ONE-DIMENSIONAL REACTION-DIffUSION EQUATIONS AND SIMPLE SOURCE-SINK ARCS ON A CIRCLE Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2018, vol. 14, no. 3, pp. 325-330. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd180303

MSC 2010: 37D15

One-Dimensional Reaction-Diffusion Equations and Simple Source-Sink Arcs on a Circle

O. V. Pochinka, A. S. Loginova, E. V. Nozdrinova

This article presents a number of models that arise in physics, biology, chemistry, etc., described by a one-dimensional reaction-diffusion equation. The local dynamics of such models for various values of the parameters is described by a rough transformation of the circle. Accordingly, the control of such dynamics reduces to the consideration of a continuous family of maps of the circle. In this connection, the question of the possibility of joining two maps of the circle by an arc without bifurcation points naturally arises. In this paper it is shown that any orientation-preserving source-sink diffeomorphism on a circle is joined by such an arc. Note that such a result is not true for multidimensional spheres.

Keywords: reaction-diffusion equation, source-sink arc

1. Introduction

A wide range of models emerging in physics, biology, chemistry, etc. is described by the one-dimensional reaction-diffusion equation. The corresponding chain of connected maps (CCM) or discrete variants of this equation can be regarded as an infinite set of copies of the local dynamical system on the phase space R or its compactification S1. If the local dynamics exhibits hyperbolic behavior, then the dynamics of the entire CCM is completely determined by local dynamics with sufficiently small parameters (see, for example, [5]).

So the Kolmogorov-Petrovsky-Piskunov (KPP) model of the development of useful genes is described by a local map f (u) = u + yu(1 — u). For any y e (0,1) the compactification map f is a Morse-Smale diffeomorphism of the circle S1 with four fixed points (see [5, Theorem 5.1.1]).

Received May 30, 2018 Accepted June 23, 2018

This study was carried out within the framework of the RFBR project 18-31-00022 and the Basic Research Program at the National Research University Higher School of Economics (HSE) in 2018.

Olga V. Pochinka olga-pochinka@yandex.ru Anastasia S. Loginova n.logi@mail.ru Elena V. Nozdrinova maati@mail.ru

National Research University Higher School of Economics (HSE) ul. B. Pecherskaya 25/12, Nizhny Novgorod, 603150, Russia

The Nagumo model of propagation of a voltage pulse through the neuron axon is described by a local map f (u) = u — Au(u — 0)(u — 1). For all sufficiently small A the compactifica-tion map f is a Morse-Smale diffeomorphism of the circle S1 with four fixed points (see [5, Theorem 5.1.1]).

The real amplitude equation of waves generated by the wind is described by a local map f (u) = Au — bu3. If b > 0 and A G (0,1) U tQ , then the compactifying map / is the Morse —

Smale diffeomorphism of the circle S*1 with four fixed points at A e and two fixed points

at A e (0,1) (see [5, Theorem 5.3.1]).

The control of the dynamics of the described models reduces to the consideration of a continuous family of maps of the circle. This raises the question of the possibility of joining two maps of the circle by an arc without bifurcation points. In this paper we show that such an arc joins any orientation-preserving source-sink diffeomorphisms on the circle. In more detail.

Let us consider the circle S1 = {(x1,x2) e R2: xf + xf = 1}. We denote by Diff (S1) a set of diffeomorphisms on a given circle and by J(S1) C Diff(S1) a set of orientation-preserving source-sink diffeomorphisms, i.e., diffeomorphisms whose nonwandering sets consist of two hyperbolic points: a source and a sink.

Theorem 1. For any diffeomorphisms f, f e J(S1) there is a smooth arc {ft e J(S1)} joining them.

We note that the multidimensional reaction-diffusion equations reduce to a source-sink diffeomorphism on a multidimensional sphere for which the result of Theorem 1 is not true (see, for example, [2]).

2. Supporting concepts and facts

Two diffeomorphisms f, f': S1 — S1 are smoothly isotopic if there is a smooth map F: S1 x [0,1] — S1 (smooth isotopy) such that ft given by the formula ft(x) = f (x,t) is a diffeomorphism for each t e [0,1] and f0 = f, f1 = f'. We say that the family {ft} is a smooth arc joining diffeomorphisms f and f'. A support of the isotopy {ft} is the closure of the set {x e S1: ft(x) = fo(x) for some t e [0,1]}.

