On Singularities of Discontinuous Vector Fields
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Bull. Sci. math. 127 (2003) 611–633 www.elsevier.com/locate/bulsci On singularities of discontinuous vector fields Alain Jacquemard a,∗, Marco-Antonio Teixeira b a Institut de mathématiques de Bourgogne, UMR CNRS 5584, Université de Bourgogne, BP 47870, 21078 Dijon, France b Instituto de Matematica e Estatistica, University of Campinas, 13081-970 Campinas SP, Brazil Received 15 May 2003; accepted 27 May 2003 Abstract The subject of this paper concerns the classification of typical singularities of a class of discontinuous vector fields in 4D. The focus is on certain discontinuous systems having some symmetric properties. 2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Résumé On classifie les singularités d’une classe de champs de vecteurs discontinus en dimension quatre. On s’intéresse particulièrement aux systèmes discontinus présentant des symétries. 2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. MSC: primary 37G10; secondary 37G40, 49J15 Keywords: Discontinuous vector fields; Singularity; Reversibility; Normal form 1. Introduction In dynamical systems, it is usual to classify objects up to C0-equivalence; in the theory of singularities of vector fields one studies the classification problem by listing the codimension k singularities. In our approach, due to the complexity of the problem, we introduce the concept of mild C0-equivalence and our main goal is to classify the codimension k (k = 0, 1) singularities of a class of discontinuous vector fields. These systems are defined in R4, and present discontinuities at a codimension one submanifold H0. * Corresponding author. E-mail addresses: [email protected] (A. Jacquemard), [email protected] (M.-A. Teixeira). 0007-4497/$ – see front matter 2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. doi:10.1016/S0007-4497(03)00047-2 612 A. Jacquemard, M.-A. Teixeira / Bull. Sci. math. 127 (2003) 611–633 There is a growing interest in such systems, also referred to as relay systems, which arise in various applications in Nonlinear Oscillations and Control Theory. Many industrial applications, for instance in mechanical and electromechanical systems are reported (see [1,3,10]). To fix thoughts first consider a basic model which is the following semi-linear vector field La (a ∈ R): ∂ ∂ ∂ ∂ x2 + x3 + x4 + a sgn(x1) ∂x1 ∂x2 ∂x3 ∂x4 (where sgn(x) = 1ifx 0andsgn(x) =−1forx<0). The approach we follow is partly inspired by the work of Anosov in [1] and by the setting of Control Theory contained in [10]. We recall that Anosov discusses the so-called relay vector fields in Rn of the form: X = Ax+ sgn(x1)k = ∈ = Rn where x (x1,x2,...,xn), A MR(n, n), k (k1,k2,...,kn) is a constant vector in . His goal was to give conditions on the local asymptotic stability at the origin which is a typical singularity of the system. He detected four different topological classes of such vector fields represented by normal forms. One of these normal forms is exactly the vector field La mentioned above and from this the instability of that class was immediately deduced. In [10] some properties of the deformations of La are discussed in a quite 2 ∂ different approach. But deformations as simple as La + λx (λ ∈ R) are not considered 4 ∂x1 there and this model presents a very interesting behavior, in particular if the presence of periodic orbits is discussed. We now look at those vector fields which are deformations of La preserving some distinguished properties of the unperturbed system. For instance, consider the set W of 3 vector fields Za,λ,µ (with (a,λ,µ)∈ R ) of the type: ∂ ∂ ∂ ∂ + 2 + 3 + + + x2 λx4 µx4 x3 x4 a sgn(x1) . ∂x1 ∂x2 ∂x3 ∂x4 A system written in this way is said to be the standard (reversible) model. We point out some interesting properties that both vector fields La and Za,λ,µ enjoy. These are: 1. There is a linear involution ϕ that fixes half the phase variables and under which the systems are invariant after time reversal (reversibility). 2. La has an integrable Hamiltonian structure (non-smooth on H0 ={x1 = 0}). We focus on the singularities of the reversible vector fields. Observe that reversibility extends the notion of symmetry used in [3]. In the following, we restrict ourselves to elements in W such that a =±1 (this is not a real restriction since it can be attained via a time-rescaling action). We then consider the set Ω of all vector fields close to the elements of W, in the sense that they have the ∞ form Za,λ,µ + R,whereR is C and small (see Section 1.2). Concerning the shape of our typical singularities, there are some significative differences between the smooth and non-smooth cases. We emphasize that a typical generic singularity p of our systems is not a critical point and sometimes the trajectory passing through it coincides with {p}. A. Jacquemard, M.