19 research outputs found
Global theory of nonlinear systems-chaos, knots and stability
In this paper we shall give a brief overview of nonlinear dynamical systems theory including the theory of chaos, knots, approximation theory and stability
Inversely Unstable Solutions of Two-Dimensional Systems on Genus-p Surfaces and the Topology of Knotted Attractors
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
behaviour. This is achieved by studying the appearance of inversely unstable
solutions within each invariant set.Comment: 19 pages with 20 figures, AMS La-TeX, to be published in
International Journal of Bifurcation and Chao
Topology of complements to real affine space line arrangements
It is shown that the diffeomorphism type of the complement to a real space
line arrangement in any dimensional affine ambient space is determined only by
the number of lines and the data on multiple points.Comment: 13 pages, 14 figures. Submitte
Global analysis of dynamical systems on low-dimensional manifolds.
The interaction of topology and dynamics has attracted a great deal of attention from
numerous mathematicians. This thesis is devoted to the study of dynamical systems
on low-dimensional manifolds.
In the order of dimensions, we first look at the case of two-manifolds (surfaces) and
derive explicit differential equations for dynamical systems defined on generic surfaces
by applying elliptic and automorphic function theory to uniformise the surfaces in
the upper half of the complex plane with the hyperbolic metric. By modifying the
definition of the standard theta series, we will determine general meromorphic systems
on a fundamental domain in the upper half plane, the solution trajectories of which
'roll up' onto an appropriate surface of any given genus. Meanwhile, 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 behaviour. This is achieved by analyzing the topology of inversely unstable
solutions contained within each invariant set.
Then the thesis concerns a study of three-dimensional systems. 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 tubular neighbourhood of the
braid. The second one, generating chaotic systems, is realized by modelling the Smale
horseshoe.
Moreover, we shall consider the analytical and topological structure of systems
on 2- and 3- manifolds. By considering surgery operations, such as Dehn surgery,
Heegaard splittings and connected sums, we shall show that it is possible to obtain
systems with 'arbitrarily strange' behaviour, Le., arbitrary numbers of chaotic regimes
which are knotted and linked in arbitrary ways.
We will also consider diffeomorphisms which are defined on closed 3-manifolds
and contain generalized Smale solenoids as the non-wandering sets. Motivated by the
result in [Jiang, Ni and Wang, 2004], we will investigate the possibility of generating
dynamical systems containing an arbitrary number of solenoids on any closed, orientable
3-manifold. This shall also include the study of branched coverings and Reeb
foliations.
Based on the intense development from four-manifold theory recently, we shall
consider four-dimensional dynamical systems at the end. However, this part of the
thesis will be mainly speculative
Bifurcations of transition states
A transition state for a Hamiltonian system is a closed, invariant, oriented, codimension-2 submanifold of an energy-level that can be spanned by two compact codimension-1 surfaces of unidirectional flux whose union, called a dividing surface, locally separates the energy-level into two components and has no local recrossings. For this to happen robustly to all smooth perturbations, the transition state must be normally hyperbolic. The dividing surface then has locally minimal geometric flux through it, giving a useful upper bound on the rate of transport in either direction.
Transition states diffeomorphic to S2mâ3 are known to exist for energies just above any index-1 critical point of a Hamiltonian of m degrees of freedom, with dividing surfaces S2mâ2. The question addressed here is what qualitative changes in the transition state, and consequently the dividing surface, may occur as the energy or other parameters are varied? We find that there is a class of systems for which the transition state becomes singular and then regains normal hyperbolicity with a change in diffeomorphism class. These are Morse bifurcations.
Continuing the dividing surfaces and transition states through Morse bifurcations allows us to compute the flux for a larger range of energies. The effect of Morse bifurcations on the flux, as a function of energy, is considered and we find a loss of differentiability in the neighbourhood of the bifurcations.
Various examples are considered. Firstly, some simple examples in which transition states connect or disconnect, and the dividing surface may become a torus or other. Then, we show that sequences of Morse bifurcations producing various interesting transition state and dividing surface are present in reacting systems, specifically bimolecular capture processes. We consider first planar reactions, for which the reduction of symmetries is easiest, and then also spatial reactions, where we find interesting Morse bifurcations involving both the attitude degrees of freedom and the angular momentum ones.
