Krych, Michał

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AstérisqueMICHAŁ KRYCHTwo remarks on structural stability of planedynamical systemsAstérisque, tome 50 (1977), p. 197-204<http://www.numdam.org/item?id=AST_1977__50__197_0>© Société mathématique de France, 1977, tous droits réservés.L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/Publications/Asterisque/) implique l’accord avec les conditions générales d’uti-lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ouimpression systématique est constitutive d’une infraction pénale. Toute copieou impression de ce fichier doit contenir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/Société Mathématique de France Astérisque 50 (1977) p.197-204 Two remarks on structural stability of plane dynamical Systems by Mi chai Krych We shall discuss some questions of structural stability of dynamical Systems on the plane. The space of diffeomorphisms or vector fields on open manifold can be endowed with C r Whitney topology, r > 0 . In the case of the plane this topology is defined as topology generated by ail sets B(f, £ ,r) ={gs [dif(x)-dig(x)/< <£(x) for i*t0,1, •••,rj], f is a diffeomorphism or 2 vector field, r is a non-negative integer, £ :R — * R is a continuous positive function* Définition 1. A diffeomorphism f is structurally stable ( Cr-structurally stable, r^0)iff there is a neighbourhood (C neighbourhood) B(f f E ,r) of f such that if geB(f, £ fr) then f and g are topologicàlly équivalent i.e # there is a homeoraorphism h: R — * R such that h*f « goh. Définition 2, A vector field X is C r structurally stable iff there is a C r neighbourhood B(X, £ ,r) such that if Y6B(X, *,r) then X and Y are topologicàlly équivalent i.e, there is a homeomorphism h: R •—ïR which carries X-orbits onto Y-orbits. There is a question raised several years ago whether C -structural stability is équivalent to C -structural stability for k^n, k V U (for dynamical Systems 197 M. KRYCH on closed manifolds). No answer is known up to now for the compact case. In the first part of the paper we shall construct (̂ -structurally stable diffeomorphism 2 2 of R which is not C structurally stable* Our diffeomorphism has a non-hyperbolic fixed point, thus hyperbolicity of fixed points is not necessary for stability on open manifolds. We recall that for Cr stability, r>1, of diffeomorphisms of closed monifolds hyperbolicity of periodic orbits is a necessary condition, see £5], but, as we know, the analogous theorem for vector fields on closed manifolds has not been proved yet and it is possible that it is not true. In the second part of the paper we shall deal with non-vanishing vector fields on the plane. We shall state a theorem analogous to the well-known theorem of Kupka and Smale and we shall use it to show that there is an open non-empty set of structurally unstable non-vanishing 2 vector fields on R . I Let P : R~-̂ R be defined by the formula P(x) = = -x - ql<x3-x5)(1+x2r2*5 and let r(x) = F(x)+x. One can easily check that |r(x)|^ y and lr'(x)| ̂  £ for every real number x. Thus P ' ( X ) = -1+r'(x) < 0. This inequality implies that P is an orientation reversing diffeomorphism of R. Therefore P has exactly one fixed point, namely 0# P'(0) = -1, i.e. { o j i s not a hyperbolic fixed poinx.Because of r(x) and x have *e same sign for for x sufficiently close to 0,{Oj is a repellor. All other non-wandering points of P are periodic points of minimal period 2. It is easy to check that P has only one periodic orbit different from [0^, namely the orbit consisting of -1 and 1. Thus Si ( P ) ={-1,0,1}. 198 STRUCTURAL STABILITY OF PLANE SYSTEMS Let P^x) = P(x) + 224^(1+x 2)" 2 # 5 and P^(x) * P(x)-- I x O + x 2 ) - 2 * 5 . It is easy to see that Jl(^) -- {-% - h 0, 1, f j , {0} and {-% f} are fcyperfcolic attractors, ̂ - lj iS a hyperbolic repellor. Analogously jl(P^) = {-tf?, 0 , { ? 3 t {0} i s a hyperbolic repellor and {-l/^f?