Morse-Smale system

Morse–Smale dynamical system

A smooth flow (continuous-time dynamical system) $ \{ S _ {t} \} $ or cascade (discrete-time dynamical system) $ \{ S ^ {n} \} $( generated by a diffeomorphism $ S $, which is called a Morse–Smale diffeomorphism in this case) on a compact (usually closed) differentiable $ m $- dimensional manifold $ M ^ {m} $, 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 (usually the definition requires an equivalent property of the corresponding linearized system). This guarantees the existence of stable and unstable invariant manifolds $ W ^ {s} $ and $ W ^ {u} $ for each such trajectory (if the trajectory is stable, or totally unstable, $ W ^ {u} $, respectively $ W ^ {s} $, reduces to the trajectory itself); the dimension of $ W ^ {u} $ is called its index. The index generalizes the notion of the Morse index of a non-degenerate critical (stationary) point $ w _ {0} $ of a smooth function $ f : M \rightarrow \mathbf R $, since the latter coincides with the index of $ w $ as an equilibrium point of the gradient dynamical system

$$ \tag{1 } \dot{w} = - \mathop{\rm grad} f ( w) , $$

where the gradient is taken with respect to any Riemannian metric on $ M $.

3) The invariant manifolds of the trajectories listed in 1) intersect transversely (that is, if $ w \in W _ {1} ^ {s} \cap W _ {2} ^ {u} $, then for the tangent spaces $ T _ {w} W _ {1} ^ {s} + T _ {w} W _ {2} ^ {u} = T _ {w} M $).

4) All remaining trajectories tend, as $ t \rightarrow \pm \infty $ or as $ n \rightarrow \pm \infty $, to one of the trajectories listed in 1).

5) If $ M $ 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, [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 [2]) and cascades on the circle (see [4]–). In the general case Morse–Smale systems were introduced by S. Smale, who considered Morse–Smale systems on a closed $ M $, for which the following Morse–Smale inequalities were proved. For a cascade, let $ m _ {i} $ be the number of periodic points of index $ i $, and for a flow it denotes the sum of the number of equilibrium positions of index $ i $ and the number of periodic trajectories of indices $ i $ and $ i + 1 $. Then for $ i = 0 \dots m $,

$$ \tag{2 } \sum_{j=0}^ { i } ( - 1 ) ^ {j} m _ {i-j} \geq \ \sum_{j=0}^ { i } ( - 1 ) ^ {j} b _ {i-j} , $$

where $ b _ {i} $ is the $ i $- th Betti number of $ M $( if some of the $ W ^ {u} $, $ W ^ {s} $ introduced in 2) are non-orientable, then the Betti number is taken over a field of characteristic two). For $ i = m $, (2) becomes an equality.

The inequalities (2) generalize the usual Morse inequalities for a smooth function $ f : M \rightarrow \mathbf R $ 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 [7], [8]).

The question of when there is a Morse–Smale diffeomorphism in a given isotopy class (see [9], [10]) and, for an $ M ^ {m} $ with Euler characteristic zero, the analogous question with respect to a homotopy class of non-singular vector fields (here for $ m \geq 4 $ the answer is always positive, see [11]) has been investigated. For flows with $ m = 2 $( see [12], [13]) and certain special types of flows with $ m \geq 3 $( see [14], [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 [3], [16]), and the case $ m = 1 $ is trivial.)

References

[1]	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
[2]	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) MR0344606
[3]	A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Qualitative theory of second-order dynamic systems" , Wiley (1973) (Translated from Russian) MR0350126 Zbl 0282.34022
[4]	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)
[5]	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)
[6a]	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
[6b]	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) Zbl 0369.58011
[7]	S. Smale, "Morse inequalities for dynamical systems" Bull. Amer. Math. Soc. , 66 (1960) pp. 43–49
[8]	S. Smale, "On gradient dynamical systems" Ann. of Math. (2) , 74 (1961) pp. 199–206 MR0133139 Zbl 0136.43702
[9]	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
[10]	M. Shub, D. Sullivan, "Homology theory and dynamical systems" Topology , 14 (1975) pp. 109–132
[11]	D. Asimov, "Homotopy of non-singular vector fields to structurally stable ones" Ann. of Math. , 102 : 1 (1975) pp. 55–65
[12]	M. Peixoto, "Sur la classification des équations différentielles" C.R. Acad. Sci. Paris , 272 (1971) pp. A262-A265
[13]	M.M. Peixoto, "Dynamical systems" M.M. Peixoto (ed.) , Dynamical Systems (Proc. Conf. Salvador, 1971) , Acad. Press (1973) pp. 389–419
[14]	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
[15]	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
[16]	D. Neumann, T. O'Brien, "Global structure of continuous flows on 2-manifolds" J. Diff. Eq. , 22 : 1 (1976) pp. 89–110

Comments

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

References

[a1]	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 MR0182020 Zbl 0142.41103
[a2]	M. Shub, "Dynamical systems, filtrations and entropy" Bull. Amer. Math. Soc. , 80 (1974) pp. 27–41 MR0334284 Zbl 0305.58014
[a3]	J. Frants, M. Shub, "The existence of Morse–Smale diffeomorphisms" Topology , 20 (1981) pp. 273–290
[a4]	M. Maller, "Fitted diffeomorphisms of non-simply connected manifolds" Topology , 19 (1980) pp. 395–410
[a5]	J. Palis, "On Morse–Smale dynamical systems" Topology , 8 (1969) pp. 385–405