For the smooth arcs {ft} and {gt} such that f1 = g0 the usually product of the respective paths is not smooth in general, but we can define the smooth arc {ht} joining diffeomorphisms f0, g1 as follows:

ht =

f'2r(t)) O^i^i

9'2r(t) — l i

where r: [0,1] —> [0,1] is a smooth monotone function such that r(t) = 0 for 0 ^ t ^

and r(t) = I for | ^ i ^ I. We will call the smooth arc {ht} a smooth product of the smooth

arcs {ft} and {gt} and denote it by {ht} = {ft * gt}.

In the present paper we will work with the class J (S1) C Diff (S1) of orientation-preserving source-sink diffeomorphisms, that is, diffeomorphisms whose nonwandering sets consist of precisely two hyperbolic points: a source and a sink. Also, we introduce a subclass NS(S1) C J(S1) for which the source is the North pole N(0,1) and the sink is the South pole S(0, —1).

Define the diffeomorphisms

tf- : S1 \{S} — R, tf+ : S1 \{N} — R (the stereographic projections) by the formulas:

'(?_(ж1,ж2)= Xl

1+ X2

ti+{Xi,X2) = Xl

1 - X2

Define the diffeomorphisms

which are given by the formulas:

tf-1 : R — S1 \{S},

tf;1 : R — S1 \{N}

\x2 + 1 x2 + 1

^K^TT'^TT

\x2 + 1 x2 + 1

Through stereographic projections, diffeomorphisms from NS (S1) are associated with the contractions on the line

S(R) = f$+-1,f G NS(S1)}

and extensions on the line

N (R) = {ïï-fïï--1, f G NS(S1)}. We define model diffeomorphisms g: S1 —> S1, g: R —> R by the formulas

4x1 5x2 — 3N

g(x1,x2) =

5 — 3.Ï2 ' 5 — 3a?2

9(x) = (J

It is easy to verify that g G NSiS1), g e S"(R), g~l e N{R), g~l = d-gdZ1 and g = d+gi)^1.

Let Eg d NS(S1) be a set of such diffeomorphisms h for which there are neighborhoods Vh{N), Vh{S) of points N, S such that h\Vh{N)uVh{S) = g\vh(N)uvh(S)- We denote by Eg С S(R) {Eg-1 С N(R)) a set of diffeomorphisms h for which there is a neighborhood V^(0) of the point О such that h\vT(0) = TS\vT(0) {h\vT(0) =g~1\v-K(0))-

3. The construction of a smooth arc is a proof of Theorem 1

It is easy to prove Theorem 1 if f and f1 smoothly conjugate by a diffeomorphism h. In this case there is a smooth isotopy ht: S1 ^ S1 such that h0 = id, h1 = h. Thus, ft = h-1fht is a required arc. In the opposite case we show how to construct a smooth arc {lt £ J(S1)} joining an arbitrary diffeomorphism f £ J(S1) with the model diffeomorphism g. Similarly,

1 1 o

2. lt is a diffeomorphism from the class NS(§ ) for all is -r, , l2 G Eg (see Proposition 2);

J9

3

we construct a smooth arc {l't G J(S1)} joining f ' and g. Then the smooth product of smooth arcs ft = lt * l'1-t is the required arc. The arc lt is a smooth product of tree smooth arcs /3t, Yt, 5t (lt = I3t * Yt * ^t), whose construction is given in the corresponding sentences below. Thus,

I. It is a diffeomorphism from the class J^S1) for all t G 0, , Iq = / and l-y is a diffeomor-

3

phism from the class NS (S1) (see Proposition 1);

I / a _ _

_3' 3.

r 2

3. It is a diffeomorphism from the class Eg for all is I , l\ = g (see Proposition 3).

Proposition 1. For any diffeomorphism f G J(S1 ) there exists a smooth arc {It G (J(S1 ) n Diff (S1))} joining |o = f with a diffeomorphism |i G NS(S1).