} is a hyperbolic attractor. Thus P1 and P are C structurally stable. It has been proved in Q?J and also in [3 J that if G is sufficiently close to P in topology, then either G is equivalent to or G is topologically equivalent to P .. ~ 1 2 Now we can describe the diffeomorphism of R which 2 2 we are looking for. Let f: R—^R be a diffeomorphism such that (i) all lines L n » {(x,y)$R : 7*n}9 n is an arbitrary integer are f-invariant and normally hyperbolic, L 2 n are repellors while I»2n+1 a r e a * ^ r a c * o r s t (ii) fj L = P, f. = psgrln for ̂  î eger n̂ O, (iii) f. is an orientation reversing diffeomorphism I 2n+1 which has exactly one non-wandering point which is a hyperbolic attractor, (iv) the stable manifold of the fixed point lying on I»2n+1 i s egu a* to the open region bounded by the lines If g is sufficiently close to f in the C -topology then there are lines L (g) which are g-invariant and 3 n 3 C close to L • It is clear that g,_ , x is C close to n l L n ( g ) f. T for any integer n. Thus g. T , v and f are lLn l Ln ( g } l Ln topologically equivalent for any n^O. If f, _ and gi_ , N 199 M. KRYCH are topologically equivalent, then it is possible to construct a cĉ Sugacy carrying L^s onto Ln(g)'s. If tj L is not topologically equivalent to S^g) then conjugacy carrying Ln, n arbitrary integer, onto *»n_2(g) can be constructed. We omit details because it is well-known how such conjugacies can be defined. 2 2 f is not C structurally stable because in any C neighbourhood of it there is a diffeomorphism which has uncountably many of periodic points and therefore it is not topologically equivalent to any Kupka-Smale diffeomorphi Similar diffeomorphisms can be constructed on many other open manifolds. The author does not know whether each open connected manifold of dimension not less than 2 admits such diffeomorphism. The author strongly believes that all 2 periodic points of C structurally stable diffeomorphism must be hyperbolic. The analogous construction can be made for vector fields, also on some two dimensional manifolds (non-orientable) • II. Let Wr be the set of all Cr vector fields on R2 which have not any non-wandering point. By the Poincare-Bendixon theorem, W1* consits of all Cr non-2 r vanishing vector fields on R . Therefore W is open in the 2 set of all vector fields on R and thus it has the Baire property. Thus it is sensible to investigate structural r stability of elements of W . Recently Z.Nitecki proved 2 that R and R are the only manifolds on which the set of all vector fields without non-wandering points is open, so on oth*r connected manifolds we are to deal with another set. Let Oj.(a) denotes the X-orb±t of a, 0*(a) and 0~(a) denote positive and negative semi-orbits of a. We write 200 STRUCTURAL STABILITY OF PLANE SYSTEMS Ox(a) < Ox(b) iff Ox(a) ̂  (>x(b) and for any neighbourhoods a and V ab there is a point c^U such that ^(c) n V / 0. The pair <01f02> of orbits of vector field X such that either 0 < 0£ or 02<01 is called a pair of séparatrices, its elements are called séparatrices, if 0^< 02, then 0̂  is called the beginning of ^ ^ t ^ ) and 02 is called the end of ̂ 0^,0^. In £2]J the following theorem is proved: Density Theorem. The set of X6 Wr, r^1, such that there are no 3 points a,b,c such that 0x(a) < ^ °2^c) is residual therefore dense in If • In the prove of this theorem the countability of the set of pairs of séparatrices is used. This is not time on other manifolds, there exist vector fields on 2-manifolds which have uncountable many pairs of séparatrices. Another difficulty in generalizing of this theorem is that W is not open on other manifolds (Nitecki's result) and therefore we must be much more carefull when perturbing vector fields. As an application of this theorem we shall shaw that x» there is an open non-empty subset of W , r ̂.1, consisting of C structurally unstable vector fields. r 2 Let XfcW be a vector field on R which has the following properties: (1) X is invariant under the translation (x,y)-—^ ~-*(x,y+1) and transversal to the x-axis , (2) the orbit of the point (—,0), n is a non-negative 