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4Russian Journal of Nonlinear Dynamics, 2021, vol. 17, no. 1, pp. 23-37. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd210103
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 37D15
Stable Arcs Connecting Polar Cascades on a Torus
O. V. Pochinka, E. V. Nozdrinova
The problem of the existence of an arc with at most countable (finite) number of bifurcations connecting structurally stable systems (Morse-Smale systems) on manifolds was included in the list of fifty Palis-Pugh problems at number 33.
In 1976 S. Newhouse, J. Palis, F. Takens introduced the concept of a stable arc connecting two structurally stable systems on a manifold. Such an arc does not change its quality properties with small changes. In the same year, S. Newhouse and M. Peixoto proved the existence of a simple arc (containing only elementary bifurcations) between any two Morse-Smale flows. From the result of the work of J. Fliteas it follows that the simple arc constructed by Newhouse and Peixoto can always be replaced by a stable one. For Morse-Smale diffeomorphisms defined on manifolds of any dimension, there are examples of systems that cannot be connected by a stable arc. In this connection, the question naturally arises of finding an invariant that uniquely determines the equivalence class of a Morse-Smale diffeomorphism with respect to the relation of connection by a stable arc (a component of a stable isotopic connection).
In the article, the components of the stable isotopic connection of polar gradient-like diffeomorphisms on a two-dimensional torus are found under the assumption that all non-wandering points are fixed and have a positive orientation type.
Keywords: stable arc, saddle-node, gradient-like diffeomorphism, two-dimensional torus
Received February 28, 2021 Accepted March 21, 2021
This work is supported by the Russian Science Foundation under grant 17-11-01041, except of study of the dynamics of diffeomorphisms of the class under consideration supported by Foundation for the Advancement of Theoretical Physics and Mathematics "BASIS" (project 19-7-1-15-1).
Olga V. Pochinka [email protected] Elena V. Nozdrinova [email protected]
Higher School of Economics — Nizhny Novgorod
ul. B. Pecherskaya 25/12, Nizhny Novgorod, 603150 Russia
1. Introduction and formulation of results
The problem of the existence of an arc with no more than a countable (finite) number of bifurcations connecting structurally stable systems (Morse-Smale systems) on manifolds is on the list of fifty Palis-Pugh problems [21] under number 33.
In 1976, S.Newhouse, J.Palis and F.Takens [15] introduced the concept of a stable arc connecting two structurally stable systems on a manifold. Such an arc does not change its quality properties with a small perturbation. In the same year, S.Newhouse and M.Peixoto [17] proved the existence of a simple arc (containing only elementary bifurcations) between any two Morse -Smale flows. It follows from the result of G. Fleitas [8] that a simple arc constructed by Newhouse and Peixoto can always be replaced by a stable one [16]. For Morse-Smale diffeomorphisms given on manifolds of any dimension, examples of systems that cannot be connected by a stable arc are known.
Obstructions appear already for orientation-preserving diffeomorphisms of the circle S1, which are connected by a stable arc only if the rotation numbers coincide [18].
Beginning with dimension two, additional obstructions appear to the existence of stable arcs between isotopic diffeomorphisms. They are associated with the existence of periodic points [6, 20], heteroclinic intersections [13], wild embeddings of separatrices [10], etc.
On the 6-dimensional sphere, examples of source-sink diffeomorphisms are known that are not connected by any smooth arc [7], which, in fact, became the source for constructing different smooth structures on a sphere of dimension 7. For n = 2, 3 the nontrivial fact of the existence of an arc without bifurcations between two source-sink diffeomorphisms was established in [7, 19].
Polar diffeomorphisms, i.e., gradient-like diffeomorphisms with a unique source and a unique sink, are a natural generalization of source-sink systems. It follows from Morse theory that such diffeomorphisms exist on any manifolds.
In this paper, we consider the class G of polar gradient-like diffeomorphisms on the two-dimensional torus T2 under the assumption that all nonwandering points are fixed and of positive orientation type. In Chapter 2 it is established that any diffeomorphism f £ G has exactly two saddle points and is isotopic to the identity. Moreover, all diffeomorphisms of the class under consideration are pairwise topologically conjugate (see, for example, [5, 9]). Moreover, the closures of stable (unstable) manifolds of saddle points of different diffeomorphisms can belong to different homotopy classes of closed curves on the torus. Therefore, in the general case there is no arc without bifurcations connecting two diffeomorphisms of the class under consideration.