Proof. Let f G J(S1 ). We denote by a the source and by u the sink of the diffeomorphism f. Let Da, D^ (Ds, Dn) be pairwise disjoint arcs of a circle containing a, u (S, N). According to [3], there exists a smooth arc {Ht G Diff (S1)} such that H0 = id, H1(DN) = Da, H1(Ds) = Dw, H1(N) = a and H1(S) = u. Then (!t = H-1fHt is the desired isotopy joining the diffeomorphism f = ¡3o with the diffeomorphism ¡3\ = H^1fHi G NSiS1). ■

Proposition 2. For any diffeomorphism I G NS(S1) there exists a smooth arc {Yt G NS(S1)} joining y0 = | with a diffeomorphism Y1 G Eg.

Proof. Define ]3+ = ■d+f3'd+~1 and /T = d-fSd-'1. Then ]3+ G S(M) and /T G N(R).

If ¡3+ = 7j in some neighborhood of the origin O, then we define {y^ = ¡3+}. In the opposite case

we show below how to construct a smooth arc {y^ G S'(R)} joining ¡3+ with a diffeomorphism

Yi G Eg such that yt f°r every t S [0,1] coincides with ¡3 out of a neighborhood V+ C B1 =

= {igR: ^ 1} of the origin O. Similarly if ¡3 = Tj~l in some neighborhood of the origin O,

then we define {y7 = P }• In the opposite case, similarly to yt we construct a smooth arc

{Yt G N(R)} joining ¡3 with a diffeomorphism yÏ S E^-i such that Yt f°r every t G [0,1]

coincides with ¡3 out of a neighborhood V~ C B1 of the origin O. Thus, the required arc yt given by the formula

ftf+-1(Yt+ (#+(w))), w G tf+-1(V+);

Yt(w) = I "&--1(y-(#-(w))), w G "&--l(V-);

[p(w),w G (S1 \ (tf+-1(V+) U V--1(V-)).

Let us construct a smooth arc {y^ G S'(R)} joining j3+ with a diffeomorphism Y\ s Eg such that y t f°r every t G [0,1] coincides with j3+ out of a neighborhood V+ C B1. The arc yt will be a smooth product, of arcs: and ^, where

f) G S*(R)} joining the diffeomorphism ¡3 with a diffeomorphism pf such that pf for every t G [0,1] coincides with ¡3+ out of a neighborhood V-^ C B1 of the origin O and ~pf coincides with differential Dj3% (denote it by Q) in a neighborhood V2+ C of the origin O;

2) {¿¡^ G S*(R)} joining the diffeomorphism p^ with a diffeomorphism G Eg such that ^ for every t G [0,1] coincides with p^ out of a neighborhood V3+ C V2+.

1) Since O is a hyperbolic sink for diffeomorphism ¡3+, it follows that

Q(x) = Ax

for some 0 < A < 1. Since the map ¡3+ in some neighborhood of the origin has the form 3 (x) = x(A + a(x)) where a(x) tends to 0 with x tending to 0, there exists a number l1 > 0 such that in the neighborhood U+ = {x £ R: \x\ ^ li} of the point O for all x = 0 there is the inequality |/3+(.r)| < Define the arc \t : R ^ R by the formula

xt = {I-t)T3+ +tQ.

Then |xt"(®)| < M f°r all nonzero x £ U^ . Note that the origin O is a fixed point for each diffeomorphism \t ancl Xt{U\) c *nt f°r every t £ [0,1]. Let = and

let us consider the isotopy ht = which connects the identity map with the diffeo-

morphism (j3+)~lQ. By construction ht{U^) C V+ for every t £ [0,1]. According to Thorn's theorem about the continuation of an isotopy (see, for example, [4, Theorem 5.8]), there is an isotopy Ht: R ^ R, which coincides with ht on U+ and is the identity outside V+. Then the required arc is given by

p+ = J3+Ht.

2) If Q = g, then we define {¿¡^ = p^}. In the opposite case we define the arc R —> R by the formula

r+ = (1 - t)Q + tg.