In order to consider these examples, we present a method of constructing dividing surfaces spanning general transition states, and also a method to approximate normally hyperbolic submanifolds due to MacKay
The physics of climate variability and climate change
The climate system is a forced, dissipative, nonlinear, complex and heterogeneous system that is out of thermodynamic equilibrium. The system exhibits natural variability on many scales of motion, in time as well as space, and it is subject to various external forcings, natural as well as anthropogenic. This paper reviews the observational evidence on climate phenomena and the governing equations of planetary-scale flow, as well as presenting the key concept of a hierarchy of models as used in the climate sciences. Recent advances in the application of dynamical systems theory, on the one hand, and of nonequilibrium statistical physics, on the other, are brought together for the first time and shown to complement each other in helping understand and predict the system's behavior. These complementary points of view permit a self-consistent handling of subgrid-scale phenomena as stochastic processes, as well as a unified handling of natural climate variability and forced climate change, along with a treatment of the crucial issues of climate sensitivity, response, and predictability
Cascades and transitions in turbulent flows
Turbulence is characterized by the non-linear cascades of energy and other
inviscid invariants across a huge range of scales, from where they are injected
to where they are dissipated. Recently, new experimental, numerical and
theoretical works have revealed that many turbulent configurations deviate from
the ideal 3D/2D isotropic cases characterized by the presence of a strictly
direct/inverse energy cascade, respectively. We review recent works from a
unified point of view and we present a classification of all known transfer
mechanisms. Beside the classical cases of direct and inverse cascades, the
different scenarios include: split cascades to small and large scales
simultaneously, multiple/dual cascades of different quantities, bi-directional
cascades where direct and inverse transfers of the same invariant coexist in
the same scale-range and finally equilibrium states where no cascades are
present, including the case when a condensate is formed. We classify all
transitions as the control parameters are changed and we analyse when and why
different configurations are observed. Our discussion is based on a set of
paradigmatic applications: helical turbulence, rotating and/or stratified
flows, MHD and passive/active scalars where the transfer properties are altered
as one changes the embedding dimensions, the thickness of the domain or other
relevant control parameters, as the Reynolds, Rossby, Froude, Peclet, or Alfven
numbers. We discuss the presence of anomalous scaling laws in connection with
the intermittent nature of the energy dissipation in configuration space. An
overview is also provided concerning cascades in other applications such as
bounded flows, quantum, relativistic and compressible turbulence, and active
matter, together with implications for turbulent modelling. Finally, we present
a series of open problems and challenges that future work needs to address.Comment: accepted for publication on Physics Reports 201
Normal Form of Equivariant Maps and Singular Symplectic Reduction in Infinite Dimensions with Applications to Gauge Field Theory
Inspired by problems in gauge field theory, this thesis is concerned with various
aspects of infinite-dimensional differential geometry.
In the first part, a local normal form theorem for smooth equivariant maps
between tame Fréchet manifolds is established. Moreover, an elliptic version of
this theorem is obtained. The proof these normal form results is inspired by
the LyapunovâSchmidt reduction for dynamical systems and by the Kuranishi
method for moduli spaces, and uses a slice theorem for Fréchet manifolds as
the main technical tool. As a consequence of this equivariant normal form
theorem, the abstract moduli space obtained by factorizing a level set of the
equivariant map with respect to the group action carries the structure of a
Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient
by a compact group of the zero set of a smooth map.
In the second part of the thesis, the theory of singular symplectic reduction
is developed in the infinite-dimensional Fréchet setting. By refining the above
construction, a normal form for momentum maps similar to the classical
MarleâGuilleminâSternberg normal form is established. Analogous to the
reasoning in finite dimensions, this normal form result is then used to show
that the reduced phase space decomposes into smooth manifolds each carrying
a natural symplectic structure.
Finally,the singular symplectic reduction scheme is further investigated in the
situation where the original phase space is an infinite-dimensional cotangent
bundle. The fibered structure of the cotangent bundle yields a refinement of
the usual orbit-momentum type strata into so-called seams. Using a suitable
normal form theorem, it is shown that these seams are manifolds. Taking
the harmonic oscillator as an example, the influence of the singular seams on
dynamics is illustrated.
The general results stated above are applied to various gauge theory models.
The moduli spaces of anti-self-dual connections in four dimensions and of
YangâMills connections in two dimensions is studied. Moreover, the stratified
structure of the reduced phase space of the YangâMillsâHiggs theory is
investigated in a Hamiltonian formulation after a (3 + 1)-splitting
Normal Form of Equivariant Maps and Singular Symplectic Reduction in Infinite Dimensions with Applications to Gauge Field Theory
Inspired by problems in gauge field theory, this thesis is concerned with various
aspects of infinite-dimensional differential geometry.
In the first part, a local normal form theorem for smooth equivariant maps
between tame Fréchet manifolds is established. Moreover, an elliptic version of
this theorem is obtained. The proof these normal form results is inspired by
the LyapunovâSchmidt reduction for dynamical systems and by the Kuranishi
method for moduli spaces, and uses a slice theorem for Fréchet manifolds as
the main technical tool. As a consequence of this equivariant normal form
theorem, the abstract moduli space obtained by factorizing a level set of the
equivariant map with respect to the group action carries the structure of a
Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient
by a compact group of the zero set of a smooth map.
In the second part of the thesis, the theory of singular symplectic reduction
is developed in the infinite-dimensional Fréchet setting. By refining the above
construction, a normal form for momentum maps similar to the classical
MarleâGuilleminâSternberg normal form is established. Analogous to the
reasoning in finite dimensions, this normal form result is then used to show
that the reduced phase space decomposes into smooth manifolds each carrying
a natural symplectic structure.
Finally,the singular symplectic reduction scheme is further investigated in the
situation where the original phase space is an infinite-dimensional cotangent
bundle. The fibered structure of the cotangent bundle yields a refinement of
the usual orbit-momentum type strata into so-called seams. Using a suitable
normal form theorem, it is shown that these seams are manifolds. Taking
the harmonic oscillator as an example, the influence of the singular seams on
dynamics is illustrated.
The general results stated above are applied to various gauge theory models.
The moduli spaces of anti-self-dual connections in four dimensions and of
YangâMills connections in two dimensions is studied. Moreover, the stratified
structure of the reduced phase space of the YangâMillsâHiggs theory is
investigated in a Hamiltonian formulation after a (3 + 1)-splitting
Notes in Pure Mathematics & Mathematical Structures in Physics
These Notes deal with various areas of mathematics, and seek reciprocal
combinations, explore mutual relations, ranging from abstract objects to
problems in physics.Comment: Small improvements and addition