The main result of this work is the proof of the following theorem.
Theorem 1. Any diffeomorphisms f, f1 £ G can be connected by a stable arc with a finite number of saddle-node bifurcations.
2. Diffeomorphisms of class G
2.1. General properties
In this section, we establish the basic dynamical properties of diffeomorphisms f: T2 ^ T2 from the class G.
Recall that a diffeomorphism f is gradient-like if its nonwandering set Q/ consists of a finite number of hyperbolic points and the invariant manifolds of different saddle points do not intersect.
A gradient-like diffeomorphism f is called polar if the set Q/ contains exactly two nodal points, namely, one sink and one source.
Fix a system of generators of the fundamental group of torus T2 = S1 x S1:
a = S1 x {0} = (1,0), b = {0} x S1 = (0,1). Recall that the algebraic torus automorphism, L: T2 ^ T2, T2 = R2/Z2 is called the diffeomor-
f a P\
phism defined by the matrix L = , belonging to the set GL(2, Z) unimodular matrices —
SJ
integer matrices with determinant ±1. That is,
L(x,y) = (ax + /3y,Yx + 5y) (modi).
The following statement follows directly from the relationship of gradient-like dynamics with the topology of the ambient surface and the homotopy properties of the torus.
Statement 2.1. Any diffeomorphism f £ G has the following properties:
1. The nonwandering set Q/ of the diffeomorphism f consists of exactly four fixed hyperbolic points: the sink u/, the source a/, and the saddles a^, af , the closures of invariant manifolds of which are closed curves:
cf = cl WSi = WSi U af, cU1 = cl WU = WU U u/,
J 7f 7 f •> f 7f 7f •>
cf = cl Ws2 = Ws2 U af, cf = cl Wu2 = Wu2 U uf. J 7f 7 f J 7f 7f
2. There is only one choice of saddle points numbering aj, a2 and the orientation of the closures of their invariant manifolds such that the curves J1, cu2 are of homotopy type (/, vh
and the curves cf, cu1 are of homotopy type (/,vf in the basis a,b; also, J/ = ^ J
V/ v/J
is a unimodular matrix with the following properties:
a) / ^ ¡2 ^ 0,
b) v) > v/ if ¡/ = ¡f,
c) v2 = 1, if / = 0.
3. The diffeomorphism f is isotopic to the identity map.
2.2. Construction of model diffeomorphisms in the class G
In this section, for any unimodular matrix J = ( I ¡2 | such that ¡j,1 ^ ¡2 ^ 0 and v1 > v2
v1 v2
if i1 = ¡i2, we construct a model diffeomorphism fj £ G for which J/3 = J.
The simplest example of a diffeomorphism from the class G is the direct product of two copies of a source-sink diffeomorphism on the circle S1, which we denote by f0. First, we construct
a source-sink diffeomorphism on the circle. In order to do this, consider the map Fo: R —>■ R given by the formula
Fq(x) = x — sin ( 2"7T (x — \ 4n V V 4
1 —
By construction, x = j and x = j are fixed points of the map Fq on the segment [0,1]
(Fig. 1)
0.0 0.2 0.4 0.6 0.8 1.0
Fig. 1. Graph of the map Fq.
Consider the projection tt: R —> 81 given by the formula tt(x) = e2mx. As Fo is strictly increasing and satisfies the condition Fo(.r + l) = Fo(.t) + 1, there is a diffeomorphism projecting it to the circle
F0 =7TiV-1: S1 -»-S1. By construction, the diffeomorphism F0 has a fixed hyperbolic sink at the point N = n
and a fixed hyperbolic source at the point S = ir , 4 ..
Define the diffeomorphism f0: T2 T2 by the formula (Fig. 2)
fo(z,w) = (Fo(z),Fo(w)), z,w e S1.
N
N
S S
Fig. 2. Cartesian square of the diffeomorphism F0.
