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УДК 517.9+513.8 https://doi.org/1Q.185QQ/Q869-6632-2019-27-6-63-72
Моделирование градиентно-подобных потоков на я-сфере
О. В. Починка, С. Ю. Галкина, Д. Д. Шубин
Национальный исследовательский университет «Высшая школа экономики» Россия, 603155 Нижний Новгород, Большая Печерская, 25/12 E-mail: olga-pochinka@yandex.ru, svetlana.u.galkina@mail.ru, schub.danil@yandex.ru Автор для переписки Данила Денисович Шубин, schub.danil@yandex.ru
Поступила в редакцию 5.09.2019, принята к публикации 28.10.2019, опубликована 2.12.2019
Общей идеей качественного изучения динамических систем, со времён работ А. Андронова, Е. Леонто-вич, А. Майера, является возможность описания динамики системы комбинаторным инвариантом. Так, например, М. Пейшото доказал структурно устойчивые потоки на плоскости определяются единственным образом, с точностью до топологической эквивалентности, классом изоморфных ориентированных графов. Многомерные структурно устойчивые потоки не позволяют ввести их классификацию в рамках общего комбинаторного инварианта. Однако для некоторых подклассов таких систем существует возможность достигнуть полного комбинаторного описания их динамики.
В настоящей работе, основанной на результатах С. Пилюгина, А. Пришляка, В. Гринеса, Е. Гуревич и О. Починка, каждое дерево с двухцветной раскраской вершин реализовано как градиентно-подобный поток на п-сфере, п > 2 без гетероклинических пересечений. Эта задача решается с помощью соответствующих операций приклеивания так называемых ячеек Черри к потоку-сдвигу. Этот результат не только завершает топологическую классификацию таких потоков, но и позволяет моделировать системы с регулярным поведением. Для таких потоков реализация особенно важна, поскольку они моделируют, например, процессы присоединения в солнечной короне.
Ключевые слова: градиентно-подобный, моделирование.
Образец цитирования: Починка О.В., Галкина С.Ю., Шубин Д.Д. Моделирование градиентно-подобных потоков на п-сфере//Известия вузов. ПНД. 2019. Т. 27, № 6. С. 63-72. https://doi.org/10.18500/0869-6632-2019-27-6-63-72
Финансовая поддержка. Доказательство основной теоремы поддержано Российским Научным Фондом (Грант № 17-11-01041), доказательство вспомогательного результата поддержано Центром фундаментальных исследований Национального исследовательского университета Высшая школа экономики в 2019 году.
© Починка О.В., Галкина С.Ю., Шубин Д.Д., 2019
63
https://doi.org/10.18500/0869-6632-2019-27-6-63-72
Modeling of gradient-like flows on w-sphere
O. V. Pochinka, S. Yu. Galkina, D. D. Shubin
National Research University Higher School of Economics 25/12, Bolshaya Pecherskaya St., 603155 Nizhny Novgorod, Russia E-mail: olga-pochinka@yandex.ru, svetlana.u.galkina@mail.ru, schub.danil@yandex.ru Correspondence should be addressed to Danila D. Shubin, schub.danil@yandex.ru Received 5.08.2019, accepted for publication 28.10.2019, published 2.12.2019
A general idea of the qualitative study of dynamical systems, going back to the works by A. Andronov, E. Leontovich, A. Mayer, is a possibility to describe dynamics of a system using combinatorial invariants. So M. Peixoto proved that the structurally stable flows on surfaces are uniquely determined, up to topological equivalence, by the isomorphic class of a directed graph. Multidimensional structurally stable flows does not allow entering their classification into the framework of a general combinatorial invariant. However, for some subclasses of such systems it is possible to achieve the complet combinatorial description of their dynamics.
In the present paper, based on classification results by S. Pilyugin, A. Prishlyak, V. Grines, E. Gurevich, O. Pochinka, any connected bi-color tree implemented as gradient-like flow of n-sphere, n > 2 without heteroclinic intersections. This problem is solved using the appropriate gluing operations of the so-called Cherry boxes to the flow-shift. This result not only completes the topological classification for such flows, but also allows to model systems with a regular behavior. For such flows, the implementation is especially important because they model, for example, the reconnection processes in the solar corona.
Key words: gradient-like flow, modeling.
Reference: Pochinka O.V., Galkina S.Yu., Shubin D.D. Modeling of gradient-like flows on n-sphere. Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, vol. 27, no. 6, pp. 63-72. https://doi.org/10.18500/0869-6632-2019-27-6-63-72
Acknowledgements. The proof of the main theorem is supported by RSF (Grant no. 17-11-01041), the proof of auxiliary results is supported by Basic Research Program at the National Research University Higher School of Economics (HSE) in 2019.
