Description of domain structures in the solarcorona by means multi-color graphs Текст научной статьи по специальности «Физика»

Динамические системы, 2016, том 6(34), №1, 3-14

MSC 2010: 37D05

Description of domain structures in the Solar Corona by means multi-color graphs1

D. Malyshev, O. Pochinka

Higher School of Economics

Nizhny Novgorod. E-mail: dmalishev@hse, olga-pochinka@yandex.ru

Abstracts. Magnetic charging topology explains many energy processes (flares, prominences, etc.) in the solar corona by changing the domain structure associated with the appearance or disappearance of the separators. It is known that at most of the nulls of the magnetic field are prone. In this paper it is proved that a topology of the domains of a field with the prone nulls is completely described by a multi-color graph. In addition, we give an efficient algorithm for distinguishing of these graphs. Keywords: magnetic fields, model of corona, photosphere magnetic reconnection, dynamics prominences, multi-color graph, polynomial-time algorithm.

1. Introduction and the formulation of the results

Understanding the energy processes in the corona of the sun is very important to explain many of the laws of nature. This paper considered a possible model to explain such effects in the photosphere as the flares and the prominences. Their origin is connected with the restructuring of regions (domains), on which the fans and the spines of the null points of the magnetic field divide the corona of the sun — reconnection. Therefore, the main questions for this approach are the qualitative partition of the solar corona into domains, as well as the existence of the separators (the lines of intersection of fans) — marks of upcoming or already occurred reconnection. There are different approaches to the study of the topology of domains, such as the construction of graphs that reflect the structure and the relative position of the domains [3] or footprints — traces of spines and fans on the photosphere [8]. We have proposed a new approach consisting in distinguishing of traces of fans on some circle on the photosphere. We describe these trace on a language of multi-color graph whose isomorphic class is a complete invariant for the topology of domains and gives information on the number of the separators. In more detail.

By the topological approach the magnetic field in the corona is believed to arise from a large number of dipoles in the solar interior. The dipoles are interpreted as locations where flux tubes originating in the solar interior break through the surface and spread out into the atmosphere (see figure 1). We use the assumptions of Magnetic Charge

1rThis work was supported by the Russian Foundation for Basic Research (projects no. 15-01-03687-a, 16-51-10005-Ko_a, 16-31-60008-mol-a-dk), RF President grant MK-4819.2016.1, the Basic Research Program at the HSE (project 98) in 2016, by LATNA laboratory, National Research University Higher School of Economics.

© D. MALYSHEV, O. POCHINKA

Topology [7], where photospheric flux patches are modeled as point sources (charges) on

the photosphere. Although this suggestion violate the solenoidal condition, but each

source is considered to represent a flux tube passing through the solar surface and

spreading out into the overlying corona, then this simplification is allowable. Following

[2] for a model of the magnetic field B with point sources the two-dimensional sphere

P = {(x,y,z,w) £ S3 | w = 0} in three-dimensional sphere S3 = {(x,y,z,w) £

R4 | x2 + y2 + z2 + w2 = 1} is used as the photosphere and the region {(x,y,z,w) £

S3 | w > 0} as solar corona. Moreover we suppose that B is symmetrically extended

to the region {(x,y,z,w) £ S3 | w < 0} being termed the mirror corona and, hence, it

k

is defined on M = S3 \ (J qi where qi,... ,qk are the points on the photosphere where

i=i

the charges are situated.

Magnetic nulls are the points where the magnitude of magnetic field vector vanishes. Due to the solenoidal condition V • B = 0 three eigenvalues Ai, X2, X3 of the critical point satisfy the equality Ai + A2 + A3 = 0. Since B is potential then all eigenvalues are real number. Generically each eigenvalue is different from 0, thus each null of B is a saddle point. Two quite distinct families of field lines tends to a null point: the spine is a line and the fan is a surface. For a null p denote by Sp the spine and by Fp the fan of p. The spines of different nulls have no intersections in general position. A null is called positive (negative) if Ai • A2 • A3 > 0 (Ai • A2 • A3 < 0). The topological structure of a magnetic field B is largely defined by null points, spines, fans, and separators, the union of which forms the so-called skeleton of the magnetic field. There are several types of nulls. A null which belongs to the photosphere is called photospheric. A photospheric null point whose spine lies in the photosphere is called prone, whereas a photospheric

null with a spine directed vertically is called upright. The coronal null is a null above the photosphere. It follows from [1] that the most nulls are prone.