Let V2+ = U^ and U.^ = Q(U^). Consider the isotopy % = Q_1Tf which connects the identity map with the diffeomorphism Q~l~g. By the construction rjtiU^) C V2+ for every t £ [0,1]. According to Thom's theorem about the continuation of an isotopy (see, for example, [4, Theorem 5.8]), there is an isotopy dt: R ^ R, which on U2+ is dt and which is the identity outside V2+. Then the required arc is given by

It = pf0t. U

Proposition 3. For any diffeomorphism y £ Eg there is a smooth arc {5t £ J(S1)} joining ¿o = Y with = g■

Proof. Since y £ Eg, there are neighborhoods VY(N) and VY(S) of the points N and S such that y\vy(n)uv^(s) = g\v.y(n)uv^(s) . We define a diffeomorphism ^: S1 \ {S} ^ S1 \ {S} by the formula (w) = gk(y-k(w)) where k £ Z such that Y-k(w) £ VY(N) for w £ S1 \ {S}. Then y = ■ If can be continued smoothly on S by the formula (S) = S, then there

exists a smooth isotopy pt: S1 ^ S1, such that p0 = p1 = id. Thus, the arc 5t = p-1gpt is the desired arc. In the opposite case we show that there exists a smooth arc {(t £ Eg} joining the diffeomorphism (0 = y with some diffeomorphism Z1, such that ^ can be continued smoothly on S1 by the formula ^ (S) = S.

Let Br = {a; G R: < r} and Kr = d(B.r \ Br) for r > 0. Let 7 = 't>+rt>+~1 and

2

= §+ip1,d+~1. Then 7 £ Eg and, therefore, there is vq > 0 such that 7 = g on B,ro and the space Kro is a fundamental domain of the restriction of the diffeomorphism g (as well as the diffeomorphism 7) on R \ {O}. By construction, the orbit space C = (R \ {0})/g is a disjoint, union of two circles C = C+ U C-. We denote by p: Bro \ {O} ^ C the natural projection.

Then the curves a+ = p(Ox+), a- = p(Ox-) are generators of fundamental groups n1(C+) and n1(C-), respectively.

Since translates orbits of the diffeomorphism g into orbits of the diffeomorphism 7 and Kro is a common fundamental area for g, 7 on R \ {O}, it follows that is designed for C± by the diffeomorphism 07± = pip p-1: C± —> C±. Moreover, the induced isomorphism ip1± *: ^1(C±) — n1(C±) is the identity. Thus, the diffeomorphism ip1± is smoothly isotopic to the identity map.

We choose an open covering U± = {U±,..., U±} of the manifold C± such that each connected component of the set is a subset of Kri for some < According to the

fragmentation lemma (see, for example, [1]), there exist diffeomorphisms w±,..., w±: C± — C± such that they are smoothly isotopic to the identity map and

i) for each i = 1 ,q and for each t G [0,1] diffeomorphism wft is the identity outside Uf1 and {w±t} is the smooth isotopy between the identity map and w±;

ii) ipY± = w± ... w±.

Let wft: R —> R be a diffeomorphism coinciding with (p\Kn)~1^tt'P 011 KTi and coinciding with the identity map outside Kri. Let (±t = 7wft...w^t: Ox± —> Ox±. We denote by (t: R —> R a diffeomorphism coinciding with (+t on Ox+, with (_t on Ox- and such that (t(0) = O. By construction (t G Eg for every t G [0,1], (0 = 7 and 0Ci = Thus, ip^ = §+~1ip£1,d+: S1 \ {S*} —> S1 \ {S*} can be continued smoothly on S1 by the formula ip^iS) = S. ■

References

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[3] Hirsch, M. W., Differential Topology, Grad. Texts in Math., vol. 33, New York: Springer, 1976.

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[5] Pesin, Ya. B. and Yurchenko, A. A., Some Physical Models Described by the Reaction-Diffusion Equation, and Coupled Map Lattices, Russian Math. Surveys, 2004, vol.59, no. 3, pp. 481-513; see also: Uspekhi Mat. Nauk,, 2004, vol.59, no. 3(357), pp. 81-114.