Table of contents

We study the Hessian of the Yang-Mills-Higgs functional as the self-interaction parameter varies and examine how convergence in the configuration space controls the sign of the first eigenvalue at the limit. As an application, we show that the spherically symmetric solutions of 't Hooft and Polyakov are stable for in the neighbourhood of 0 and that the kernels of their Hessians are precisely the spaces generated by their spatial derivatives.

The degenerate Ginzburg-Landau equation gives a description of patterns which arise in the case of weakly unstable PDEs with a unbounded spatial direction when the Landau constant (characterizing the influence of the nonlinearity) is small. This equation possesses a family of periodic solutions, moreover there exists a band of stable periodic solutions among them. We give the full description of the possible behaviour of the system just outside this stable band. This is done through derivation of the so-called modulated modulation equations. We also study some solutions of these equations among which stationary periodic and heteroclinic solutions, moving solitons, cnoidal waves and front-like solutions are found.

An approach to the centre-focus problem for homogeneous perturbations proposed by Cherkas yields a transformation to periodic Abel equations of degree 3. In this paper we consider both the polynomial and periodic Abel equations of any degree. We define the Bautin ideal for these two classes of Abel equations. Recently an approach based on the use of a 1-parameter integrating factor allowed to find the successive derivatives of the return map for a polynomial system which is a homogeneous perturbation of the rotation at the origin. We present the same type of results for the Abel equations. For the polynomial Abel equations, we show that there is an integrating factor defined by a convergent series expansion with polynomial coefficients which satisfy a simple linear recurrency relation. We solve this recurrency relation for low degrees of the perturbation and compute the Bautin index. We then use our previous findings based on the Bernstein inequality and Bautin index to bound the number of complex periodic solutions on a neighbourhood of prescribed size. For the periodic Abel equations, we show that the existence of the integrating factor is equivalent to the periodicity of all other orbits.

Bogomol'nyi vortices on a disk with Dirichlet boundary conditions are investigated and solutions are shown to exist. The dimension of the moduli space of solutions is twice the vortex number. For one vortex near the boundary of the disk, the moduli space looks like a hyperbolic plane. To check these findings various numerical tests are made on square domains with/without holes.

Spatially quasiperiodic patterns with q independent wavenumbers in each spatial direction, which satisfy a nonlinear partial differential equation (PDE), are considered. For a class of toral-equivariant PDEs, a framework for the existence and linear stability analysis of such patterns is developed. The idea is to reformulate the PDE on the jet space, which for a large class of PDEs is finite dimensional. This reformulation induces an orthogonal action of the toral group on the finite-dimensional jet space. A non-trivial generalization of relative equilibria is introduced and used to characterize quasiperiodic patterns. The geometric framework associated with this generalization of relative equilibria on the jet space leads to a rigourous analysis of the stability equation, for quasiperiodic patterns of arbitrary amplitude, and a general instability criterion for a class of quasiperiodic patterns. We show that this framework is a natural setting for a geometric analysis of patterns, and show how the theory extends to pattern formation PDEs without explicit toral symmetry. The theory is applied to the two-toral pattern solutions of a system of Ginzburg-Landau equations in one and two space dimensions, and to the Swift-Hohenberg equation and its generalizations.

We study a Maxwell-Bloch system describing the dynamics of single-longitudinal Raman lasers in the two transverse space dimensions. Raman lasing is generated by a coherent external pump laser, as a three-wave interaction involving two optical and one material wave. On the other hand, a two-level laser involves incoherent external pumping through, for example, an electrical discharge or flashlamp. Raman lasers have the advantage of being tunable and display a novel explicit nonlinear detuning between the pump and laser emission frequencies. Consequently these lasers exhibit much richer nonlinear dynamics. We establish the global existence of classical solutions, and show that for periodic domains the dynamics is governed by a global smooth attractor of finite dimensions. We explain the structures of nonlinear interactions and couplings that lead to the time asymptotic smoothing. We also construct mild solutions with the dispersive Strichartz inequality for rough but spatially decaying data (in , ) on the whole plane, which physically corresponds to the absorbing boundary condition.

It is known that neither one- nor two-dimensional ordinary differential equations can exhibit chaotic phenomena, while such behaviour is possible in three-dimensional systems. Here it is shown that the solutions of any dissipative equation can be approximated arbitrarily closely by the trajectories of an appropriate three-dimensional system. This class includes the Navier-Stokes equations, showing that the dynamical system which may describe fully developed turbulence can be approximated using just three degrees of freedom.