By construction, the diffeomorphism f0 contains a fixed hyperbolic sink at the point w = (N, N), a hyperbolic source a = (S, S) and has two saddle points a1 = (N, S), a2 = (S, N) (Fig. 3). Moreover, the closures of their invariant manifolds lie in the classes of generators a and b, namely,
f = clWS1 = S1 x{S}, 4 = clWS2 = {S} x S1,
et = cl WU1 = {N} x S1, 4 = cl WU2 = S1 x {N}.
Let fj = Jf0 J 1. We will call the diffeomorphism fj a model diffeomorphism. By construction, f& = f0.
Using the methods of [20], one can construct an arc without bifurcations from the diffeomorphism f e G to the model one, namely, to prove the following statement:
Statement 2.2. Every diffeomorphism f e G is connected by an arc without bifurcations Hf,t with the diffeomorphism fjf.
Fig. 3. Diffeomorphism f0.
3. On stable arcs of diffeomorphisms
Consider a 1-parametric family of diffeomorphisms (an arc) pt: M — M, t e [0,1]. An arc pt is called smooth if the map F: M x [0,1] — M defined by the formula F(x, t) = pt(x) is smooth. The smooth arc pt is called a smooth product of the smooth arcs p^ and p"2 such that
{p\T[t), 0
p1 = p0, if pt = < where t : [0,1] — [0,1] is a smooth monotone map such
that, r(t) = 0 for 0 < t < | and r(t) = 1 for | < t < 1. We will write pt = pj * p\.
Following [16], an arc pt is called stable if it is an inner point of the equivalence class with respect to the following relation: two arcs pt, p't are called conjugate if there are homeomorphisms h: [0,1] — [0,1], Ht: M — M such that Htpt = p'h{tt)Ht, t e [0,1] and Ht continuously depend on t.
In [16] it is also established that the arc {pt}, consisting of diffeomorphisms with a finite limit set, is stable iff all its points are structurally stable diffeomorphisms with the exception of a finite number of bifurcation points, pbi, i = 1,...,q such that pbi:
1) has no cycles;
2) has a unique nonhyperbolic periodic orbit which is a noncritical saddle-node or flip;
3) the invariant manifolds of all periodic points of the diffeomorphismpbi intersect transver-sally;
4) the transition through pbi is a generically unfolded saddle-node or period-doubling bifurcation, with the saddle-node point being noncritical.
Recall the definition of the generically unfolding arc pt through the saddle-node or flip. We give the definition for a fixed nonhyperbolic point in the case where it has a period k > 1. A similar definition is given for the arc pk.
An arc {pt} £ Q unfolds generically through a saddle-node bifurcation (Fig. 4) if in some neighborhood of the nonhyperbolic point (p,bi) the arc pt is conjugate to
ipjix 1,X2, Xi +nu , x2+nu Xn) = [xi + + t, ±2x2
2
,± 2x
1+nu
±X2+n
where (xi,...,Xn) e Rn, \xi\ < 1/2, \t\ < 1/10.
\ k \ k
Yi A \
\ \ k
Yt y \
Fig. 4. Saddle-node bifurcation.
In the local coordinates (x1,... ,xn,t) the bifurcation occurs at time t = 0 and the origin O e Rn is a saddle-node point. The axis Ox1 is called a central manifold WO, the half-space {(xi,x2, ...,xn) e Rn : xi ^ 0, x2+nu — ... — xn — 0} is the unstable manifold WO, and the half-space {(x1,x2,... ,xn) e Rn : x1 ^ 0, x2 — ... — x1+nu — 0} is the stable manifold WO of the point O.
If p is a saddle-node point of the diffeomorphism , then there exists a unique - invariant foliation Fpss with smooth leaves such that dWpS is a leave of this foliation [11]. Fs is called a strongly stable foliation (Fig. 5). A similar strongly unstable foliation is denoted by Fuu. A point p is called s-critical if there exists some hyperbolic periodic point q such that WU nontransversally intersect some leaf of the foliation F^s; u-criticality is defined similarly. Point p is called
- semicritical if it is either s- or u-critical;
- bicritical if it is s- and u-critical;
- noncritical if it is not semicritical1.
Fig. 5. Strongly stable and unstable foliations.
xFor the first time, the effect of arc instability in a neighborhood of a critical saddle was discovered in 1974 by V. Afraimovich and L. Shilnikov [1, 2]. The existence of invariant foliations Fs
proved earlier in the work of V. Lukyanov and L. Shilnikov[12].