When two fans have intersection they form a separator, which joins two oppositely signed null points. Fans divide the corona into different regions which called domains. Appearance and disappearance of separators change the topology of domains splitting. Such situation is called separator reconnection, which is one of the major reconnection mechanisms [15]. Much papers [3], [10], [11], [12] were devoted to classification of the magnetic field configurations that arise from such point-source models. It is naturally to introduce the following definition which goes back to the classic paper [14], see also [16].

Definition 1. One says that two coronal magnetic fields B, B' are topologically equivalent if there is a homeomorphism H : M ^ M sending magnetic lines of B to magnetic lines of B' with preserving orientation on the lines.

Denote by B the set of the magnetic fields B with the following properties:

1) each null of B is prone;

2) if two fans of B are intersected then they are either coincide, either have contact along one curve on the photosphere or have transversal intersection along two symmetric with respect to the photosphere curves;

3) the closures of the spines of different nulls have no intersection. Now let B e B.

Theorem 1. For each magnetic field B e B there is a circle C C P v)hich is transversal to the flow generated by B on P and such that each fan intersects C at exactly two points.

We will called such circle C by photosphere section. Denote by N the set of nulls of B. Set W = P \ U Sp, F = U Fp and X = C n F. Denote by Nu (Ns) the set of

p&N p&N

positive (negative) nulls of B. Set Fu = (J Fp (Fs = (J Fp), Xu = C n Fu (Xs =

peNu peNs

C n Fs) and X* = Xu n Xs.

In order to introduce a combinatorial topological invariant of the magnetic field B e B we recall the following definitions.

A finite graph r is an ordered pair (V,E), such that the following conditions hold: V is a non-empty finite set of vertices; E is a set of pairs of vertices called edges.

If a graph contains an edge e = (a,b), then each of the vertices a, b is said to be incident to the edge e and the vertices a and b are said to be connected by the edge e.

A path in a graph is a finite sequence of its vertices and edges of the form: b0, (b0,bl),bl, ■ ■ ■ , bi-i, (bi-i, bi),bi, ■ ■ ■ bk-l, (bk-l, bk),bk, k > 1. The number k is called the length of the path, it is equal to the number of edges involved in the path.

A cycle of length k, k e N in a graph is a finite subset of vertices and edges of the form {bo, (bo,bl),bl, ■■■ ,bi-l, (bi-l ,bi),bi, ■■■ bk-l, (bk-l,b0)}. A simple cycle is a cycle all of whose vertices and edges are pairwise distinct.

A graph T is called multi-color graph if the set of vertices or edges of T is the union of finite number subsets each of which consists of the vertices or edges of the same color.

Two multi-color graphs T and r" are said to be isomorphic if there exists a one-to-one correspondence ^ between the sets of their vertices which preserve the relations of incidence and the color.

For our invariant we will use three colors, we denote these colors by the letters s, t, u and, for brevity, refer to these vertices or edges as s-, ¿-, «-vertices or s-, ¿-, u-edges. We construct a multi-color graph corresponding to a magnetic field B G B as follows (see figure 2 where s, ¿, u are green, blue, red, accordingly):

Рис. 2. Magnetic fields and their multi-color graphs

1) the t-vertices are in a one-to-one correspondence with the points of the set

2) the s-vertices (u-vertices) are in a one-to-one correspondence with the points of the set X8 \ Xt (Xй \ X4);

3) the ¿-edges are in a one-to-one correspondence with the connected components of С \ X and two vertices of the graph are incident to an ¿-edge if the corresponding points are boundary points for corresponding connected component;

4) two vertices of the graph are incident to an s-edge («-edge) if the corresponding points are exactly Fp n C for some null p e Ns (p e Nu).

