Two-fold Singularity in Piecewise Smooth Flows

A vector field is piecewise smooth if its value jumps across a hypersurface, and a two-fold singularity is a point where the flow is tangent to the hypersurface from both sides. Two-folds are generic in piecewise smooth systems of three or more dimensions. We derive the local dynamics of all possible two-folds in three dimensions, including nonlinear effects around certain bifurcations, finding that they admit a flow exhibiting chaotic but nondeterministic dynamics. In cases where the flow passes through the two-fold, upon reaching the singularity it is unique in neither forward nor backward time, meaning the causal link between inward and outward dynamics is severed. In one scenario this occurs recurrently. The resulting flow makes repeated, but nonperiodic, excursions from the singularity, whose path and amplitude is not determined by previous excursions. We show that this behavior is robust and has many of the properties associated with chaos. Local geometry reveals that the chaotic behavior can be eliminated by varying a single parameter: the angular jump of the vector field across the two-fold.

Two-folds in nonsmooth dynamical systems

In a three dimensional dynamical system with a discontinuity along a codimension one switching manifold, orbits of the system may be tangent to both sides of the switching manifold generically at isolated points. It is perhaps surprising, then, that examples of such ‘two-fold’ singularities are difficult to find amongst physical models. They occur where the relative curvature between the flow field and the switching manifold is nonsymmetric about the discontinuity. Here we motivate their study with a local form model of nonlinear control that exhibits the two-fold singularity, where the flow is constant either side of a curved switching manifold. We describe the local dynamics around general two-fold singularities, then consider their effect on global dynamics via one parameter bifurcations of limit cycles.

The Two-Fold Singularity of Discontinuous Vector Fields

When a vector field in $\mathbb{R}^3$ is discontinuous on a smooth codimension one surface, it may be simultaneously tangent to both sides of the surface at generic isolated points (singularities). For a piecewise-smooth dynamical system governed by the vector field, we show that the local dynamics depends on a single quantity—the jump in direction of the vector field through the singularity. This quantity controls a bifurcation, in which the initially repelling singularity becomes the apex of a pair of parabolic invariant surfaces. The surfaces are smooth except where they intersect the discontinuity surface, and they divide local space into regions of attraction to, and repulsion from, the singularity.