F^u was also
2
u
2
2
In particular, for M2, the noncriticality of the saddle-node point means that the saddle-node separatrix does not intersect with the one-dimensional manifold of the saddle-node point. The two-dimensional manifold of a saddle-node point must intersect the transversally invariant layer (Fig. 6).
0"H
Vi
N
y
TPUU bpl
Fig. 6. pi point.
s-critical saddle-node point, p2 — «-critical saddle-node point, p3 — noncritical saddle-node
4. Construction of a stable arc between model torus diffeomorphisms
4.1. Construction of auxiliary functions
In this section, we construct model functions that will later be used to construct a stable arc. The construction is based on the principle of gluing infinitely smooth functions by means of the following sigmoid function.
Let a < b and 5a;b: R ^ [0,1] be a sigmoid function defined by the formula (Fig. 7)
Í0,
1
àa;b(x) = <
1 + exp
(o + b)/2 - X (x — a)2(x — b)2
x ^ a, a < x < b,
x > b.
1
Fig. 7. Graph of the sigmoid function.
Define the function <f>l : R —>■ R by the formula (Fig. 8)
faix) = sin (67r ~ \
Define the function g1: R —>■ R by the formula (Fig. 9)
0*0,
0 < x < 0.26,
9i(x) = <
(1 - $0.26-,0.27{x))<f>0{x) + 60.26-,0.27{x)<f>i{x), 0.26 < X < 0.27,
faix), 0.27 < X < 0.76,
(1 - $0.76;0.77(x))(f)i(x) + 60.76;0.77(x)(t)0(x), 0.76 < X < 0.77,
j>o(x), 0.77 < X < 1.
Define the function <j)2: R —;► R by the formula (Fig. 10)
$2(x) = x + -^ Sin Qtt ^ - y2
Define the function g2: R —;► R by the formula (Fig. 11)
'9i(x), 0 < x < 0.42,
(1 - $0A2-,0As{x))gi{x) + 6oA2-,OAs{x)<f>2(x), °-42 < x <
Mx),
92(x) =
0.43 < x < 0.98,
(1 - ¿0.98;0.99 0*0)02 0*0 + $0.98;0.99(x)gi(x), 0.98 < X < 0.99,
Jl(x),
4.2. Construction of the model arcs
0.99 < x < 1.
In this section, we will construct arcs which are the main components making up an arc HJt. (i 0\
For n e Z let Jn = .
Vn V
Lemma 1. The diffeomorphism f0 is connected with the diffeomorphism fj1 by a stable arc H01, t with two of genetically unfolding noncritical saddle-node bifurcations.
Proof. In this proof, the bar-free mappings are projections on S1 by n of the bar-mappings given on the line R. The stable arc H0t1 ,t, connecting the diffeomorphism f0 with the diffeomorphism fj1 is the product of the arcs r£, r2, constructed in step 1 and step 2 below, and the arc Hr2, t.
Step 1. First saddle-node bifurcation.
1. The birth of a saddle-node point. We start with the diffeomorphism f0: T2 — T2 defined by the formula
fo(z,w) = (0o(z),0o(w)), z,w e S1.
Let and
= (! - t)Mx) + Wi(x), x E R, te [0,1]
^0?) = (l-r)r?tV)+T</>o0r), xeR, t E [0,1], t E [0,1].
Fig. 8. Graph of the map 4>i{x)- Fig- 9. Graph of the map g 1(x).
1.0
1.0
0.8 0.6 0.4 0.2 0.0
0.0 0.2 0.4 0.6 0.8 1.0
Fig. 10. Graph of the map <f>2(x).
0.8 0.6 0.4 0.2 0.0
0.0 0.2 0.4 0.6 0.8 1.0
Fig. 11. Graph of the map
Define a smooth arc Hj: T2 — T2, t £ [0,1] by the formula
(mz),vl\8x-2\(w))> z = 7t(x), w£§\
H (z,w) =
fo(z,ui), z = tt(x), weS1.
