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''Morse–Smale dynamical system''

A smooth [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m0649901.png" /> or [[Cascade|cascade]] (discrete-time dynamical system) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m0649902.png" /> (generated by a diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m0649903.png" />, which is called a Morse–Smale diffeomorphism in this case) on a compact (usually closed) differentiable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m0649904.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m0649905.png" />, having the following properties:

1) The system has a finite number of periodic trajectories (including fixed points in the case of a cascade) and (in the case of a flow) equilibrium states.

2) Each periodic trajectory listed in 1) has [[Local structural stability|local structural stability]] (usually the definition requires an equivalent property of the corresponding linearized system). This guarantees the existence of stable and unstable invariant manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m0649906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m0649907.png" /> for each such trajectory (if the trajectory is stable, or totally unstable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m0649908.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m0649909.png" />, reduces to the trajectory itself); the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499010.png" /> is called its index. The index generalizes the notion of the [[Morse index|Morse index]] of a non-degenerate critical (stationary) point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499011.png" /> of a smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499012.png" />, since the latter coincides with the index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499013.png" /> as an equilibrium point of the [[Gradient dynamical system|gradient dynamical system]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where the gradient is taken with respect to any Riemannian metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499015.png" />.

3) The invariant manifolds of the trajectories listed in 1) intersect transversely (that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499016.png" />, then for the tangent spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499017.png" />).

4) All remaining trajectories tend, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499018.png" /> or as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499019.png" />, to one of the trajectories listed in 1).

5) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499020.png" /> has a boundary, then some condition must be placed on the behaviour of the system near the boundary. For flows (up to now, only this case has been considered) it is usual to require that the phase velocity vector always be transversal to the boundary.

Morse–Smale systems are structurally stable systems (cf. [[Rough system|Rough system]], [[#References|[1]]]). It was precisely in connection with the study of the latter that special cases of Morse–Smale systems were first discussed — flows in plane domains (see the more detailed account in [[#References|[2]]]) and cascades on the circle (see [[#References|[4]]]–). In the general case Morse–Smale systems were introduced by S. Smale, who considered Morse–Smale systems on a closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499021.png" />, for which the following Morse–Smale inequalities were proved. For a cascade, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499022.png" /> be the number of periodic points of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499023.png" />, and for a flow it denotes the sum of the number of equilibrium positions of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499024.png" /> and the number of periodic trajectories of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499026.png" />. Then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499027.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499029.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499030.png" />-th [[Betti number|Betti number]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499031.png" /> (if some of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499033.png" /> introduced in 2) are non-orientable, then the Betti number is taken over a field of characteristic two). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499034.png" />, (2) becomes an equality.

The inequalities (2) generalize the usual [[Morse inequalities|Morse inequalities]] for a smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499035.png" /> with non-degenerate critical points. Namely, the Morse inequalities can be obtained by an application of (2) to the system (1) (which, in reality, need not be a Morse–Smale system, so minor additional arguments are required, see [[#References|[7]]], [[#References|[8]]]).

The question of when there is a Morse–Smale diffeomorphism in a given isotopy class (see [[#References|[9]]], [[#References|[10]]]) and, for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499036.png" /> with Euler characteristic zero, the analogous question with respect to a homotopy class of non-singular vector fields (here for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499037.png" /> the answer is always positive, see [[#References|[11]]]) has been investigated. For flows with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499038.png" /> (see [[#References|[12]]], [[#References|[13]]]) and certain special types of flows with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499039.png" /> (see [[#References|[14]]], [[#References|[15]]]) it has been clarified which topological invariants determine the topological equivalence of two Morse–Smale systems. (In the two-dimensional case this question has been solved for a broader class of flows (see [[#References|[3]]], [[#References|[16]]]), and the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064990/m06499040.png" /> is trivial.)

