Despite the lack of a universal mathematical framework for the analysis of chaotic systems, development of analysis techniques has been swift due both to the research interest in chaos theory from many diverse subjects and to the vast computing resources now available. However, many potential applications remain in the realm of theory. This thesis provides a link between chaos theory and communication systems, with the aim of developing a modulation technique capable of implementation with current technology. Although potentially very simple, chaotic systems result in extremely complex behaviour, resulting in loss of predictability in the long term. Their sensitive dependence on initial conditions enables large-scale changes to be effected with small control perturbations, and their aperiodic behaviour results in broad power spectra. The potential for an electronically simple chaos-based transmitter architecture, employing direct modulation of the passband chaotic oscillator, provides the motivation for this research. Building on the fundamental properties and analysis tools of chaotic systems, an intuitive exploration of the chaos generation mechanisms of several single-scroll chaotic attractors is developed, leading to the creation of several novel chaotic attractors and a discussion of their applications. The concept of a flexible chaotic oscillator-based transmitter architecture which is compatible with standard synchronisation and demodulation techniques is considered, and the resulting noise performance of both simulation and hardware implementations found to be competitive with standard modulation techniques. Additionally, a powerful baseband simulation technique is proposed and implemented, leading to the suggestion of a digital baseband implementation of the transmitter and its potential applications. , Avenues of further work are identified, providing direction for improvements of the proposed system and other related branches worthy of further study.

Complex systems play an important role across science and technology. However, their interwoven structures and nonlinear dynamics makes analysis and control a challenge. We developed a range of mathematical measures and computational tools to help understand these systems. This was based on the formalism of an Evolving Dynamical Network (EDN) designed to coherently capture their networked structure, dynamics and evolution. Bringing these aspects together, we investigated how synchronisation of a dynamical network can be enhanced by computational evolution of the underlying structure. Unlike other methods that solely consider topological features during evolution, we allowed for simulated dynamics to guide this process. Standard network analysis of the resultant topologies illustrated many similarities between the networks, but also highlighted important differences for those evolved using simulated dynamics. We found the existence of a topological bifurcation as connection strengths were increased, and the robust emergence of localised structures called motifs. To bridge the gap between localised structures and system-level architecture, we explored how motifs became pieced together and clustered in a broad range of real-world systems. Clustering was found to be pervasive, but striking differences were seen in the precise forms exhibited. We showed how these biases were linked to architectural features of the networks and in some cases remnants of their evolution by duplication-based processes. Furthermore, because duplication could not fully account for the biases seen, motif clustering may also have functional benefits that are positively selected for during evolution. These findings provide new opportunities to extract rules for the ways that motifs interact and evolve. This has implications for the analysis of existing systems and is of great use in the development of new complex systems. This thesis is organised into two parts. The first considers the theory and methods, and the second applies these to a range of complex systems.

The process of blending observations and numerical models is called in the environmental sciences community, data assimilation. Data assimilation schemes produce an analysis state, which is the best estimate to the state of the system. Error in the analysis state, which is due to errors in the observations and numerical models, is called the analysis error. In this thesis we formulate an expression for the analysis error as the data assimilation procedure is cycled in time and derive results on the boundedness of the analysis error for a number of different data assimilation schemes. Our work is focused on infinite dimensional dynamical systems where the equation which we solve is ill-posed. We present stability results for diagonal dynamical systems for a three-dimensional variational data assimilation scheme. We show that increasing the assumed uncertainty in the background state, which is an a priori estimate to the state of the system, leads to a bounded analysis error. We demonstrate for general linear dynamical systems that if there is uniform dissipation in the model dynamics with respect to the observation operator, then regularization can be used to ensure stability of many cycled data assimilation schemes. Under certain conditions we show that cycled data assimilation schemes that update the background error covariance in a general way remain stable for all time and demonstrate that many of these conditions hold for the Kalman filter. Our results are extended to dynamical system where the model dynamics are nonlinear and the observation operator is linear. Under certain Lipschitz continuous and dissipativity assumptions we demonstrate that the assumed uncertainty in the background state can be increased to ensure stability of cycled data assimilation schemes that update the background error covariance. The results are demonstrated numerically using the two-dimensional Eady model and the Lorenz 1963 model.