1. Introduction and statement of results
A general idea of a qualitative study of dynamical systems, going back to the works by A. Andronov [1], E. Leontovich, A. Mayer [2,3], is a possibility to describe dynamics of a system using combinatorial invariants. A brilliant example of the implementation of such approach is the topological classification of Morse-Smale flows on surfaces obtained by M. Peixoto [4]. He proved that structurally stable flows on surfaces are uniquely determined, up to topological equivalence, by the isomorphic class of a directed graph.
Multidimensional structurally stable flows do not allow to classify their into the framework of a general combinatorial invariant. However, for some subclasses of such systems it is possible to achieve a completely combinatorial description of their dynamics. Thus, according to the results by S. Pilyugin [5], A. Prishlyak [6], V. Grines, E. Gurevich, O. Pochinka [7], the topological equivalence class of a gradient-like flow (Morse-Smale flow without periodic orbits) on n-sphere, n > 2, without heteroclinic intersections is completely determined by a bi-color tree corresponding to a skeleton of co-dimensional one saddle invariant manifolds. A problem of the realization of an abstract invariant is an integral part of the topological classification. For such flows, it is especially important because they model regular processes in various natural sciences (see, for example, [8]). In particular, such flows model reconnection processes in the solar corona (see, for example, [9,10]).
In such processes the corona of the Sun is divided on domains by fans and spines (2- and 1-dimensional invariant manifolds) of null points of the magnetic field (the points, at which the strength of magnetic field is null). Restructuring of this domains underlie such effects as the flares and prominences. This energy processes are very important for explanation of many nature laws. The topo-logical structure of a magnetic field is defined by null points, spines, fans and separators, the union of those forming the so-called skeleton of the magnetic field. Experiments and observations show that the evolution of the structure of the magnetic field is similar to relaxation processes. At first plasma evolves slowly for some considerable time but at some point there occurs a topological restructuring of the magnetic configuration (reconnection) [11].
Indeed below we model different stable states of the magnetic field.
Let us recall that n-ball or n-disk is a manifold with a boundary homeomorphic to a standard
n-ball
Bn = {(xi, ...,Xn) e Rn | x\ + ... + x2n < 1}.
An open n-ball or n-disk we call a manifold homeomorphic to an interior of Bn. We call by (n—1)-sphere a manifold Sn-i homeomorphic to a standard (n — 1)-sphere
Sn-i = dBn
Let us consider a class G of gradient-like flows without heteroclinic intersections on n-dimensio-nal sphere Sn,n > 3, that is, flows whose the non-wandering set consists of a finite many hyperbolic fixed points such that the invariant manifolds of saddle points have no intersections.
Let f e G. According to the result by S. Pilyugin [5, Lemma 2.2], the dimensions of the invariant saddle manifolds of f4 have to be only (n — 1) and 1. Let us denote by Q f t the non-wandering set of f4, and let
Qft = {p G Qft | dim Wu = i}
By [12, Theorem 2.3],
sn = U wpu = U
pçQft pea ft
It follows from [13, Proposition 2.3] that for any saddle point o of a flow f the closure of its invariant manifold WO with dimension (n — 1) contains, except the manifold itself, exactly one fixed point. That point is a sink if ô = u and a source if ô = s. Then the set cl WO is a sphere with dimension (n — 1). By [14] and [15] this sphere is cylindrically embedded1. Denote by mf t the number of saddle points of a flow ft. Then the union
Wf t = y cl Wps U y cl wqu
pea1ft
of closures of all invariant manifolds of dimension (n — 1) divides a sphere Sn into kf t = mf t + 1 connected components. Denote such components by D\,..., Dkf t, and let
kft
Vft = y Di. i= 1
^A sphere Sn 1 C Mn is called cylindrically embedded in Mn, if there exists a topological embedding h : x [-1; +1] ^ Mn, such that h(Sn-1 x {0}) = Sn-1.
1 x
А А Д
Fig. 1. Example of a flow and its bi-color graph Рис. 1. Пример потока и его двухцветного графа
A bi-colour graph of a flow ft e G is a graph r f t, such that:
1) the set r0t of vertices of r ft bijectively corresponds to Dft by a bijection
: r01 ^ Dft;
2) two vertices vi5 Vj are connected by an edge e^j iff domains Di = '%0(vi), Dj = "%0(vj) have a common boundary;
3) an edge ei,j has a colour u (resp. s) if the common boundary of Di and Dj is the closure of an unstable (resp. stable) saddle manifold (see Fig. 1).