We investigate a model of a charged particle in a uniform magnetic field with resonant nonlinear forcing. The resonance is chosen so that the problem is invariant under a rank 2 lattice of translations in velocities and thus reduces to iteration of a torus map. The rotation set of this map, at least for well-pronounced nonlinearities, has a positive (and easy to estimate) area. This is an indication of `toroidal chaos', and physically corresponds to stochastical heating of the plasma. Our main result is the monotonicity of the rotation set as a function of the amplitude of the wave in the case when the forcing is `saw-tooth-like'.

We study quadratic, volume-preserving diffeomorphisms whose inverse is also quadratic. Such maps generalize the Hénon area-preserving map and the family of symplectic quadratic maps studied by Moser. In particular, we investigate a family of quadratic volume-preserving maps in three-space for which we find a normal form and study invariant sets. We also give an alternative proof of a theorem by Moser classifying quadratic symplectic maps.

It is proved that periodic and homoclinic trajectories which are tangent to the boundary of any scattering (ergodic) billiard produce elliptic islands in the `nearby' Hamiltonian flows i.e. in a family of two-degrees-of-freedom smooth Hamiltonian flows which converge to the singular billiard flow smoothly where the billiard flow is smooth and continuously where it is continuous. Such Hamiltonians exist; indeed, sufficient conditions are supplied, and thus it is proved that a large class of smooth Hamiltonians converges to billiard flows in this manner. These results imply that ergodicity may be lost in the physical setting, where smooth Hamiltonians which are arbitrarily close to the ergodic billiards, arise.

A diffeomorphism is called a (reversing) k-symmetry of a dynamical system in represented by the diffeomorphism if k is the smallest positive integer for which U is a (reversing) symmetry of (the k-times iterate of f), i.e. .

In this paper we show how k-symmetry naturally arises in the context of return maps of flows with spacetime symmetries. We discuss the connection between periodic orbits of k-symmetric maps and symmetric periodic orbits of the flows they represent, and illustrate the application of our results in local bifurcation theory. We also provide a geometric interpretation of formal (Birkhoff) normal form symmetries for diffeomorphisms as a time-shift symmetry of a locally constructed spacetime symmetric suspension flow.

Finally, we explain the occurrence of dual (representations of) reversing k-symmetry groups in k-symmetric maps in relation to different choices for the position of the surface of section for return maps of spacetime symmetric flows.

We study some aspects of the Bäcklund transformations for the purely bosonic and supersymmetric models in two dimensions. We first construct the Bäcklund transformations for the models as reduced models from the purely bosonic models. We obtain solutions of the considered purely bosonic and supersymmetric linear systems, and give some explicit examples. These solutions will be useful in obtaining new solutions of the models via the Bäcklund transformations. Conclusions and a few comments are given.

We consider the planar three-body problem (3BP) with a Newtonian-like potential of the form with and also the Newtonian case . We study the least action principle on the manifold defined by the vanishing of the total angular momentum. Using the topology of the reduced configuration space we are able to find a class of trajectories on which a generalization of the Poincaré inequality holds, this then enables us to apply the direct method of calculus of variations. We prove the existence of a periodic solution: for the Newtonian potential , this periodic solution has at most a finite number of collisions. For the solution is in the homotopic class of the trajectories that go through at least three different collinear configurations. This result is a direct consequence of the first homotopy group of the reduced configuration space without coincidence set, this group is isomorphic to the free-generated group .

It is commonly believed that drift waves and drift-wave turbulence play a major role in the understanding of anomalous transport at the plasma edge of a tokamak fusion reactor. A one-field equation describing the electrostatic potential fluctuations in this regime is the so-called Hasegawa-Mima equation. If this equation is driven by some instability and damped by some hyperviscous term, the energy grows exponentially in time which is not consistent with the approximations made in the derivation of the equation. Numerical simulations of a perturbed Hasegawa-Mima equation which includes in addition a so-called nonlinearity showed that the energy saturates at a finite level. In this paper this numerical observation is proven analytically.

A representative model of a return map near homoclinic bifurcation is studied. This model is the so-called fattened Arnold map, a diffeomorphism of the annulus. The dynamics is extremely rich, involving periodicity, quasiperiodicity and chaos.

The method of study is a mixture of analytic perturbation theory, numerical continuation, iteration to an attractor and experiments, in which the guesses are inspired by the theory. In turn the results lead to fine-tuning of the theory. This approach is a natural paradigm for the study of complicated dynamical systems.

By following generic bifurcations, both local and homoclinic, various routes to chaos and strange attractors are detected. Here, particularly, the `large' strange attractors which wind around the annulus are of interest. Furthermore, a global phenomenon regarding Arnold tongues is important. This concerns the accumulation of tongues on lines of homoclinic bifurcation. This phenomenon sheds some new light on the occurrence of infinitely many sinks in certain cases, as predicted by the theory.