Existence and Multiplicity of Fast Homoclinic Solutions for a Class of Damped Vibration Problems with Impulsive Effects

This paper is concerned with the existence and multiplicity of fast homoclinic solutions for a class of damped vibration problems with impulsive effects. Some new results are obtained under more relaxed conditions by using Mountain Pass Theorem and Symmetric Mountain Pass Theorem in critical point theory. The results obtained in this paper generalize and improve some existing works in the literature

Nearly inviscid Faraday waves

Many powerful techniques from Hamiltonian mechanics are available for the study of ideal hydrodynamics. This article explores some of the consequences of including small viscosity in a study of surface gravity-capillary waves excited by the vertical vibration of a container. It is shown that in this system, as in others, the addition of small viscosity provides a singular perturbation of the ideal fluid system, and that as a result its effects are nontrivial. The relevance of existing studies of ideal fluid problems is discussed from this point of view

Nonlinear resonance and excitability in interconnected systems

Engineering design amounts to develop components and interconnect them to obtain a desired behaviour. While in the context of equilibrium dynamics there is a well-developed theory that can account for robustness and optimality in this process, we still lack a corresponding methodology for nonequilibrium dynamics and in particular oscillatory behaviours. With the aim of fostering such a theory, this thesis studies two basic interconnections in the contexts of nonlinear resonance and excitability, two phenomena with the potential of encompassing a large number of applications. The first interconnection is considered in the context of vibration absorption. It corresponds to coupling two Duffing oscillators, the prototypical example of nonlinear resonator. Of primary interest is the frequency response of the system, which quantifies the behaviour in presence of harmonic forces. The analysis focuses on how isolated families of solutions appear and merge with a main one. Using singularity theory it is possible to organise these solutions in the space of parameters and delimit their presence through numerical methods. The second interconnection studied in this dissertation appears in the context of excitable circuits. Combining a fast excitable system and a slower oscillatory system that share a similar structure naturally leads to bursting. The resulting system has a slow-fast structure that can be leveraged in the analysis. The first step of this analysis is a novel slow-fast model of bistability between a rest state and a spiking attractor. Following this, the analysis moves to the complete interconnection, and in particular on how it can generate different patterns of bursting activity

Predicting incipent instabilities and bifurcations of nonlinear dynamical systems modelling compliant off-shore structures

For engineers, the two most important aspects of dynamical analysis are high amplitude resonance vibrations and structural stability, i.e. whether a steady state solution is stable under small perturbations. For the former case, a novel and simple method based on Poincare mapping technique has been devised to predict an imminent flip bifurcation. This bifurcation represents the beginning of the second order subharmonic response. For the latter case, we discovered that while classical quantitative analytical techniques work well in establishing the 'local' structural stability of a steady state solution, the global geometric structure of the catchment region can alter dramatically such that even an initial condition close to the steady state can diverge from it rather than being attracted. This phenomenon known as fractal basin boundary occurs when the invariant manifolds of the saddle separating the steady state solution from any remote attractor cross. The critical point in which the invariant manifolds just touch can be accurately predict by the Melinkov's method. Because of the complicated interwoven nature of the invariant manifolds, it is called a tangle. If the invariant manifolds are originated from the same saddle, the crossing is known as a homoclinic tangle, if originated from different saddle, a heteroclinic tangle. The critical point is then known as homoclinic or heteroclinic tangency. Tangles are also intimately related to chaotic behaviour. The creation and destruction of chaotic attractors have been observed through a series of homoclinic and heteroclinic tangency. In fact, after the invariant manifolds of an inverting saddle cross, the unstable manifold becomes the chaotic attractor. This leads us to believe that all chaotic attractors are topologically the same