Two graphs rf t and rfit of some flows ft, f ;t are called isomorphic if there exists an isomorphism n: rft ^ rfit mapping vertices of rf t into vertices of rfit preserving adjacency and coloring.
It follows from [7], that the flows ft, f;t e G are topologically equivalent iff their graphs rft and rfit are isomorphic. Indeed, for any flow ft e G its bi-color graph is a tree, i.e. connected graph without cycles.
The main result of the present paper is the following theorem.
Theorem 1. For every bi-color tree r there is a flow f teG whose graph If t is isomorphic to graph r.
Notice that flows of the considered class, under the assumption that they have a unique sink, were classified and realized in [16] by means of a directed graph.
2. Realization of a flow by a bi-color tree
2.1. Description of bi-color tree. Recall some definitions from the graph theory (see, for example, [17] for details).
A graph is a pair (V, E), where V is a set of vertices and E is a set of pairs of vertices, which are called edges. If E contains ordered pairs, then the graph is called a directed one. A k-edge-colouring of a graph is an assignment of k colours to its edges.
Two vertices are called adjacent if they are connected by an edge (i.e. they constitute the edge), and the edge is incident to each of the vertices. A loop is an edge, whose end-vertices coincide. A simple graph is an undirected graph without loops.
The number of edges incident to a vertex is called degree of the vertex.
A set (vi, (v1,v2),v2,..., Vfc_i, (vfc_i, vk), vk} is called a path of the length k. A path is called a cycle if v1 = vk. A graph is called connected if every two its vertices are joined by a path.
A tree is a connected acyclic graph. It means that any two its vertices are connected by exactly one path.
Any tree with at least 2 vertices has at least two pendant vertex, that is a vertex of the degree 1.
Any tree becomes a out-tree if arbitrary its vertex r is selected, as a root. In the other words a planted tree is a tree in which one vertex r has been designated as the root and every edge is directed away from the root.
If v is a vertex in a planted tree other than the root, the parent of v is the unique vertex w such that there is a directed edge (w, v). If w is the parent of v, then v is called a child of w.
The rooted vertex r by definition has a level 0. The level d of any other vertex v in a such planted tree is the number of edges in the unique path between the vertex v and the root r. The depth of a tree D is the maximum level of any vertex there.
An ordered out-tree is an out-tree where the children of each vertex are ordered.
2.2. Construction of the flow by the graph. To construct a required flow on the n-sphere Sn for the given bi-color tree r choose a pendant vertex r of r as a root and an order all children to get from the tree r an ordered out-tree. Denote by N the number of all vertices of r.
To realize of a flow from the bi-color tree r, we will use the idea of embedding of N — 1 pairwise disjoint Cherry boxes Bv in a flow-shift g0 : Rn ^ Rn, given by the formula
for some av, |v e R, Sv > 0 which depends on parameters of v. The dynamics in Bv coincides with the flow-shift dynamics on the boundary of Bv and differs from one inside the box due to the appearance of a saddle and a node. We will say that the dynamics in Bv has a type u (s), if the saddle point has (n — 1)-dimensional unstable (stable) manifold and the node point is a source (sink) (see Fig. 2).
Below we give formulas for the following things:
1) The calculation of a position and a size of the Cherry-box Bv;
2) The definition of a flow gfv in Bv;
3) The embedding of a resulting dynamics in Sn.
1. The calculation of a position and a size of the Cherry-box Bv. For the vertex y which is a unique child of the root r we put
go (Ж1,...,ЖП) = (xi + i,...
and the cherry box Bv has a form
Bv = {(xi,. ..,ж„) e Rn : |xi - av | < bv, - ßv )2 + x^ + ... x£ < ö^}
Fig. 2. An embedding of Cherry-boxes of the types u and s to the flow-shift Рис. 2. Вложение ячеек Черри типа u и s в поток-сдвиг
where py equals 1 (—1) if the edge (r,y) has a colour s (u) and D is the depth of the tree r. For any other vertex v with the level dv > 2 the parameters of the box Bv are determined through the parameters aw, |w, Sw of its parent's box Bw, the order kv of v as a child and a number pv which equals 1 (—1) if the edge (w, v) has a colour s (u) by the following way:
8
8w
8w
V = 2N _ 4, av - Pv (|aw| - 8W - 8V), ßv = ßw + — - (2hv - 1)8v
So, the size and position are defined for each Cherry-box corresponding to every vertex of r except the root.
2. The definition of a flow gfv in Bv. Let
2v = (xi — av )2 + (x2 — |v )2 + x3 +-----+ x2n.