Theorem 2. Magnetic fields B, B' from B are topologically equivalent if and only if their multi-color graphs , are isomorphic.

Theorem 2 motivates to ask the question about the computational complexity of distinguishing two multi-color graphs corresponding to magnetic fields. An algorithm solving the graph isomorphism problem is considered to be efficient if its running time is bounded by a polynomial on the number of vertices of input graphs. This problem can really be solved in polynomial time for the graphs of magnetic fields.

Theorem 3. Isomorphism of multi-color graphs corresponding to Solar magnetic fields can be recognized in polynomial time.

2. Necessary and Sufficient conditions for the topological equivalence of magnetic fields from B

To prove the results we compactify the magnetic field lines in the places of pointcharge by the bundle of straight lines, such idea was used in [6] for the finding of the separators of magnetic fields in electrically conducting fluids. Then the magnetic lines of the field B coincide geometrically on M with trajectories of a three-dimensional flow fT : S3 ^ S3 with the following properties:

1) the non-wandering set Q(fT) of fT consists of finite number hyperbolic equilibrium states2 all of them belong to the photosphere P;

2) all trajectories of fT are symmetric with respect the photosphere P and number of sinks coincide with number of sources;

3) the closures of one-dimensional invariant manifolds of different saddle points are disjoint;

4) if two-dimensional invariant manifolds of different saddle points are intersected then they are either coincide, either have contact along one curve on the photosphere or have transversal intersection along two symmetric with respect to the photosphere curves.

Denote by G the set of flows with properties above. By the construction we see the following interrelation between magnetic field B e B and its compactification fT e G:

2An equilibrium state w of the flow fT is called hyperbolic if it has no eigenvalues with zero real part. Any hyperbolic equilibrium state w of the flow fT possesses invariant manifolds: stable manifold W = {y e S3 : lim d(fT(y),w)=0}, unstable manifold = {y e S3 : lim d(fT(y),w) = 0}

T ^ + Ж T ^ — ж

which are homeomorphic to l"s, I™", where ns, nu — the numbers of the eigenvalues with negative and positive real parts, correspondingly, d — a metric on S3. We will denote by dim WS = ns, dim WU = nu the dimensions of WS and WU.

- the charges coincide with the sink and source equilibrium states,

- the null points coincide with the saddle equilibrium states,

- the fan (spine) of each null coincides with two-dimensional (one-dimensional) invariant manifold of the corresponding saddle,

- the separators coincide with heteroclinic curves — connected component of the intersection of two-dimensional invariant manifolds of the saddle points,

- the magnetic lines of B coincide with the trajectories of fT on M

- magnetic fields B, B' are equivalent if and only if corresponding flows fT, f 'T are equivalent.

Let fT G G and a be a saddle point of fT with the unstable manifold WU and the stable manifold W®. Denote by Q1 (Q2) the set of saddle points a of fT such that dim WU = 1 (dim WU = 2) and by Qq (Q3) the set of sinks (sources). Let us set

A = У clW^, R = У elWc

a£Q 1 <г€П2

The following proposition is due to [16] (see also [5] for details). Proposition 1. For each flow fT £ G the following statements hold:

i) S3 = |J WS = U W'U and each invariant manifold WS (WU) is a submanifold3 of S3;

ii) cl Wu n = 0 if and only if Wu n W.s = 0;

iii) the sets A, R are pairwise disjoint and each of them is connected.

Proof of Theorem 1

Theorem 1 follows from lemma below.

Lemma 1. For each flow fT £ G there is a circle C d P which is transversal to the flow fT|p and such that two-dimensional invariant manifold of each saddle point intersects C at exactly two points.

