The matrix Riccati equation which arises from optimal control and filtering problems is a quadratic differential equation on the space of real symmetric $n \times n$ matrices. It is closely related, via compactification of the phase space, to the differential equations on the Grassmann manifold and the Lagrange–Grassmann manifold whose flows are generated by the action of one-parameter subgroups of the general linear group and of the symplectic group respectively. We. determine the complete phase portraits of the Riccati equations on all three spaces. The asymptotic behavior of every solution is described. The phase portraits are characterized topologically as well as set-theoretically. Although the Riccati equation is not generally a Morse–Smale vector field, we are able to show that it possesses suitable generalizations of many of the important properties of Morse–Smale vector fields. In particular, the Riccati equation satisfies a generalized version of the Morse inequalities for a Morse–Smale dynamica...

Phase portrait of the matrix Riccati equation

This paper explores the relations between the matrix Riccati equation and the standard matrix eigenvalue methods. It is demonstrated that the mathematics of the analysis of the two objects is essentially the same; consisting of the analysis of flows on the homogeneous spaces of various Lie groups.

The geometry of matrix eigenvalue methods

Let $\g$ be a semisimple Lie algebra and $G = \Int(\g)$. In this article, we relate the Jordan decomposition of $X \in \g$ (or $g \in G$) with the dynamics induced on generalized flag manifolds by the right invariant continuous-time flow generated by $X$ (or the discrete-time flow generated by $g$). We characterize the recurrent set and the finest Morse decomposition (including its stable sets) of these flows and show that its entropy always vanishes. We characterize the structurally stable ones and compute the Conley index of the attractor Morse component. When the nilpotent part of $X$ is trivial, we compute the Conley indexes of all Morse components. Finally, we consider the dynamical aspects of linear differential equations with periodic coefficients in $\g$, which can be regarded as an extension of the dynamics generated by an element $X \in \g$. In this context, we generalize Floquet theory and extend the previous results to this case.

Jordan decomposition and dynamics on flag manifolds

This paper is a survey of relatively recent results on the classification of Morse-Smale dynamical systems on closed manifolds. It also contains both old and relatively recent results on the relationship between the topology of the ambient manifold and the dynamical characteristics of Morse-Smale systems.

Global dynamics of Morse-Smale systems

For orientation preserving diffeomorphisms on the torus necessary and sufficient conditions are given for an isotopy class to admit a Morse-Smale diffeomorphism with a specified periodic behavior. The Morse-Smale diffeomorphisms have been shown to be exactly the structurally stable diffeomorphisms with finite nonwandering set [8]. Several recent papers have analyzed the relationship between the dynamics of a Morse-Smale diffeomorphism and the topology of the manifold on which it is defined ([2], [3], [6], and [11]). In [2] Franks related the periodic behavior of a Morse-Smale diffeomorphism to the action of the map on homology. For a given manifold and homotopy (or isotopy) class of maps on that manifold, this result provides a necessary condition for the Morse-Smale dynamics which can occur in the homotopy (isotopy) class. In [6] Narasimhan showed that in the homotopy class of the identity on a compact surface, a given set of Morse-Smale dynamics can be realized provided it satisfies Franks' condition along with two other necessary conditions. We shall prove this result for orientation preserving isotopy classes on the torus (T2 = S1 x S 1). THEOREM. Let g E Diff(T2) and suppose g preserves orientation. There exists a Morse-Smale diffeomorphism f on T2 isotopic to g and with peridoic data {(pi, U, Ai)}7=1 if and only if (l)u, = Oanduj = 2for some i andj; (2) Hg(t) = lI Ai( tP)( (3) A, = + I for each i with u, E (0, 21. I would like to thank John Franks and Ken Mandelberg for valuable conversations and suggestions. Definitions and background. Let Diff(M) be the space of diffeomorphisms on a compact manifold M. A point x E M is a nonwandering point of f E Diff(M) provided that for any neighborhood U of x there exists a nonzero integer m such that fm(U) n U =# 0. The set of nonwandering points of f is denoted O(f). Two diffeomorphismsf, g E Diff(M) are topologiReceived by the editors August 22, 1978 and, in revised form, November 22, 1978. AMS (MOS) subject classifications (1970). Primary 58F20; Secondary 55C20. ? 1979 American Mathematical Society 0002-9947/79/0000-0569/$03.25 395 This content downloaded from 157.55.39.78 on Mon, 20 Jun 2016 07:28:16 UTC All use subject to http://about.jstor.org/terms