Nonlinear dynamics is prevalent in chemical engineering systems and gives rise to mique characteristics, such as bistability, hysteresis, oscillations and chaos. Chaos or aperiodicity is a unique nonlinear dynamical phenomenon which induces unpredictable oscillations and extreme sensitivity to reaction conditions. Analysis of nonlinear reaction dynamics is therefore necessary to ensure reliable and robust performance of systems.

The main focus of this thesis is the dynamics of a two-degree-of-freedom Hamiltonian system near an elliptic equilibrium point in 1 : ±2 resonance, as described by its integrable resonant normal form approximation. After applying singular reduction, the system is studied on the reduced phase space that Kummer called the "orbit manifold". We give a complete description of the critical values of the energy–momentum mapping which than enables us to study the topology of the regular, and more importantly, the singular fibres. We then derive equations for the period of the reduced system, for the rotation number and the non-trivial action. Although some of these results are not new, our approach does not rely on compact fibration and is based on a similar approach introduced earlier by S. Vũ Ngoc for focus–focus points. The non-trivial action of the system enables us to establish fractional monodromy very elegantly by deriving the transition matrix for the actions directly. Both the isoenergetic non-degeneracy condition and the Kolmogorov non-degeneracy condition of KAM theory are derived and analysed for the resonant case. It tuns out that the twist vanishes in a neighbourhood of the equilibrium point for the sign-indefinite case. The Kolmogorov condition, however, is always satisfied near 1 : ±2 resonant equilibria.

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This thesis deals with two aspects of dynamics in systems described by multi-symplectic partial differential equations. The first part is devoted to the study of heteroclinic intersections in systems which govern the dynamics of travelling waves in multi-symplectic partial differential equations with perturbations. In this study a version of the Melnikov method is developed which takes into account the symmetry of the systems under consideration. The presence of the symmetry leads to various interesting differences between the method we develop and the standard approach. In particular, a result about persistence of the fixed point of the Poincare map under perturbations has to be amended since the unperturbed fixed point is non-hyperbolic. The symmetry also results in the necessity to consider separately two cases: when the perturbation has no component in the group direction at all, or when on average it has no component in the group direction when evaluated on the unperturbed solutions. For each of those cases we discuss persistence of the fixed points of the Poincare map and persistence of invariant manifolds, where the knowledge of the symmetry in incorporated in the geometrical constructions. Finally, we derive Melnikov-type conditions in both aforementioned cases which guarantee the existence of transverse intersections of the stable and unstable manifolds. We discuss some possible areas of applications of the Melnikov-type method derived and illustrate the method on the examples of a perturbed Korteweg-de Vries equation and a perturbed nonlinear Schrodinger equation. Implications of the transverse or topological intersections of the manifolds for possible chaotic behaviour in the systems are discussed together with directions of further investigation. The second part of this thesis considers the stability of solitary waves with respect to perturbations which are transverse to the basic direction of propagation of these waves. Using various analytical and numerical techniques, we study this problem for the solitary waves of the (2+l)-dimensional Boussinesq equation and the generalised fifth-order Kadomtsev-Petviashvili equation. For both equations we use a geometric condition for transverse instability based on the multi-symplectic formulation of the equation to derive a condition for transverse instability in the long-wavelength regime. Then an Evans function approach is employed to determine the dependence of the instability growth rate on the transverse wavenumber for all possible wavenumbers. In the case of the (2+l)-dimensional Boussinesq equation this is done analytically, while for the generalised fifth-order Kadomtsev-Petviashvili equation we have to resort to numerical simulations. Finally, for the (2+l)-dimensional Boussinesq equation we perform direct numerical simulations of the full equation to investigate the nonlinear stage of the evolution of the transversely unstable solitary waves, and the result is that the instability leads to the collapse of the solitary wave. The thesis is concluded by a discussion of some open problems.