Proof. Let us set $>T = fT|p. It follows from the description of class G that fT is a flow on S3 with finite hyperbolic non-wandering set, then by Lefschetz formula |^o| — |Hi| + |^2| — |^3| = 0, where | • | is the cardinality. In the other side $>T is a flow on S3 with the same non-wandering set, then |Q0| — |Qi| — |Q2| + |Q31 = 2. Thus

|^o| —|^i| = 1.

Let us choose neighbourhood U(A) of the set A on P such that dU(A) is transversal all trajectories in (Wi \ A) П P (see figure 3). Due to item iii) of Proposition 1, U(A) has euler characteristic 1, it means that U(A) is 2-disk. By item i) of Proposition 1, Wi \ A = WR \ R. Set Q = Wi \ A and C = dU (A). By item i) of Proposition 1 and symmetry property of fT, each two-dimensional manifold of saddle point intersect Q П P along exactly two trajectories. Thus C is required photospheric section. □

Proof of Theorem 2

We assign a flow fT E G for each magnetic field B E B, also we have a graph Г в corresponding to B. Then theorem 2 follows from the next lemma.

Lemma 2. Flows fT,f'T are topologically equivalent if and only if multi-color graphs Гв, Гв/ are isomorphic.

Proof. First, we prove necessity. Suppose that fT and f'T from G are topologically equivalent, that is, there exists a homeomorphism h : S3 ^ S3 which sends the trajectories of fT to trajectories of f'T with preservation of orientation. Let us prove that multi-color graphs Гв, ГВ' are isomorphic. We assume without loss of generality that the graph Гв was constructed by using the photospheric section C = h(C). Since the conjugating homeomorphism h takes invariant manifolds of fixed points of fT to invariant manifolds of f'T with preservation of the stability, it follows that this homeomorphism takes Xs, X1, Xu to X's, Xn, X'u. Then the requaired isomorphism £ : Гв ^ ГВ' is defined by the formula £ = nf' hn-1 where nf, nf' are one-to-one maps of the set X, X' onto the sets of vertices of the graph Гв, Гв, accordingly.

Let us prove sufficiency. Consider the multi-colour graphs Гв, Гв' of the flows fT, f'T E G, respectively. Suppose that there exists an isomorphism £ between the sets of vertices of Гв, Гв' which preserve the relations of incidence and the color. We construct step by step a homeomorphism h : S3 ^ S3 conjugating fT and f'T.

3Let ц E {0,1, 2, 3}. A subset Y of S3 is said to be its ц-dimensional submanifold if for every point y of the set Y there is a neighbourhood Uy of y and a homeomorphism фу : Uy ^ R3 for which фу (Uy П Y) = RM where RM С R3 is the set of points whose last (3 — /л) coordinates are zero.

Step 1. Set V = S3 \(AUR). Similar to proof of Lemma 1, for each flow fT £ G there is a 2-sphere £ d M which is transversal to the flow fT ^ and such that two-dimensional invariant manifold of each saddle point intersects £ at exactly one circles. Moreover, it is possible to construct £ such that £ n P = C .Set Cu = £ n Fu (Cs = £ n Fs), Ct = Yu n Ys and do the same for f 'T.

By the construction all vertices and all i-edges the multi-color graph form a simple cycle and £ preserves such cycle with the preserving of the color of the vertices than there exist an orientation-preserving homeomorphism hs : £ — £' such that hs(Cu) = C'u, hs(Cs) = C's and hs(Ct) = Crt We denote by lx (l'x) the trajectory of fT (of f'T) passing through x £ S3. According to Proposition 1 there are unique pair of the equilibrium states a(lx), u(lx) (a(l'x), u(lx)) such that lx d (WU(lx) n Ws{lx)) (lx d (WU(lf) n W^, ))). By Proposition 1 we have the following possibilities for point x £ £:

- a(lx) £ ^3, u(lx) £ ^o for x £ £ \ (Cu U Cs);

- a(lx) £ ^2, u(lx) £ ^o for x £ (Cu \ Cs);

- a(lx) £ ^3, u(lx) £ ^i for x £ (Cs \ Cu);

- a(lx) £ Q2, u(lx) £ Qi for x £ Ct.