-A. Teixeira / Bull. Sci. math. 127 (2003) 611–633 613 In our setting the classification of those singularities are made up the contact between the vector field and H0. This classification generalizes some results presented in [2,5]. The main scope of this paper is to provide a generic classification of the elements in Ω which are reversible. The paper is organized as follows. In Section 1.4 we discuss some reversibility properties of the system. In Section 2, we state the main result of this article where we give a precise classification (see Theorem 1) of the topological types of the singularities of reversible vector fields. From these local models for regular and singular points, we derive a sufficient condition for the unicity of solutions for such vector fields; moreover we show that generic typical singularities fill up a 2 dimensional manifold of H0. Section 3 is dedicated to the proof of Proposition 1.1. In Section 4 we prove Theorem 1. In Section 5 we exhibit bifurcation diagrams and discuss the stratification of the singularities (Theorem 2). In Section 6 we discuss the behavior of some examples of non-reversible vector fields and non-generic vector fields. 1.1. First notations 4 4 Assume that H0 ={x ∈ R | h(x) = 0} is a non-singular hypersurface at 0 ∈ R .For 4 ¯ ε =±1, we denote Hε ={x ∈ R | h(x) ε > 0} and Hε = Hε ∪ H0. Without loss of generality we can fix the coordinates (x1,x2,x3,x4) in such a way that H0 is locally defined by x1 = 0. In the following 4 1. we fix a coordinate system x = (x1,x2,x3,x4) in R ; 2. we denote by h the map defined by h(x1,x2,x3,x4) = x1. Let Z be a vector field defined on Hε, ε =±1. We denote by Zε the restriction of Z ¯ ¯ to the half-space Hε. If the vector field Zε extends to Hε we denote by Zε the resulting smooth vector field. We recall that if X is a C∞ vector field, and h is a C∞ function, −−→ we can compute the function X(h) = X, grad h,where , is the standard scalar product of R4. We then iterate: Xk+1(h) = X(Xk(h)) for k 1. In the following we just write Xkh instead of Xk(h). Throughout the paper we may sometimes use the germ terminology without any explicit mention to it. 1.2. The set Ω Our aim is to consider small deformations of a standard vector field. Put weight 4 − i on = α1 α2 α3 α4 the variable xi (1 i 4). For each monomial m x1 x2 x3 x4 , we define the number = + + + R4 → R ∞ d0(m) 4α1 3α2 2α3 α4 as the weighted degree of m.Letf : be a C = =+∞ function such that f(0) 0. If all derivatives of f at 0 vanish, we put ν0(f) . Otherwise, we can expand f near 0 ∈ R4, ordering the non-zero terms of our Taylor expansion with respect to our weighted degree. For any Taylor expansion with non-zero = term, there is a minimal monomial m in the expansion and we define ν0(f) d0(m).We call ν0(f) the valuation (at 0) of f . 614 A. Jacquemard, M.-A. Teixeira / Bull. Sci. math. 127 (2003) 611–633 For each Z ∈ Ω, we write in a unique way: ∂ ∂ ∂ ∂ = + + + + Z Za,λ,µ ξ1 ξ2 ξ3 ξ4 (1.1) ∂x1 ∂x2 ∂x3 ∂x4 − with ν0(ξi ) 5 i for 1 i 4. We say that (1.1) is a primitive normal form for Z.For simplicity the elements of Ω will be called deformations of the standard vector field. One should observe that La (see [10]) and Za,λ,µ have an hamiltonian structure. We replace the hamiltonian structure by the reversible property in Section 1.4. 1.3. Orbits A solution of Z ∈ Ω is interpreted in the sense of Filippov [4]. Observe that the manifold H0 ={x1 = 0} is the discontinuity set of such vector fields. We address our attention to the H0-singularities (or simply singularities) of Z. They are those points in H0 where the vector field is tangent to H0 (defined by x1 = x2 = 0). The orbit-solutions passing through such points in this hypersurface may be not well defined (see [4] for more details). We give in Theorem 1 conditions which guarantee convenient definitions of singular orbits. Definition 1.1. Let Z ∈ Ω. We say that γ : t ∈ I → γ(t), with I open interval in R,isa simple solution of Z if: 1. t → γ(t)is continuous and piecewise C1. 2. For all t ∈ I such that γ(t)∈ /H0, Z(γ(t)) =˙γ(t).