Elliptic and Automorphic Dynamical Systems on Surfaces

2008

In this paper, we will show that a periodic nonlinear, time-varying dissipative system that is defined on a genus-p surface contains one or more invariant sets which act as attractors. Moreover, we shall generalize a result in [Martins, 2004] and give conditions under which these invariant sets are not homeomorphic to a circle individually, which implies the existence of chaotic behavior. This is achieved by studying the appearance of inversely unstable solutions within each invariant set.

Section: Systems On Genus-p Surfacesmentioning

confidence: 99%

INVERSELY UNSTABLE SOLUTIONS OF TWO-DIMENSIONAL SYSTEMS ON GENUS-p SURFACES AND THE TOPOLOGY OF KNOTTED ATTRACTORS

2008

“…In the second part of the paper we shall consider a more general approach to the study of dynamical systems on three manifolds, using Heegaard splittings and the theory developed recently in [Banks & Song, 2006] for the structure of general dynamical systems on surfaces, using automorphic function theory.…”

Section: Figure 1: Constructing Knots From Braids By Gluing the Two Ementioning

confidence: 99%

Dynamical Systems on Three Manifolds Part I: Knots, Links and Chaos

Diaz

2007

In this paper, we give an explicit construction of dynamical systems (defined within a solid torus) containing any knot (or link) and arbitrarily knotted chaos. The first is achieved by expressing the knots in terms of braids, defining a system containing the braids and extending periodically to obtain a system naturally defined on a torus and which contains the given knotted trajectories. To get explicit differential equations for dynamical systems containing the braids, we will use a certain function to define a tube neigbourhood of the braid. The second one, generating chaotic systems, is realized by modelling the Smale horseshoe.

“…Modifying systems along links to generate Pseudo-Anosov diffeomorphisms has been considered in [Lozano, 1997]. Here we will apply the results of [Banks and Song, 2006] to generate an analytic (automorphic) system on one manifold and induce a twisted version on the other manifold by using the so-called C-homeomorphisms of [Lickorish, 1962].…”

Section: Gluing Two Systemsmentioning

confidence: 99%

“…In a recent paper ( [Banks & Song, 2006]), we have shown how to define general (analytic) systems on 2-manifolds by using the theory of automorphic functions. The importance of this theory to dynamical systems is that, globally, they are defined not on 'flat' Euclidean spaces, but on manifolds.…”

Section: Introductionmentioning

confidence: 99%

“…The importance of this theory to dynamical systems is that, globally, they are defined not on 'flat' Euclidean spaces, but on manifolds. In fact, it was shown in ( [Banks & Song, 2006]) that the simple pendulum 'sits' naturally on a Klein bottle. In this paper, we consider the case of three-dimensional systems and derive some results on the nature of threedimensional dynamical systems and the 3-manifolds on which they 'live'.…”

Section: Introductionmentioning

confidence: 99%

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DYNAMICAL SYSTEMS ON THREE MANIFOLDS PART II: THREE-MANIFOLDS, HEEGAARD SPLITTINGS AND THREE-DIMENSIONAL SYSTEMS

2007

The global behaviour of nonlinear systems is extremely important in control and systems theory since the usual local theories will only give information about a system in some neighbourhood of an operating point. Away from that point, the system may have totally different behaviour and so the theory developed for the local system will be useless for the global one.In this paper we shall consider the analytical and topological structure of systems on 2-and 3-manifolds and show that it is possible to obtain systems with 'arbitrarily strange' behaviour, i.e., arbitrary numbers of chaotic regimes which are knotted and linked in arbitrary ways. We shall do this by considering Heegaard Splittings of these manifolds and the resulting systems defined on the boundaries.