For points yi,y2 £ cl (lx) denote by [yi,y2] the length of arc [yi,y2] d lx. For each point y £ lx situated between x and a(lx) (u(lx)) set p(y) = [¿y^ (p(y) = xxJjt)]). Similar situation is for points from £'. For any point x £ £, we set x' = hs(x). As hs(Cs) = C's, hs (Cu) = C'u then on the set lx a homeomorphism hlx : lx — l'x, is well-defined by the formula

hlx (y) = y' where p' (y') = p(y).

Denote by hV : V — V' a map composed from hlx ,x £ £. By the construction hV is a homeomorphism which sends two-dimensional invariant manifolds of the saddle point a of fT to the two-dimensional invariant manifolds of the saddle point a' of f 'T. Let us show that hV(u(lx)) = u(l'xi) for each x £ £.

Step 2. Denote by Q d S3 compact 3-ball bounded by £ and containing Q0. Then Q d Wq0uQl and the set Da = W% n Q is a 2-disk for each a £ Qi. Denote by Y a connected component of the set Q \ . Then there is a unique sink u £ such that u £ Y d W^. Simultaneously there is a unique connected component KY of the set £ \ Cs belonging Y and such that Y \ A = (J (lx n Y) U u. Similar situation is

x£Ky

for flow f'T. Since hs(£ \ Cs) = £' \ C's then hs(KY) is a connected component of £' \ C's belonging to a connected component Y' of the set Q' \ Ws 1 containing a sink u' £ Q0. By the construction hV(Y \ A) = Y' \ A' and, hence hV(u(lx)) = u(l'xi) for each x £ (£ \ Cs). By the continuously hV(u(lx)) = u(l'xi) for each x £ Cs.

Thus hV can be uniquely extended to the sets Q0, Qi. We keep the notation hV for the homeomorphism thus obtained and set p' = HV(p) for each p £ (Q0 U Qi).

Step 3. Let a G f^. Denote AT flow in R3 generated by a system of linear equations

{OC OC j

y = y,

z = — z.

This flow has a unique equilibrium state — hyperbolic saddle located at the origin O.

Рис. 4. Linearization of saddle equilibrium state neighborhood

Stable manifold of this saddle is plane XOY, unstable — axis O. Set

U={(x, y, ;) £ K3 : (i-2 + y2)z2 < 1}.

It is immediately verified that U is invariant with respect to the flow AT. Due to [13] there is a neighborhood Va С §3 of the saddle equilibrium state a and a homeomorphism Ha : Va —>■ U such that the homeomorphism sends the trajectories of flow /r|yCT to the trajectories of flow Ar\u (see figure 4). Similar neighborhood Va' and a homeomorphism Ha, : Va, ->• U exist for flow fT. Set = Я~}Яа : Va ->• Vj. Without loss of generality we can assume that homeomorphism Ha a/ sends one-dimensional separatrix of a which contains a sink uj in its closure to one-dimensional separatrix of a' which contains a sink u' in its closure (in opposite case we use (Ha instead Ha where С (x,y,z) = (x,y,-z)).

Step 4. For fi e (0,1) let us set

U,l = {(x,y,z)eRs:(x2 + y2)z2<fi}

and Va4l = H~l{U^). Choose ц such that Ha^,(Va4l) \ С Hv{Va). Set Z = cl (Va \ Va4l) and Z' = d (Hy{ya) \ (Va4l)). By the construction the sets Z,Z' consists of

two connected components Z+, Z-, Z+, Z- each of them is homeomorphic to W = S1 x R1 x [0,1]. Denote by Hz+ : Z+ — W, Hz_ : Z- — W, HZ+ : Z+ — W, HZ- : Z- — W corresponding homeomorphisms sending trajectories of flows to lines {s} x R1 x {t}. For t E [0,1], 8 E {+,-} set Wt = S1 x R1 x {t} and

Hs,o = HzsHayH-1\wo : Wo - Wo, HSA = HzsHvH-^w, : W1 - W1.