Define the flow glv : Rn ^ Rn by the formulas:
я, - ^ (2v - 82)2 , 2v < 82
1, otherwise
' x2 - ßv
si^-^ - зп -1 82
x2 I -(x2 - ßv), iv < f 0, otherwise
- \
2-1 (sb(- - зп -1
lv < — v < 2
0, otherwise
82 2
^ < iv < 8v
82
82 < iv < 82
By construction flow gv has exactly two hyperbolic fixed points: the saddle (source) point Pv (av + pvSv/2, |v, 0,..., 0) and the sink (saddle) point Qv (av — pvSv/2, |v, 0,..., 0) for pv = 1 (pv = —1). Define g£ : Rn ^ Rn in such a way that it coincides with gfv in Bv and is g0 outside all Cherry-boxes (see figure 3).
Ox2.....x„k
x i
D 0
D 2
A
Fig. 3. An example of a tree Г and the flow дГ Рис. 3. Пример дерева Г и потока дГ
Also in we have
Let us notice that the flow gf has no heteroclinic intersections. Indeed, by the construction the interiors of the Cherry-boxes are pairwise disjoint. Moreover, in the hyperplane xi = av we have
xi < 0 if (x2 - Pv)2 + x3 + ...xn < (Sv/2)2,
x i > 0 if (Sv/2)2 < (X2 - Pv )2 + x3 + ...xn < S2.
X2 < 0 if x2 > pv, x2 > 0 if x2 < pv, xi < 0 if xi > 0, xi > 0 if xi < 0, i = 3,..., n.
Thus, the invariant (n — 1)-manifold of saddle point from Bv outside Bv coincides with a cylinder
Cv = {(xi,...,xn) G Rn : (x2 — Pv)2 + x2 + ...xn < V2},
where Sv/2 < vv < Sv .By the construction these cylinders are pairwise disjoint, that proves the fact. 3. The embedding of a resulting dynamics in Sn. Let us define a flow h : Rn ^ Rn by the formula:
h*(xi, x2,..., xn) = (24xi, 2*x2,..., 2*xn) .
Let R+ = {(xi ,...,xn) G Rn : xi > 0} and C = {(xi,. ..,x„) G Rn : x2 + ••• + xn < 1}. It is easy to verify that a diffeomorphism Z : R+ \ O ^ C given by the formula
Z(X1, . . . ,X„) = (log2 0, —, . . . , —) , 0 = \JX2 +-----+ X,
conjugates the diffeomorphisms h|dRn \O and glsc. It allows to define a flow ^: Rn —► Rn in such a way that ^ coincides with h outside int R+ and coincides with Z-igfZ on R+.
Let us project the flow ^ to the n-sphere by means the stereographic projection.
Denote by N(0,..., 0,1) the North Pole
n
of the sphere Sn. For every point x = = (xi,...,xn+i) in Sn C Rn+i there is the unique line passing through the points N and x. This line intersects Rn = Oxi... xn at exactly one point fl(x) (see figure 4), which is called the stereographic projection of the point x. One can easily to check that fl : Sn \ {N} ^ Rn is a
diffeomorphism, given by the formula
ft(xi, . . . ,Xn+l) =
X1
1 - Xra+i
Fig. 4. The stereographic projection Рис. 4. Стереографическая проекция
Xra— 1
1 - Xra+i 1 - Xra+i
As flow coincides with h in some neighborhoods of the origin O and of the infinity point, hence, it induces on Sn the required flow
f (X) =
А-1(ф*(ft(X))), X = N;
N, x = N.
X
n
References
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10. Grines V., Zhuzhoma E.V., Pochinka O., Medvedev T.V. On heteroclinic separators of magnetic fields in electrically conducting fluids. Physica D: Nonlinear Phenomena, 2015, vol. 294, pp. 1-5.
11. Priest E., Forbes T. Magnetic Reconnection: MHD Theory and Applications. Cambridge Univ. Prees, Cambridge, 2000.
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Библиографический список
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3. Леонтович Е.А., Майер А.Г. О схеме, определяющей топологическую структуру разбиения на траектории // ДАН СССР. 1955. Т. 103, № 4. C. 557-560.
4. Peixoto M. On the Classification of Flows on Two Manifolds. Dynamical systems Proc., 1971.
5. Пилюгин С.Ю. Фазовые диаграммы, определяющие системы Морса-Смейла без периодических траекторий на сферах // Дифференц. уравнения. 1978. Vol. 14, № 2. C. 245-254.
6. Пришляк А.О. Векторные поля Морса-Смейла без замкнутых траекторий на трехмерных многообразиях // Матем. заметки. 2002. Vol. 71, № 2. C. 230-235.
