44 research outputs found
New lower bounds for the Hilbert numbers using reversible centers
Altres ajuts: UNAB13-4E-1604 (FEDER)In this paper we provide the best lower bounds, that are known up to now, for the Hilbert numbers of polynomial vector fields of degree N,, for small values of N. These limit cycles appear bifurcating from symmetric Darboux reversible centers with very high simultaneous cyclicity. The considered systems have, at least, three centers, one on the reversibility straight line and two symmetric outside it. More concretely, the limit cycles are in a three nests configuration and the total number of limit cycles is at least 2n + m, for some values of n and m. The new lower bounds are obtained using simultaneous degenerate Hopf bifurcations. In particular, H(4) ≥ 28, H(5) ≥ 37, H(6) ≥ 53, H(7) ≥ 74, H(8) ≥ 96, H(9) ≥ 120 and H(10) ≥ 142
Bifurcation of Limit Cycles in Smooth and Non-smooth Dynamical Systems with Normal Form Computation
This thesis contains two parts. In the first part, we investigate bifurcation of limit cycles around a singular point in planar cubic systems and quadratic switching systems. For planar cubic systems, we study cubic perturbations of a quadratic Hamiltonian system and obtain 10 small-amplitude limit cycles bifurcating from an elementary center, for which up to 5th-order Melnikov functions are used. Moreover, we prove the existence of 12 small-amplitude limit cycles around a singular point in a cubic system by computing focus values. For quadratic switching system, we develop a recursive algorithm for computing Lyapunov constants. With this efficient algorithm, we obtain a complete classification of the center conditions for a switching Bautin system. Moreover, we construct a concrete example of switching system to obtain 10 small-amplitude limit cycles bifurcating from a center.
In the second part, we derive two explicit, computationally explicit, recursive formulas for computing the normal forms, center manifolds and nonlinear transformations for general n-dimensional systems, associated with Hopf and semisimple singularities, respectively. Based on the formulas, we develop Maple programs, which are very convenient for an end-user who only needs to prepare an input file and then execute the program to “automatically” generate the results. Several examples are presented to demonstrate the computational efficiency of the algorithms. In addition, we show that a simple 3-dimensional quadratic vector field can have 7 small-amplitude limit cycles, bifurcating from a Hopf singular point. This result is surprising higher than the Bautin’s result for quadratic planar vector fields which can only have 3 small-amplitude limit cycles around an elementary focus or an elementary center
Quanta of Maths
The work of Alain Connes has cut a wide swath across several areas of math- ematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics
Brane Tilings and Their Applications
We review recent developments in the theory of brane tilings and
four-dimensional N=1 supersymmetric quiver gauge theories. This review consists
of two parts. In part I, we describe foundations of brane tilings, emphasizing
the physical interpretation of brane tilings as fivebrane systems. In part II,
we discuss application of brane tilings to AdS/CFT correspondence and
homological mirror symmetry. More topics, such as orientifold of brane tilings,
phenomenological model building, similarities with BPS solitons in
supersymmetric gauge theories, are also briefly discussed.
This paper is a revised version of the author's master's thesis submitted to
Department of Physics, Faculty of Science, the University of Tokyo on January
2008, and is based on his several papers: math.AG/0605780, math.AG/0606548,
hep-th/0702049, math.AG/0703267, arXiv:0801.3528 and some works in progress.Comment: 208 pages, 92 figures, based on master's thesis; v2: minor
corrections, to appear in Fortschr. Phy
Topological Strings and Quantum Curves
This thesis presents several new insights on the interface between
mathematics and theoretical physics, with a central role for fermions on
Riemann surfaces. First of all, the duality between Vafa-Witten theory and WZW
models is embedded into string theory. Secondly, this model is generalized to a
web of dualities connecting topological string theory and N=2 supersymmetric
gauge theories to a configuration of D-branes that intersect over a Riemann
surface. This description yields a new perspective on topological string theory
in terms of a KP integrable system based on a quantum curve. Thirdly, this
thesis describes a geometric analysis of wall-crossing in N=4 string theory.
And lastly, it offers a novel approach to construct metastable vacua in type
IIB string theory.Comment: PhD thesis, July 2009, 308 pages, 65 figure
New Directions for Contact Integrators
Contact integrators are a family of geometric numerical schemes which
guarantee the conservation of the contact structure. In this work we review the
construction of both the variational and Hamiltonian versions of these methods.
We illustrate some of the advantages of geometric integration in the
dissipative setting by focusing on models inspired by recent studies in
celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282
Dynamics of Patterns
Patterns and nonlinear waves arise in many applications. Mathematical descriptions and analyses draw from a variety of fields such as partial differential equations of various types, differential and difference equations on networks and lattices, multi-particle systems, time-delayed systems, and numerical analysis. This workshop brought together researchers from these diverse areas to bridge existing gaps and to facilitate interaction
Topological Interpretation of Non-Linear Transonic Aeroelastic Phenomena
Recent computational studies based on numerical solutions of the unsteady Euler equations have revealed hitherto unanticipated transonic aeroelastic phenomena characterized by non-linear flutter and divergence and interactions between divergence and flutter. In the absence of extensive parametric searches, however, such quantitative studies provide little insight into the nature and extent of such bifurcational behaviour, particularly when multiple parameters are involved. Qualitative dynamical systems theory offers a complementary approach to the analysis of bifurcation problems. In the vicinity of bifurcation, the qualitative behaviour of complex dynamical systems can often be characterized by simple ordinary differential equation models. Of particular interest are the simplest models which exhibit complex interactions characteristic of those observed in non-linear aeroelastic systems. Such models offer scope for attaining greater insight into the nature of complex aeroelastic bifurcations and for systematically predicting qualitative changes resulting from parameter variations. The present work describes the elements of a qualitative, or topological, model identification strategy for a general class of aerodynamically non-linear hereditary aeroelastic systems. The methodology is motivated, principally, by a desire to circumvent the need for detailed knowledge of the unsteady aerodynamic environment. The approach employs a notional non-linear functional description of the aerodynamic force response free from any low-frequency or quasisteady aerodynamic assumptions. Application of the scheme to the transonic aeroelastic problem demonstrates the feasibility and limitations of qualitative techniques. Based on partial bifurcational information derived from published numerical solutions of the coupled aerodynamic and structural equations of motion, a simplified model is identified which captures the (local) bifurcational behaviour of a structurally linear typical section aerofoil in 2-D transonic flow. The model facilitates a new interpretation of divergence/flutter interaction phenomena in transonic flow, including the effects of structural asymmetry, and highlights some difficulties of definition and interpretation of non-linear flutter. Evidence is presented which suggests the existence of new transonic aeroelastic phenomena not previously encountered in computational studies