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Palis,   S. Smale,   "Structural stability theorems"  S.-S. Chern (ed.)  S. Smale (ed.) , ''Global analysis'' , ''Proc. Symp. Pure Math.'' , '''14''' , Amer. Math. Soc.  (1970)  pp. 223–232</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Andronov,   E.A. Leontovich,   I.I. Gordon,   A.G. Maier,   "Theory of bifurcations of dynamic systems on a plane" , Israel Program Sci. Transl.  (1971)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Andronov,   E.A. Leontovich,   I.I. Gordon,   A.G. Maier,   "Qualitative theory of second-order dynamic systems" , Wiley  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.G. Maier,   "A structurally stable map of the circle onto itself"  ''Uchen. Zap. Gor'k. Gos. Inst.'' , '''12'''  (1939)  pp. 215–229  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.A. Pliss,   "On the structural stability of differential equations on the torus"  ''Vestnik Leningrad. Univ. Ser. Mat.'' , '''15''' :  13  (1960)  pp. 15–23  (In Russian)</TD></TR><TR><TD valign="top">[6a]</TD> <TD valign="top">  V.I. Arnol'd,   "Small denominators I. Mapping the circle onto itself"  ''Transl. Amer. Math. Soc. (2)'' , '''46'''  (1965)  pp. 213–284  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''25''' :  1  (1961)  pp. 21–86</TD></TR><TR><TD valign="top">[6b]</TD> <TD valign="top">  V.I. Arnol'd,   "Correction to  "Small denominators, I. Mapping the circle onto itself" "  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''28''' :  2  (1964)  pp. 479–480  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S. Smale,   "Morse inequalities for dynamical systems"  ''Bull. Amer. Math. Soc.'' , '''66'''  (1960)  pp. 43–49</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  S. Smale,   "On gradient dynamical systems"  ''Ann. of Math. (2)'' , '''74'''  (1961)  pp. 199–206</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  M. Shub,   "Morse–Smale diffeomorphisms are unipotent in homology"  M.M. Peixoto (ed.) , ''Dynamical Systems (Proc. Conf. Salvador, 1971)'' , Acad. Press  (1973)  pp. 489–491</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M. Shub,   D. Sullivan,   "Homology theory and dynamical systems"  ''Topology'' , '''14'''  (1975)  pp. 109–132</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  D. Asimov,   "Homotopy of non-singular vector fields to structurally stable ones"  ''Ann. of Math.'' , '''102''' :  1  (1975)  pp. 55–65</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  M. Peixoto,   "Sur la classification des équations différentielles"  ''C.R. Acad. Sci. Paris'' , '''272'''  (1971)  pp. A262-A265</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  M.M. Peixoto,   "Dynamical systems"  M.M. Peixoto (ed.) , ''Dynamical Systems (Proc. Conf. Salvador, 1971)'' , Acad. Press  (1973)  pp. 389–419</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  Ya.L. Umanskii,   "The scheme of a 3-dimensional Morse–Smale dynamical system without closed trajectories"  ''Soviet Math. Dokl.'' , '''17'''  (1976)  pp. 1479–1482  ''Dokl. Akad. Nauk SSSR'' , '''230''' :  6  (1976)  pp. 1286–1289</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  S.Yu. Pilyugin,   "Phase diagrams that determine Morse–Smale systems without periodic trajectories on spheres"  ''Diff. Eq.'' , '''14''' :  2  (1978)  pp. 170–177  ''Diff. Uravnen.'' , '''14''' :  2  (1978)  pp. 245–254</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top">  D. Neumann,   T. O'Brien,   "Global structure of continuous flows on 2-manifolds"  ''J. Diff. Eq.'' , '''22''' :  1  (1976)  pp. 89–110</TD></TR></table>



====Comments====
Condition (2) above is often called hyperbolicity of the periodic trajectories and invariant points (cf. [[Hyperbolic set|Hyperbolic set]]). Condition , together with condition (1), is often formulated as: The non-wandering set (cf. [[Non-wandering point|Non-wandering point]]) consists only of a finite number of periodic trajectories and invariant points (each of which is, by (2), hyperbolic).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Smale,   "Diffeomorphisms with many periodic points"  S.S. Cairns (ed.) , ''Differential and Combinatorial Topol. (Symp. in honor of M. Morse)'' , Princeton Univ. Press  (1965)  pp. 63–80</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Shub,   "Dynamical systems, filtrations and entropy"  ''Bull. Amer. Math. Soc.'' , '''80'''  (1974)  pp. 27–41</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Frants,   M. Shub,   "The existence of Morse–Smale diffeomorphisms"  ''Topology'' , '''20'''  (1981)  pp. 273–290</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Maller,   "Fitted diffeomorphisms of non-simply connected manifolds"  ''Topology'' , '''19'''  (1980)  pp. 395–410</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Palis,   "On Morse–Smale dynamical systems"  ''Topology'' , '''8'''  (1969)  pp. 385–405</TD></TR></table>