7. Grines V.Z., Gurevich E.Ya., Pochinka O.V A combinatorial invariant of Morse-Smale diffeomor-phisms without heteroclinic intersections on the sphere Sn, n > 4 // Math. Notes. 2019. Vol. 105, no. 1. P. 132-136.
8. Песин Я.Б., Юрченко А.А. Некоторые физические модели, описываемые уравнением реакции-диффузии, и цепочки связанных отображений // УМН. 2004. 59:3(357). C. 81-114.
9. Browns D.S., Priest E.R. The topological behaviour of stable magnetic separators // Sol. Phys. 1999. Vol. 190. P. 25-33.
10. Grines V., Zhuzhoma E.V., Pochinka O., Medvedev T.V. On heteroclinic separators of magnetic fields in electrically conducting fluids // Physica D: Nonlinear Phenomena. 2015. Vol. 294. P. 1-5.
11. Priest E., Forbes T. Magnetic Reconnection: MHD Theory and Applications. Cambridge univ. Prees, Cambridge, 2000.
12. Smale S. Differentiable dynamical systems // Bull. Amer. Soc. 1967. Vol. 73. P. 747-817.
13. Grines V.Z., Medvedev T.V., Pochinka O.V Dynamical Systems on 2- and 3- Manifolds // Dev. Math., 46, Springer, Cham, 2016, xxvi+295 pp.
14. Cantrell J.C. Almost locally flat sphere Sn-1 in Sn // Proceeding of the American Mathematical society. 1964. Vol. 15. P. 574-578.
15. Brown M. Locally Flat Imbeddings of Topological Manifolds // Annals of Mathematics Second Series. 1962. Vol. 75. P. 331-341.
16. Grines V.Z., Gurevich E.Ya., Medvedev VS. Classification of Morse-Smale diffeomorphisms with one-dimensional set of unstable separatrices // Proc. Steklov Inst. Math. 2010. Vol. 270. P. 57-79.
17. Harary F. Graph Theory. Addison-Wesley, Reading, MA, 1969.
Починка Ольга Витальевна - родилась в 1972 году. Окончила механико-математический факультет Нижегородского государственного университета им. Н.И. Лобачевского (1994), аспирантуру (НГСХА, 2004) и докторантуру (ННГУ, 2011). Защитила диссертации на соискание ученой степени кандидата физико-математических наук (НГСХА, 2004) и доктора физико-математических наук (ННГУ, 2012). С 1997 по 2014 год работала на кафедре теории функций механико-математического факультета ННГУ им. Н.И. Лобачевского: сначала в должности ассистента, затем старшего преподавателя, доцента. С 2014 года по настоящее время - профессор, заведующая кафедрой фундаментальной математики Национального исследовательского университета Высшая школа экономики. Профессиональные интересы - качественная теория динамических систем.
Россия, 603155 Нижний Новгород, Большая Печерская, 25/12 Лаборатория топологических методов в динамике, Национальный исследовательский университет «Высшая школа экономики» E-mail: olga-pochinka@yandex.ru
Галкина Светлана Юрьевна - родилась в 1963 году. Окончила механико-математический факультет Московского государственного университета им. М.В. Ломоносова (1986) и аспирантуру (МГУ, 1991). Кандидат физико-математических наук (МГУ, 2002). С 1991 по 2005 год работала в Нижегородском государственном педагогическом университете, на кафедре математического анализа: сначала в должности ассистента, затем старшего преподавателя, доцента, и.о. заведующего кафедрой математического анализа. С 2005 по 2019 год - доцент кафедры прикладной математики Нижегородского государственного университета им. Н.И. Лобачевского. C 2019 г. по настоящее время - доцент кафедры фундаментальной математики Национального исследовательского университета Высшая школа экономики. Научные интересы: функциональный анализ, теория функций действительного переменного, теория аппроксимации функций.
Россия, 603155 Нижний Новгород, Большая Печерская, 25/12 Национальный исследовательский университет «Высшая школа экономики» E-mail: svetlana.u.galkina@mail.ru
Шубин Данила Денисович - родился в городе Павлово Нижегородской области (2000). С 2018 г. по настоящее время - студент факультета математики, информатики и компьютерных наук НИУ ВШЭ (Нижний Новгород), стажёр-исследователь лаборатории топологических методов в динамике (ИМиКН НИУ ВШЭ).
Россия, 603155 Нижний Новгород, Большая Печерская, 25/12 Лаборатория топологических методов в динамике, Национальный исследовательский университет «Высшая школа экономики» E-mail: schub.danil@yandex.ru