Mode interactions arise when two types of bifurcation come closer and meet at a codimension 2 point and cross each other on the branch. It happens when multiple eigenvalues involving more than one parameter pass through their critical values together. A cascade happens when these mode interactions go through, in this case, successive period doubling. In this thesis we study cascades of codimension 2 mode interactions in discrete dynamical systems. The underlying focus is on understanding mode interaction cascades in iterated maps when their solutions go through successive period doubling. We consider a large system which has a mode interaction involving a period doubling bifurcation and a symmetry breaking bifurcation. If the system is more then two dimensional we perform a Centre Manifold reduction to get a two dimensional reduced form. We then reformulate the period doubling bifurcation as a symmetry breaking bifurcation and use Liapunov-Schmidt reduction to get two bifurcation equations which describe the solutions in a neighbourhood of the mode interaction. We give a detailed description of solutions of the bifurcation equations. We calculate the values of the parameters for primary and secondary bifurcations from the trivial solution and primary branches respectively and derive conditions for their sub(super) criticality. We also consider whether there is any tertiary Hopf bifurcation on the mixed mode solutions and derive conditions for it to exist. We study the conditions for a mode interaction cascade for a class of iterated maps which satisfy a few basic conditions. We list all possible mode interactions and carefully study which of these mode interactions go through a cascade and categorise them. At higher periods we study the relationship among the bifurcation equations of different points in the cycle and derive equations to find all the sets of bifurcation equations from one. We apply these techniques to different examples. We study the limiting behaviour of the cascade using renormalisation theory first studying a sequence of one dimensional functions and then a two dimensional problem which includes the one dimensional functions. We derive the parameter scaling for each function and then for the two dimensional problem. We used a second parameter mu in the mode interaction cascade. We calculate the limiting values muinfinity and universal constant for mu similar to Feigenbaum point and Feigenbaum Number.

This thesis studies counter-rotating terms part of an Hamiltonian usually neglected in the so-called Rotating Wave Approximation. Generalising the usual description of light-matter interactions and open quantum systems it is demonstrated that normally neglected counter-rotating terms have the potential to allocate energy among different system degrees of freedom. It is pointed out in examples that this aspect can affect the energy concentration in quantum systems. Initially, a composite quantum system is considered, i.e. bipartite systems like atom-cavity systems and coupled optical resonators without decay. By resorting to methods from quantum field theory it is shown that for such bosonic systems, the Rotating Wave Approximation cannot be applied far off resonance. In fact, the counter-rotating terms are related to an entropy operator that is capable of generating an irreversible time evolution. The vacuum state of the system is shown to evolve into a generalised coherent state exhibiting entanglement of the modes in which the counter-rotating terms are expressed. Furthermore, it is demonstrated that a non-trivial behaviour of the photon emission rate of such composite quantum system can occur when the counter-rotating terms are not dropped. In such a system, there is a coupling to infinitely many modes with most of them being far off resonance. An energy concentrating mechanism is discussed which cannot be described by the Rotating Wave Approximation. Its result is the continuous leakage of photons from open quantum system, even in the absence of external driving. Finally, a model is proposed to explain the origin of the sudden energy concentration in the intriguing phenomenon of sonoluminescence. The model is based on the quantum dynamics of trapped particles and assumes the presence of a weak but highly inhomogeneous electric field. It is shown that the counter-rotating terms can significantly contribute to the energy focussing mechanism in terms of quantum coherences.

Multifractal properties of a chaotic attractor are usefully quantified by its spectrum of singularities, f(α). Here, α is the pointwise dimension of the natural measure at a point on the attractor, and f(α) is the Hausdorff dimension of all points with pointwise dimension α. Within a more general thermodynamic formalism, the singularity spectrum is one of several ways in which the properties of an attractor can be quantified. The technique used to realize the singularity spectrum is orbit theory. This theory tells one how to take properties of finite time solutions and combine them to approximate the infinite time behaviour, thereby allowing qualitative and quantitative predictions to be made. These techniques are first applied to the Lorenz system, where it is also shown that the variation in the pointwise dimension on a surface of section has self-similar structure. The general idea of studying the properties of a nonlinear system through the periodic orbits it supports has, to date, been primarily applied to low-dimensional dynamical systems. In the thesis we develop the technique so that it can be applied to the infinite-dimensional Kuramoto-Sivashinsky equation. The continuation and bifurcation package Auto is used to investigate stability and bifurcation properties of different types of special solutions to the Kuramoto-Sivashinsky equation, following an expansion in Fourier modes. One such class of solutions is defined by the Michelson equation, to which a very detailed numerical bifurcation analysis is given. Orbit theory is applied to regimes of an asymmetric Kuramoto-Sivashinsky equation where complicated behaviour is observed in a manner similar to that used in lowdimensional systems. Each periodic orbit can be considered as a spatiotemporal pattern, in which both qualitative (the structure and bifurcations of) and quantitative (the dimension and spectrum of Lyapunov exponents) aspects are discussed.