As Hv and Ha,a' send trajectories of fT to trajectories of f 'T then Hs,0, Hs1 have view

Hs,o(s, r, 0) = (Hs,o,s(s),Hs,o,r(r), 0), HsA(s,r, 1) = (Hs, 1 ,s(s), Hs, 1 ,r(r), 1).

Let us define homeomorphism Hs,t : Wt — Wt by formula

Hs,t(s, r, t) = ((1 - t)Hs,o,s(s) + tHsA,s(s), (1 - t)Hs,o,r(r) + tHs^r(r),t).

Denote by HZg,Z>s : Zs — Zs< homeomorphism composed for each t E [0,1] by H—Hs,tHZs\H-i(W). Let us define homeomorphism HVa by formula

Hv (x) = J HZ5Z(x), x E Zs, (X) \ Haj(x), x E V^.

By similar way we can define homeomorphism HVa for each a E Q2. The required homeomorphism h : S3 — S3 is defined by

!

Hv(x), x e S3 \ ( U Va),

h(x) = { ae( П un 2)

HVa (x), x e Va, a e (^1 и П2).

□

3. Algorithm to solve the distinguishing problem for multi-color graphs

In this section, we consider the distinguishing problem for multi-color graphs and present an efficient algorithm for its solution. An algorithm to solve the problem is considered to be efficient if it occupies polynomial time on the number of vertices of a given graph. The notion of an efficiently solvable problem rises to A. Cobham, who asserts that a problem can be feasibly computed on some computational device only if it can be computed in time, bounded by a polynomial on the length of input data [4]. The complexity status of the general graph isomorphism problem, i.e. for graphs of the general type, is unknown. That is, neither polynomial-time solvability neither intractability was proved for it. The graphs, associated with Solar magnetic fields, have some peculiar combinatorial properties. Namely, they have bounded degrees of vertices. Recall that degree of a vertex of a graph is the number of edges incident to it. A finite graph is called simple if it does not contain coloured vertices, loops, multiple and directed edges, coloured edges, simultaneously.

Proof of Theorem 3

It is known that for some concrete constant c* and function f (•) the isomorphism problem can be solved in O(f(A)nc*Aln(A)) time for simple n-vertex graphs with maximum degree A [9]. For each fixed k, this result gives a polynomial-time algorithm to solve the isomorphism problem in the class of all simple graphs having degrees of all vertices at most k. This observation and the facts that the graphs of Solar magnetic fields have degrees of all vertices at most three, the three colors are used to color their vertices and edges lead to the following idea. By the graphs rBl and rB2 of magnetic fields Bi and B2, we construct simple graphs PBl and r'B2 such that rBl and rB2 are isomorphic if and only if PBl and r'B2 are isomorphic. The graphs T'Bl and r'B2 will have degrees of all vertices at most 9, which implies polynomial complexity of their distinguishing, by the result of Luks.

Recall that a multi-color graph is a graph r, equipped by two functions ci : V(r) —> {1, 2,...,ki} and c2 : E (r) —> {1, 2,...,k2}. Let A(r) be the maximum degree of vertices of the graph r. By r, we construct a simple graph r' as follows. An s-star implantation into an edge (a,b) of a graph is to delete the edge from the graph, add vertices c,ci,... ,cs and the edges (a, c), (c, b), (c, ci), (c, c2),..., (c, cs). Inscribing an s-cycle in a vertex v of a graph is to add vertices vi,v2,... ,vs-i and the edges (v,vi), (vi,v2,),..., (vs-2,

