Mathematical Problems in Engineering

Volume 2014, Article ID 892948, 8 pages

http://dx.doi.org/10.1155/2014/892948

## Structural Stability of Planar Bimodal Linear Systems

Departament de Matemàtica Aplicada I, Escola Tècnica Superior d’Enginyeria Industrial de Barcelona, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain

Received 14 July 2014; Revised 21 October 2014; Accepted 9 November 2014; Published 23 December 2014

Academic Editor: Do Wan Kim

Copyright © 2014 Josep Ferrer et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Structural stability ensures that the qualitative behavior of a system is preserved under small perturbations. We study it for planar bimodal linear dynamical systems, that is, systems consisting of two linear dynamics acting on each side of a given hyperplane and assuming continuity along the separating hyperplane. We describe which one of these systems is structurally stable when (real) spiral does not appear and when it does we give necessary and sufficient conditions concerning finite periodic orbits and saddle connections. In particular, we study the finite periodic orbits and the homoclinic orbits in the saddle/spiral case.

#### 1. Introduction

Structural stability ensures that the qualitative behavior of a system is preserved under small perturbations: a system is structurally stable if anyone in some neighborhood is equivalent to it (in particular, they have the same dynamical behavior). We study this property for a class of piecewise linear systems. Piecewise linear systems have attracted the interest of the researchers in recent years by their wide range of applications, as well as the possible theoretical approaches. See, for example, [1–8]. In particular, bimodal linear systems consist of two subsystems acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. We focus on the planar case. Indeed, it is very commonly found in applications (see the above references).

As we have pointed out, a definition of structural stability involves a topology in the set of the considered systems (which defines the “small perturbations”) and an equivalence relation (which defines the “preservation of the behavior”). For piecewise linear systems, the natural topology is the one of the Euclidean space formed by the coefficients of the matrices which determine each subsystem. Concerning the equivalence relation, there are some different natural options. For example, for single linear systems, those having positive trace and positive determinant form a unique -class, whereas they are partitioned in four -classes (spirals, nodes, improper nodes, and starred nodes). Anyway, when a topology and an equivalence relation are fixed, the structural stability points are those belonging to an open equivalence class.

Alternative approaches are possible. For example, in [9], one asks about generic properties, which are verified by “almost all” piecewise linear systems. From a topological point of view, it is a matter of density instead of openness. Indeed, the properties there are both generic and stable. Also, Arnold’s techniques [10] can be partially applied because although the equivalence relation is not defined by the action of a Lie group, the equivalence classes are probably differentiable manifolds.

Here, we focus on structural stability in the sense in [11], where a list of necessary and sufficient conditions is given for planar piecewise linear systems. Our aim is to specify these criteria in terms of the coefficients of the matrices, in the particular case of bimodal linear systems. The first step is collected in Theorem 6. However, further specific studies are necessary in several cases. As a second step, we tackle (Theorem 7) the existence of homoclinic orbits and finite periodic orbits in the saddle/spiral case. It allows us (Corollary 11) to ensure its structural stability for certain values of the parameters. We expect that, for bimodal systems, a full characterization of the structural stability in terms of the coefficients of the matrices is possible.

Even more, we expect that also a systematic study of the bifurcations is possible. Bifurcations are the frontier points of an open class, so that they come out of their class by small perturbations. Again, it depends on the considered equivalence relation. For example, the improper nodes and the starred nodes are -bifurcation between spirals and nodes but not -bifurcation because all of them are -equivalent. Indeed, the -frontier of spirals/nodes is stratified as follows: a 1-codimensional manifold formed by the improper nodes and a 3-codimensional manifold formed by the starred nodes. (Hence, improper nodes appear generically in 1-parameterized families of linear systems, whereas starred nodes appear only in 3-parameterized families.) Here, three bifurcations are presented in Corollary 11: 1-codimensional (two of them) and 2-codimensional (the third one).

In Section 3, we adapt the conditions stated in [11] for piecewise linear planar dynamical systems to the particular class of bimodal ones. We conclude that if some subsystem is a starred node, a center, or a degenerate node, then the bimodal system is not structurally stable. Moreover, we list the remaining possible cases, and we ensure that the bimodal system is structurally stable if none of the subsystems is a (real) spiral. The other cases need further specific analysis.

In particular, when a (real) spiral appears, it is necessary to study the finite periodic orbits and the homoclinic orbits. In Section 4, we study the structural stability of bimodal systems for the saddle/spiral case. We conclude that this bimodal system is structurally stable if , where is the trace of the spiral matrix and is the only value where a homoclinic orbit appears. The study will be continued in future works (see [12].)

Throughout the paper, will denote the set of real numbers, the set of matrices having rows and columns and entries in (in the case where , we will simply write ), and the group of nonsingular matrices in . Finally, we will denote by the natural basis of the Euclidean space .

#### 2. Structurally Stable BLDS: Definitions

We consider where , ; ; . We assume that the dynamic is continuous along the separating hyperplane ; that is to say, both subsystems coincide with .

By means of a linear change in the state variable , we can consider . Hence, and continuity along is equivalent to We will write from now on .

*Definition 1. *In the above conditions, one says that the triplet of matrices defines a bimodal linear dynamical system (BLDS).

The placement of the equilibrium points will play a significative role in the dynamics of a BLDS. So, one defines the following.

*Definition 2. *Let one assume that a subsystem of a BLDS has a unique equilibrium point, not lying in the separating hyperplane. One says that this equilibrium point is real if it is located in the half-space corresponding to the considered subsystem. Otherwise, one says that the equilibrium point is virtual.

It is clear that not any pair of equilibrium points are compatible. For example, two real saddles are not possible. (Table 1 lists the compatible pairs, excluding centers, starred nodes, and degenerate nodes.)