For t = j, the diffeomorphism H\ has a saddle-node point p = (N, 7r(0)) whose stable ^ d
manifold is diffeomorphic to a half-plane whose boundary is arc jp (Fig. 12). 2. Rotation of the separatrix of the saddle
Consider the fundamental domain K =
n(0),n
4n
x S1, a restriction of the diffeomor-
phism f0 to V =
1
7ri"lJ'7rV4
x S1. Let V = V/f0. Then V is a torus obtained from K
1
Fig. 12. Isotopy H on the torus.
by identifying the boundaries with the map f0. Denote by q: V ^ V the natural projection. Let 72 = q(WU2 n V) and Vi = q(WS1 n V). Since for all t G [0,1] the diffeomorphism Ht coincides
with f0 on the annulus correctly.
Let W =
.vrll
x S1, it follows that the circle VP = q(YP n K) is defined
n -
n
n -
n
and W = p(W). By construction, the circle
Vp divides the annulus W into two annuli, the closures of which are denoted byW1, W2, assuming that 71 C W1 and V2 C W2 (Fig. 13).
Choose a circle V C int W1 that is not homotopic to zero, transversal to the projection of a strongly stable foliation of a saddle-node point. Such a curve always exists, since the projection of each layer of this foliation is a curve wrapped around a knot 71 (Fig. 13). According to [3, 14], there exists a diffeomorphism h1: V ^ V smoothly isotopic to the identity such that h-1 (V2) = V.
For Xi <E
let Ki =
7r(xi)] (tt^q1 (x^) xS1. Choose an open cover D = {D\,..., D^}
of the torus T such that, the connected component Di of the set q (A) is a subset of I<i for some Xi < (f)Q1(xi-i). According to the fragmentation lemma [4], there exist diffeomorphisms w1,..., wkl: T2 ^ T2 smoothly isotopic to the identity, with the following properties:
i) for each i G {1,...,k1} there exists a smooth isotopy {iVi,t} which is the identity outside Di and which joins the identity and toi;
ii) h = t?1 ...WVk1.
Let wiit: R2 ^ R2 be a diffeomorphism that coincides with (q\Ki)-1wi,tq on Ki and coincides with the identity map outside K. Let
Zt = W1,t . ..Wki,tf
G1 =
C2t,
Z1,
^ t ^ 1.
x
4
1
2
cu
Fig. 13. Curve 7.
3. Combining isotopies Hj and Gj.
Define a smooth arc : T2 — T2, t e [0,1] by the formula (Fig. 16)
H^(z,w), z = n(x), x G
1 3
8'8
we S1,
r1 (z, w) = <
Gl(z,w), z = tt(x), ,T6 oj, w e S1,
fo(z,w), z = n(x), x G
1
4
5 1 ~8'~4
U
, w G S1
Step 2. Second saddle-node bifurcation. 1. Merging saddle and node points
For all t G [0; 1] let rj'f(x) = tgi(x) + (1 - t)Jn(x), .teR and
t(.T) = (l-r)#)lr4W, xeR, t g [0, l], re [0,1].
1
8
Fig. 15. Isotopy G\ on the torus. ш
02 02
Fig. 16. Isotopy Г on the torus
Define a smooth arc Hf: T2 ^ T2, t £ [0,1] by the formula
(Фо (z) j Vt,\8x—2\ (z))j Z = 7T(X), WES1,
H?(z,w) = { , 5 1 ^ V 7 ^
Ti(z,w), z = tt(x), ^(^-g'gj' w eS .
The arc Hf realizes the merging of the sink uj and the saddle a\ into the saddle-node point p and its further disappearance. Denote by (3p the boundary of the stable manifold of a saddle-node p.
2. Rotation of the separatrix of the saddle a2.
Since for all t £ [0,1] the diffeomorphism Hf coincides with f0 on the annulus K, it follows that the circles (32 = q(WU2 П K), /3i = q(W~ П K) and = я(вр П K) are defined correctly.
Let W3 be a neighborhood of the curve 71, then choose a smooth nonzero homotopic curve 7 C W3, transversal to the projection of a strongly stable foliation of a saddle-node point. Such a curve always exists, since the projection of each layer of this foliation is a curve wrapped around the knot /1 (we construct in the same way as in Step 1). According to [3] and [14], there exists a diffeomorphism h2: 7 ^ 7 smoothly isotopic to the identity such that h2(/2) = 7 and hz(7i) = 71.