vs-i), (vs,v) to the graph. For each v E V(r), we inscribe a ci(v) + 2-cycle in v. For each e E E(r), we implant a c2(e) + A(r)-star into e. Clearly, the number of vertices of r' is at most (ki + 2)\V(r)| + (k2 + A(r) + 1)\E(r)| and degrees of all its vertices are at most k2 + A(r) + 2. As the sum of degrees of vertices of r is equal to 2\E(r) \, \E(r) \ < 2A(r)\V(r)\. Hence, \V(r')\ < 1 (k + A(r) + 1)A(r) + 2ki + 4)\V(r)\. Given r', one can uniquely restore r as follows. All vertices of r' having degrees at least A(r) + 3 are the central vertices of the implanted stars. This observation permits to restore all edges of r with their colors. Deleting all vertices of all stars from r' produces a disjoint sum of \V(r)\ simple cycles. The number of vertices in each of the cycles determines the color of the corresponding vertex of r. Therefore, two multi-color graphs r and r2 are isomorphic if and only if the corresponding simple graphs ri and r'2 are isomorphic. We may consider that \V(ri)\ = \V(r2)\ = \V\ and A(ri) = A(r2) = A, ci : V(ri) —► {1, 2,... ,ki} and c2 : E (ri) —> {1, 2,..., k2} for each i = 1, 2, otherwise r and r2 are not isomorphic. Therefore, isomorphism of r and r2 can be tested in O(f (k2 + A + 2)(1 (A(k2 + A + 1) + 2ki + 4))c*(k2+A+2)in(k2+A+2)\V\c*(k2+A+2)in(k2+A+2)) time. For the graphs of magnetic fields, A = ki = k2 = 3.

References

1. Beveridge C, Priest E. R, Brown D. S. Magnetic topologies in the solar corona due to four discrete photospheric flux regions // Geophysical and Astrophysical Fluid Dynamics. — 2005. — V. 98.

2. Beveridge C, Priest E. R., Brown D. S. Magnetic topologies due to two bipolar regions // Solar Physics. — 2002. — V. 209.

3. Close R. M, Parnell C. E, Priest E. R. Domain structures in complex 3D magnetic fields // Geophys. Astrophys. Fluid Dynam. — 2005. — V. 99.

4. Cobham A. The intrinsic computational difficulty of functions // Proc. 1964 International Congress for Logic, Methodology, and Philosophy of Science, North-Holland, Amsterdam. —1964. —P. 24-30.

5. Grines V., Pochinka O. Morse-Smale cascades on 3-manifolds // Russian Mathematical Surveys. — 2-13. — V. 68, № 1. — P. 117-173.

6. Grines V., MedvedeРj T, Pochinka, O, Zhuzhoma E. On heteroclinic separators of magnetic fields in electrically conducting fluids //Physica D: Nonlinear Phenomena. — 2015. — V. 294.

7. Longcope D. W. Topology and current ribbons: a model for current, reconnection and flaring in a complex, evolving corona // Solar Phys. — 1996. — V. 169.

8. Longcope D. W. Topological Methods for the Analysis of Solar Magnetic Fields // Solar Phys. — 2002. — V. 2.

9. Luks L. Isomorphism of graphs of bounded valence can be tested in polynomial time // Jornal of computer and system sciences. — 1982. — V. 25.

10. Maclean R. C, Beveridge C, Hornig G, Priest E. R. Coronal magnetic topologies in a spherical geometry I. Two bipolar flux sources // Solar Phys. — 2006. — V. 235.

11. Maclean R.C., Beveridge C, Priest E.R. Coronal magnetic topologies in a spherical geometry II. Four balanced flux sources // Solar Phys. — 2006. — V. 238.

12. Maclean R. C., Priest E. R. Topological aspects of global magnetic field behaviour in the solar corona // Solar Phys. — 2007. — V. 243.

13. Palis J., de Melo W. Geometric theory of dynamical systems: An introduction //Springer, New York. — 1982.

14. Poincare H. Sur les courbes definies par une equation differentielle // Journal de mathematiques pures et appliquees. —1882. —V. III, № 8.

15. Priest E. R, Titov V. S. Magnetic reconnection at three-dimensional null points // Phil. Trans. Rog. Soc. Lond. A. —1996. —V. 354.

16. Smale S. Differentiable dynamical systems // Bull. Amer. Math. Soc. —1967. —V. 74, № 6.

Получена 15.03.2016