/pdf/the-dynamics-of-morse-smale-diffeomorphisms-on-the-torus-486iqbb0hs.pdf

The dynamics of Morse-Smale diffeomorphisms on the torus

For orientation-reversing diffeomorphisms on the torus necessary and sufficient conditions are given for an isotopy class to admit a Morse-Smale diffeomorphism with a specified periodic behavior. A diffeomorphism is Morse-Smale provided it is structurally stable and has a finite nonwandering set [P-S]. Several recent papers have explored the relationship between the topology of these maps and their dynamics. In [F] Franks showed that the periodic behavior of a Morse-Smale diffeomorphism was restricted by its homology zeta function. For the homotopy class of the identity on a compact surface Narasimhan proved that virtually any periodic behavior consistent with the homology zeta function does indeed occur [N]. For orientation-reversing maps there are obstructions other than the homology zeta function. Blanchard and Franks [B-F] have shown that if an orientation-reversing homeomorphism of S2 has periodic orbits which include two distinct odd periods, then the entropy of that map is positive. This implies that no orientationreversing Morse-Smale diffeomorphism on S2 can have distinct odd periods. The following theorem was conjectured in [B-F] and has been proven by Handel. THEOREM [H]. If f: M2 __ M2 is an orientation-reversing homeomorphism of a compact oriented surface of genus g, and iff has orbits with g + 2 distinct oddperiods, then the entropy off is positive. Thus an orientation-reversing Morse-Smale diffeomorphism on the torus has orbits with at most two different odd periods. In this paper we investigate whether there are any further restrictions on its periodic behavior. In [B] we showed that it suffices to consider the isotopy classes of the toral diffeomorphisms induced by (i 1) and (_1 ). We will show that there is a further obstruction in the former class but not in the latter. I would like to thank John Franks and Lynn Narasimhan for their contributions to this paper. Preliminaries. In this section we state our main result following some necessary definitions and background. We assume the reader is familiar with various standard terms and notation from dynamical systems. Further details are available in [B], [N] and [Ni]. Received by the editors November 26, 1979 and, in revised form, February 25, 1980. AMS (MOS) subject classifications (1970). Primary 58F09, 58F20. 'Research supported in part by Emory University Summer Research Grant. ( 1981 American Mathematical Society 0002-9947/81 /0000-0101 /$03.25

/pdf/orientation-reversing-morse-smale-diffeomorphisms-on-the-8i78olmjda.pdf

Orientation-reversing Morse-Smale diffeomorphisms on the torus

A necessary condition for an isotopy class on a compact surface to admit a Smale diffeomorphism whose dynamics are a specified set of subshifts of finite type is that the Euler characteristic of the manifold be equal to a sum and difference of certain numbers obtained from the matrices representing the subshifts. In this paper it is shown that this condition is sufficient up to a finite power of the subshifts. A diffeomorphism / on a compact manifold is a Smale diffeomorphism if and only if / satisfies Axiom A, the strong transversality condition, and has only zero dimensional basic sets (see (3) or (10) for definitions). Smale diffeomorphisms are structurally stable, dense in the C° topology on Diff(M) and appear in every isotopy class (9), (10), (12). For a Smale diffeomorphism the restriction of the map to a basic set is topologically conjugate to a subshift of finite type and thus can be represen- ted by a nonnegative integer matrix. The reduced degree of an integer matrix A is the degree of the mod 2 reduction of the polynomial det(I — At). In (3) Franks presented results which for a large class of manifolds of dimension greater than 2, related the topology of the manifold to the dynamical behavior of Smale diffeomorphisms on the manifold. In this paper we consider the problem for the 2-dimensional case. More specifically, given irreducible matrices corresponding to subshifts of finite type, we present techniques for isotoping the identity map to a Smale diffeomorphism with powers of the subshifts occurring as basic sets. Combining this with the generahzed Morse inequahties yields:

/pdf/constructing-smale-diffeomorphisms-on-compact-surfaces-3lhk8qeby2.pdf

Constructing smale diffeomorphisms on compact surfaces

A Grassmann transformation is a diffeomorphism on a Grassmann manifold which is induced by an n x n nonsingular matrix. In this paper the structurally stable Grassmann transformations are characterized to be the maps which are induced by matrices whose eigenvalues have distinct moduli. There is exactly one topological conjugacy class of complex structurally stable Grassmann transformations. For the real case the topological classification is determined by the ordering (relative to modulus) of the signs of the eigenvalues of the inducing matrix.

/pdf/structurally-stable-grassmann-transformations-12acr9digo.pdf

Structurally stable Grassmann transformations

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The dynamics of Morse-Smale diffeomorphisms on the torus

Orientation-reversing Morse-Smale diffeomorphisms on the torus

Constructing smale diffeomorphisms on compact surfaces

Structurally stable Grassmann transformations