Planar Piecewise Research Articles

AN EXACT STABILITY TEST FOR PLANAR AND MULTI-MODAL PIECEWISE LINEAR SYSTEMS

A necessary and sufficient stability condition for planar and multimodal piecewise linear systems is derived. The condition is given in terms of poles and zeros of subsystems, and it is computationally tractable. In addition, three numerical examples are illustrated in order to clarify differences between a class of linear time-invariant systems and a class of piecewise linear systems from the view point of stability.

Length Preserving Multiresolution Editing of Curves

In this paper a method for multiresolution deformation of planar piecewise linear curves that preserves the curve length is presented. In a wavelet based multiresolution editing framework, the curve can be deformed at any level of resolution through its control points. Enforcing the length constraint is carried out in two steps. In a first step the multiresolution decomposition of the curve is used in order to approximate the initial curve length. In a second step the length constraint is satisfied exactly by iteratively smoothing the deformed curve. Wrinkle generation is an application the paper particularly focuses on. It is shown how the multiresolution definition of the curve allows to explicitly and intuitively control the scale of the generated wrinkles.

One-Parameter Bifurcations in Planar Filippov Systems

We give an overview of all codim 1 bifurcations in generic planar discontinuous piecewise smooth autonomous systems, here called Filippov systems. Bifurcations are defined using the classical approach of topological equivalence. This allows the development of a simple geometric criterion for classifying sliding bifurcations, i.e. bifurcations in which some sliding on the discontinuity boundary is critically involved. The full catalog of local and global bifurcations is given, together with explicit topological normal forms for the local ones. Moreover, for each bifurcation, a defining system is proposed that can be used to numerically compute the corresponding bifurcation curve with standard continuation techniques. A problem of exploitation of a predator–prey community is analyzed with the proposed methods.

PIECEWISE LINEAR FEEDBACK SYSTEMS WITH ARBITRARY NUMBER OF LIMIT CYCLES

Given an arbitrary positive integer n, it is shown that there exist planar piecewise linear differential systems with at least n limit cycles.

Symbolic analysis for some planar piecewise linear maps

In this paper a class of linear maps on the 2-torus and some planar piecewise isometries are discussed. For these discontinuous maps, by introducing codings underlying the map operations, symbolic descriptions of the dynamics and admissibility conditions for itineraries are given, and explicit expressions in terms of the codings for periodic points are presented.

Quasi-invariant attractors of piecewise isometric systems

We describe new examples of global attractors arising in planar piecewise rotations with two convex atoms. The dynamics inside these attractors is proved to be equivalent to that from models of digital filters. We also discuss some subtleties on the definition of piecewise isometric attractors and their properties, motivated not only by our new examples but also by others existing in the literature. More precisely, we require a minimality condition so that uniqueness is guaranteed and also, we establish equivalent forms of attractivity under some regularity assumption. The notion of quasi-invariance is introduced as it proves to be a suitable concept in the context of planar piecewise rotations.

On the Geometry of Orientation-Preserving Planar Piecewise Isometries

{Planar piecewise isometries (PWIs) are iterated mappings of subsets of the plane that preserve length (and hence angle and area) on each of a number of disjoint regions. They arise naturally in several applications and are a natural generalization of the well-studied interval exchange transformations. {The aim of this paper is to propose and investigate basic properties of orientation-preserving PWIs. We develop a framework with which one can classify PWIs of a polygonal region of the plane with polygonal partition. Basic dynamical properties of such maps are discussed and a number of results are proved that relate dynamical properties of the maps to the geometry of the partition. It is shown that the set of such mappings on a given number of polygons splits into a finite number of families; we call these classes. These classes may be of varying dimension and may or may not be connected. } {The classification of PWIs on n triangles for n up to 3 is discussed in some detail, and several specific cases where n is larger than three are examined. To perform this classification, equivalence under similarity is considered, and an associated perturbation dimension is defined as the dimension of a class of maps modulo this equivalence. A class of PWIs is said to be rigid if this perturbation dimension is zero.} A variety of rigid and nonrigid classes and several of these rigid classes of PWI s are found. In particular, those with angles that are multiples of π/n for n=3 , 4 , and 5 give rise to self-similar structures in their dynamical refinements that are considerably simpler than those observed for other angles.}

Limit cycles for planar piecewise linear differential systems via first integrals

Using first integrals we introduce an alternative tool to the Poincare maps in order to study the limit cycles of piecewise linear differential systems. It is also shown that an usual relaxation oscillation phenomenon occurrs in a two-parameter family of planar piecewise linear vector fields.

Tangencies in invariant disc packings for certain planar piecewise isometries are rare

For planar piecewise isometries (PWIs) (two-dimensional maps that restrict to isometries on some partition) there is a natural coding given by the itinerary of a trajectory between the pieces (atoms) of the partition on which it is defined. The set of points with the same coding is referred to as a cell and under certain general conditions the periodically coded cells define an invariant set that is a disjoint union of discs. In this paper properties of this invariant disc packing are investigated. For a one-parameter family of PWI on a torus, it is proved that tangencies between discs in this packing are rare. More precisely it is shown, using algebraic constraints on the geometry of the centres of the discs, that tangencies between any two discs can only occur at a finite number of parameter values; hence all tangencies will occur at a set of parameter values that is (at most) countably infinite. If such packings are dense it can be shown that they are maximal in a sense of measure. Examples are provided to show that the packing may not be dense if there is continuity over boundaries in the partition, and also that the absence of tangencies in the packing does not necessarily imply that the complement of the packing has positive Lebesgue measure.

Bifurcation of a periodic orbit from infinity in planar piecewise linear vector fields

Bifurcation of a periodic orbit from infinity in planar piecewise linear vector fields

Hopf-like bifurcations in planar piecewise linear systems

Continuous planar piecewise linear systems with two linear zones are considered. Due to their low differentiability specific techniques of analysis must be developed. Several bifurcations giving rise to limit cycles are pointed out.

Hopf-like bifurcations in planar piecewise linear systems

Hopf-like bifurcations in planar piecewise linear systems

Planar piecewise algebraic curves

An approach is described for piecing together segments of planar algebraic curves with derivative continuity. The application of piecewise algebraic curves to area modelling (the two-dimensional analogue of solid modelling) is discussed. A technique is presented for expressing a planar rational parametric curve as an algebraic curve segment. An upper bound is derived for the farthest distance between two algebraic curves (one of which may also be a parametric curve) within a specified region.