2n integer, k is an odd integer, is the beginning of the pair of séparatrices the end e of which lies in the nt k 2 half-plane y\n, the component G of R \ e which does not contain 0Y((—,0)) is convex and it is symmetric with 2** 201 M. KRYCH respect to the vertical straight line L . which contains n y le the unique lowest point of e , n , JtC (3) X restricted to an arbitrary G ,is diffeomorphically n fK equivalent to X restricted to |(x,y): x^Oj, (4) 0x((^f0)) is the end of a pair of séparatrices the beginning of which is contained in J(x,y): y^ (5) X restricted to [(x,y): y<0} is diff eomorphically equivalent to X restricted to G Q Q. Let B be the union of the beginnings of all pairs of séparatrices and let E be the union of the ends of all pairs of séparatrices. By the definition of X the set BA{(x,y): y=o} is dense in x-axis, thus B is dense in R 2. It can be proved, see £23, that pairs of séparatrices are stable in the following sense: if T̂  and T g are open transversal sections of vector field Y é W r. <>0. a e l . beT 2, 0y(a) < 0y(b) , C &CO Y(a) is a compact arc, C b c dOy(b) is a compact arc, then there is a C1 neighbourhood N of Y such that if Z £N, then there are points a(Z) £ T̂  and b(Z)£T2 such that (>z(a(Z)) < Oz(b(Z)), (\a( Z)-al| <£ , l(b(Z)-bl!<£ , C & is £~C1-close to Oz(a(Z)), C b is ^-C1-close to 0_(b(z)). Using this fact and applying the method found by Peixoto and Pugh one easily proves that for any Z sufficiently close to X in the C topology B(Z)i^|(x,y): y=OJ is dense in x-axis, B(Z) is theunion of all beginnings of pairs of séparatrices of vector field Z and that at least one end of a pair of séparatrices meets x-axis. This implies that in some neighbourhood of X the vector fields satisfying the assumptions of Density Theorem are dense and the vector fields which do not satisfy these assumptions are also dense in this neighbourhood. Of course no vector field of the first type 202 STRUCTURAL STABILITY OF PLANE SYSTEMS is équivalent to the vector field of a second type and therefore no vector field in this neighbourhood of X is structurally stable. The author has almost proved that X ^ W f r ^,1, is structurally stable iff B(X) A E(X) = 0 (necessity is trivial). Partial resuit in this direction has been obtained by J.F.Collins in his thesis, who proved under the additional assumption sufficiency of our condition (it was done independently). Another question is whether C structurally stable vector fields (or even)C° structurally stable) vector fields are C dense in The author believes that it is true, note that C 1 and C° structural stability are not 2 —1 5 2 équivalent e.g. the vector field (1+x ) " ((1-x ) |^ -- x(x -4) — ) is C structurally stable but not o C structurally stable, because the trajectory passing through the origin is the beginning of two différent pairs of séparatrices. Eeferences. 1 J.F.Collins. Structural Stability of Completely Unstable Flows in the Plane, dissertation, Tufts University, October 1 9 7 7 . 2 M.Krych , A Generic Property of Non-vanishing Vector Fields on R . Bull. Acad.Polon. Sci, Ser.Sci .Math. Astr,Phys., XXV.4,361-368,1977. 3 M.Krych, C^-Structurally Stable Diffeomorphism of the Plane which is net C -Structurally Stable, to be published in Bull.Acad.Polon.Sci.Ser.Sci.Math.Astr.Phys.,1977. 4 Z.Nitecki, Explosions in Completely Unstable Flows, preprint 5 C.R.Robinson, C Structural Stability Implies Kupka -Smale, Dynamical Systems, Proceedings of the Symposium 203 M. KRYCH in Salvador 1971, New York and London 1973, Academic Press. 6 J.Sotomayor, Generic one-parameter families of vector fields on two-dimensional manifolds, Publications Mathématique,№ 43,1973,IHES, 5 - 4 6 . Michal Krych Department of Mathematics Patac Kultury i Nauki IXp 00-901 Warsaw Poland 204 

Two remarks on structural stability of plane dynamical systems

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Two remarks on structural stability of plane dynamical systems

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