TITLE

# Novikov-Morse theory for dynamical systems

AUTHOR(S)
H. Fan; J. Jost
PUB. DATE
May 2003
SOURCE
Calculus of Variations & Partial Differential Equations;May2003, Vol. 17 Issue 1, p29
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
Abstract. The present paper contains an interpretation and generalization of Novikov's theory for Morse type inequalities for closed 1-forms in terms of concepts from Conley's theory for dynamical systems. We introduce the concept of a flow carrying a cocycle FORMULA>, (generalized) FORMULA>-flow for short, where FORMULA> is a continuous cocycle in bounded Alexander-Spanier cohomology theory. Gradient-like flows can then be characterized as flows carrying a trivial cocycle. We also define FORMULA>-Morse-Smale flows that allow the existence of "cycles" in contrast to the usual Morse-Smale flows. FORMULA>-flows without fixed points carry not only a cocycle, but a cohomology class, in the sense of [8], and we shall deduce a vanishing theorem for generalized Novikov numbers in that situation. By passing to a suitable cover of the underlying compact polyhedron adapted to the cocycle FORMULA>, we construct a so-called $\pi$-Morse decomposition for an FORMULA>-flow. On this basis, we can use the Conley index to derive generalized Novikov-Morse inequalitites, extending those of M. Farber [12]. In particular, these inequalities include both the classical Morse type inequalities (corresponding to the case when FORMULA> is a coboundary) as well as the Novikov type inequalities ( when FORMULA> is a nontrivial cocycle).
ACCESSION #
12008084

## Related Articles

• Short-time dynamics of a packing of polyhedral grains under horizontal vibrations. Az�ma, E.; Radja�, F.; Peyroux, R.; Richefeu, V.; Saussine, G. // European Physical Journal E -- Soft Matter;Jul2008, Vol. 26 Issue 3, p327

We analyze the dynamics of a 3D granular packing composed of particles of irregular polyhedral shape confined inside a rectangular box with a retaining wall subjected to horizontal harmonic forcing. The simulations are performed by means of the contact dynamics method for a broad set of loading...

• Two-edge connected subgraphs with bounded rings: Polyhedral results and Branch-and-Cut. Fortz, B.; Mahjoub, A. R.; McCormick, S. T.; Pesneau, P. // Mathematical Programming;Jan2006, Vol. 105 Issue 1, p85

We consider the network design problem which consists in determining at minimum cost a 2-edge connected network such that the shortest cycle (a â€œringâ€) to which each edge belongs, does not exceed a given length K. We identify a class of inequalities, called cycle inequalities, valid...

• Optimization of Output Feedback Control Under Set-Membership Uncertainty. Kurzhanski, A.; Varaiya, P. // Journal of Optimization Theory & Applications;Oct2011, Vol. 151 Issue 1, p11

This paper presents a description of solution approaches to the problem of output feedback control under unknown but bounded disturbances with hard bounds on the controls and the uncertain items. The problem is treated within a finite horizon which requires to track the system dynamics...

• Superintegrable Systems and Higher Rank Factorizations. Negro, Javier; Calzada, Juan A.; del Olmo, Mariano A. // AIP Conference Proceedings;2006, Vol. 809 Issue 1, p86

We consider a class of two-dimensional super-integrable systems that can be considered as the natural generalization of some well known one-dimensional factorized systems. Using standard methods to find the shape-invariant intertwining operators we find an so(6) dynamical algebra and its...

• Generating equidistant representations in biobjective programming. Faulkenberg, Stacey; Wiecek, Margaret // Computational Optimization & Applications;Apr2012, Vol. 51 Issue 3, p1173

In recent years, emphasis has been placed on generating quality representations of the nondominated set of multiobjective optimization problems. This paper presents two methods for generating discrete representations with equidistant points for biobjective problems with solution sets determined...

• POLYHEDRAL PROGRAMMING IN LINEAR MULTISTEP DYNAMIC PURSUIT GAMES. Filimonov, Nikolaj B. // Proceedings of the International Conference on Systems for Autom;2003, p1

The class of linear multistep dynamic pursuit games is discussed. The method of attack to the solution of the given problems on the basis of the principle of the guaranteed predicted error with use of the polyhedral programming methodology is suggested Key words: dynamic pursuit game; polyhedral...

• On Polyhedral Projection and Parametric Programming. Jones, C. N.; Kerrigan, E. C.; Maciejowski, J. M. // Journal of Optimization Theory & Applications;Aug2008, Vol. 138 Issue 2, p207

This paper brings together two fundamental topics: polyhedral projection and parametric linear programming. First, it is shown that, given a parametric linear program (PLP), a polyhedron exists whose projection provides the solution to the PLP. Second, the converse is tackled and it is shown how...

• On the structure of the optimal server control for fluid networks. Gajrat, S.; Hordijk, A. // Mathematical Methods of Operations Research;2005, Vol. 62 Issue 1, p55

This paper derives the optimal trajectories in a general fluid network with server control. The stationary optimal policy in the complete state space is constructed. The optimal policy is constant on polyhedral convex cones. An algorithm is derived that computes these cones and the optimal...

• Convex duality and the Skorokhod Problem. II. Dupuis, Paul; Ramanan, Kavita // Probability Theory & Related Fields;1999, Vol. 115 Issue 2, p197

In this paper we consider Skorokhod Problems on polyhedral domains with a constant and possibly oblique constraint direction specified on each face of the domain, and with a corresponding cone of constraint directions at the intersection of faces. In part one of this paper we used convex duality...

Share