Choose an open cover U = {U1,...,Uk2} of the torus T2 such that the connected component Ui of the set q~l(Ui) is a subset of Ki for some Xi < <fi0 (xi-1). According to the fragmentation lemma [4], there exist diffeomorphisms vj1,...,vjk2: T2 ^ T2 smoothly isotopic to the identity, with the following properties:
i) for each i e{1,..., k2} there exists a smooth isotopy {7ji,t} which is the identity outside Ui and which joins the identity and 7i;
ii) h2 = 71 ...7fc2.
Let vi,t: R2 ^ R2 be a diffeomorphism that coincides with (q\Ki)-17i,tq on Ki and coincides with the identity map outside Ki. Let
Ct = vltt ...vk2 ,tri, G\ =
6,
3. Combining isotopies and
Define a smooth arc Tt2: T2 ^ T2, t e [0,1] by the formula (Fig. 17)
Ht(z,w), z = n(x), x G
1 3
8'8
we S1,
r?(z,w) =
Gt(z,w), z = tt(x), w e S1,
fo(z,w), z = n(x), x G
1
4
5 1 ~8'~4
U
1
8
, w G S1.
According to statement 2.2, the diffeomorphism r2 can be connected by an arc without bifurcations Hr2 t with the diffeomorphism fj1. □
Denote by Hn,n+1 ,t the arc with two saddle-node bifurcations connecting the diffeomorphisms fJn, fjn+1 and given by the formula
Hn,n+1, t = Jn Ho ,1 , tJn 1.
4.3. Arc construction algorithm Hj,t
In this section, using the model arcs constructed above, we will prove the following lemma.
Lemma 2. The diffeomorphism fj is joined by a stable arc Hjtt with a finite number of generically unfolding noncritical saddle-node bifurcations with the diffeomorphism f0.
Proof. Let J = {-. be a unimodular matrix such that ¡i1 ^ ¡i2 ^ 0 and v1 > v2 \ v1 v2 I
if i1 = ¡2. Consider the following possibilities for the matrix J: 1) ¡2 = 0; 2) i1 = ¡2 = 1; 3) i2 > i1 > 0. Construct the arc Hjtt in each case separately.
(b )t-
Fig. 17. Isotopy Tl on the torus.
In case 1) J = Jn. If n > 0, then HJn,t = Hn-l,ntl-t * ... * H0llll-t is the required arc. If n < 0, then HJn,t = JnHj_n>l-t J-1 is the required arc.
In case 2) Hjt = JHj_1l-tJ 1 * Hj 2 is the required arc.
In case 3) applying Euclid's algorithm to the pair ¡l,j2 generates a sequence of natural numbers nl,..., nm, kl,...,km such that ¡l = nlj2 + kl, j2 = n2kl + k2, kl = n3k2 + + k3,..., km-2 = nmkm-l + km, where km-l = 1, km = 0. Let k-l = ¡l, ko = ¡j2. Then the sequence k-l,k0,kl, ...,km satisfies the recurrence relation
ki+l = ni+lki - ki-l, i = 0,...,m - 1.
Let l-l = vl, l0 = v2 and define the sequence l-l,l0,ll,...,lm by the recurrent relation
li+l = ni+lli - li-l, i = 0,...,m - 1.
(ki-1 kA ^ l
, i = 0,...,m. Then the arc Fitt = J^i-lHJ_n,,tJi-l, i = 1,...,m joins
li-1 li J
diffeomorphisms fLi_ 1 and fLi and contains 2ni noncritical saddle-node bifurcations. Since
¡Lm = fj ,, it follows that Hjt = Fi,t * ... * Fm,t * Hj1 is the required arc.
□
References
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[17] Newhouse, S. and Peixoto, M.M., There Is a Simple Arc Joining Any Two Morse-Smale Flows, in Trois études en dynamique qualitative, Astérisque, vol. 31, Paris: Soc. Math. France, 1976, pp. 15-41.
[18] Nozdrinova, E. V., Rotation Number As a Complete Topological Invariant of a Simple Isotopic Class of Rough Transformations of a Circle, Russian J. Nonlinear Dyn., 2018, vol. 14, no. 4, pp. 543-551.
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