This paper is based on the proposition that the choice of different product tactics is influenced by the nature of the firm’s external environment. It illustrates the type of product activities suggested for a complex and turbulent environment, when viewing the environment through a chaos and complexity theory lens. A qualitative, case method, using depth interviews,investigated the product activities in two companies to identify the product activities adopted in a more successful, versus a less successful, firm in a complex/turbulent environment. The results showed that the more successful company uses some destabilizing product activities but also partially uses stabilizing product activities. These findings are of benefit to marketers as they emphasize a new way to consider future product activities in their firms. Since businesses and markets are complex adaptive systems, using complexity theory to understand how to cope in complex, turbulent environments is necessary, but has not been widely researched, with even less emphasis on individual components of the marketing mix.

Solid-fluid two-phase flows, where the solid volume fraction is large either by geometry or by population (as in slurry flows), are ubiquitous in nature and industry. The interaction between the fluid and the suspended solids, in such flows, are too strongly coupled rendering the assumption of a single-way interaction (flow influences particle motion alone but not vice-versa) invalid and inaccurate. Most commercial flow solvers do not account for twoway interactions between fluid and immersed solids. The current state-of-art is restricted to two-way coupling between spherical particles (of very small diameters, such that the particlediameter to the characteristic flow domain length scale ratio is less than 0.01) and flow. These solvers are not suitable for solving several industrial slurry flow problems such as those of hydrates which is crucial to the oil-gas industry and rheology of slurries, flows in highly constrained geometries like microchannels or sessile drops that are laden with micro-PIV beads at concentrations significant for two-way interactions to become prominent. It is therefore necessary to develop direct numerical simulation flow solvers employing rigorous two-way coupling in order to accurately characterise the flow profiles between large immersed solids and fluid. It is necessary that such a solution takes into account the full 3D governing equations of flow (Navier-Stokes and continuity equations), solid translation (Newton’s second law) and solid rotation (equation of angular momentum) while simultaneously enabling interaction at every time step between the forces in the fluid and solid domains. This thesis concerns with development and rigorous validation of a 3D solid-fluid solver based on a novel variant of immersed-boundary method (IBM). The solver takes into account full two-way fluid-solid interaction with 6 degrees-of-freedom (6DOF). The solid motion solver is seamlessly integrated into the Gerris flow solver hence called Gerris Immersed Solid Solver (GISS). The IBM developed treats both fluid and solid in the manner of “fluid fraction” such that any number of immersed solids of arbitrary geometry can be realised. Our IBM method also allows transient local mesh adaption in the fluid domain around the moving solid boundary, thereby avoiding problems caused by the mesh skewness (as seen in common mesh-adaption algorithms) and significantly improves the simulation efficiency. The solver is rigorously validated at levels of increasing complexity against theory and experiment at low to moderate flow Reynolds number. At low Reynolds numbers (Re 1) these include: the drag force and terminal settling velocities of spherical bodies (validating translational degrees of freedom), Jeffrey’s orbits tracked by elliptical solids under shear flow (validating rotational and translational degrees of freedom) and hydrodynamic interaction between a solid and wall. Studies are also carried out to understand hydrodynamic interaction between multiple solid bodies under shear flow. It is found that initial distance between bodies is crucial towards the nature of hydrodynamic interaction between them: at a distance smaller than a critical value the solid bodies cluster together (hydrodynamic attraction) and at a distance greater than this value the solid bodies travel away from each other (hydrodynamic repulsion). At moderately high flow rates (Re O(100)), the solver is validated against migratory motion of an eccentrically placed solid sphere in Poisuelle flow. Under inviscid conditions (at very high Reynolds number) the solver is validated against chaotic motion of an asymmetric solid body. These validations not only give us confidence but also demonstrate the versatility of the GISS towards tackling complex solid-fluid flows. This work demonstrates the first important step towards ultra-high resolution direct numerical simulations of solid-fluid flows. The GISS will be available as opensource code from February 2015.

Thesis (MA)--University of Stellenbosch, 2004. / ENGLISH ABSTRACT: The question of what it means to say one knows something, or has knowledge of something, triggered an epistemological study after the nature of knowledge and its acquisition. There are many different ways in which one can go about acquiring knowledge, manydifferent frameworks that one can use to search after truth. Because most real systems about which one could desire knowledge (organic, social, economic etc.) are non-linear, an understanding of non-linear systems is important for the process of acquiring knowledge. Knowledge exhibits the characteristics of a dynamic, adaptive system, and as such could be approached via a dynamic theory of adaptive systems. Therefore, chaos theory and complexity theory are two theoretical (non-linear) frameworks that can facilitate the knowledge acquisition process. As a modernist instrument for acquiring knowledge, chaos theory provides one with deterministic rules that make mathematical understanding of non-linear phenomenaa bit easier, but it is limited in that it can only provide one with certain knowledge up until the (system's) next bifurcation (i.e. when chaos sets in). After this, it is near impossible to predict what a chaotic system will do. Complexity theory, as a postmodern tool for knowledge acquisition, gives one insight into the dynamic, self-organising nature of the non-linear systems around one. By analysing the global stability complex systems produce during punctuated equilibrium, one can learn much about how these systems adapt, evolve and survive. Complexity and chaos, therefore, together can provide one with a useful framework for understanding the nature and workings of non-linear systems. However, it should be remembered that every observer of knowledge does so out of his/her own personal framework of beliefs, circumstances and history, and that knowledge therefore can never be 100 percent objective. Knowledge and truth can never be entirely relative either, however, for this would mean that all knowledge (and thereby all opposing claims and statements) is equally correct or true. This is clearly not possible. What is possible, though, is the fulfilling and successful pursuit of knowledge for the sake of the journey of learning and understandi ng. / AFRIKAANSE OPSOMMING: Die vraag na wat dit eintlik beteken om te sê mens weet iets, of dra kennis van iets, het na 'n epistemologiese soeke na die wese van kennis en die verwerwing daarvan toe gelei. Daar is baie maniere waarop mens kennis kan verwerf, baie verskillende raamwerke wat mens kan gebruik om te soek na waarheid. Omdat die meeste wesenlike stelsels waarvan mens kennis sou wou verkry (organies, sosiaal, ekonomies ens.) nie-lineêr is, is 'n verstaan van nie-lineêre stelsels belangrik vir die kennisverwerwingsproses. Kennis vertoon die eienskappe van I n dinamiese, aanpassende stelsel, en kan dus via 'n dinamiese teorie van aanpassendestelsels benader word. Daarom is chaosteorie en kompleksiteitsteorie twee teoretiese (nie-lineêre) raamwerke wat die proses van kennisverwerwing kan vergemaklik. As I n modernistiese instrument vir kennisverwerwing, verskaf chaosteorie deterministiese reëls wat die wiskundige verstaan van nie-lineêre verskynsels bietjie vergemaklik, maar dit is beperk deurdat dit net sekere kennis tot op die (stelsel se) volgende splitsing (d.w.s. waar chaos begin) verskaf. Hierna, word dit naasonmoontlik om te voorspel wat I n chaotiese stelsel gaandoen. Kompleksiteitsteorie, as I n postmodernistiese gereedskap vir kennisverwerwing, gee mens insig in die dinamiese, selforganiserende aard van die nie-lineêre stelsels om mens. Deur die globale stabiliteit wat komplekse stelsels gedurende onderbreekte ewewig ("punctuated equi/ibrium"}toon te analiseer, kan mens baie leer van hoe hierdie stelsels aanpas, ontwikkel en oorleef. Kompleksiteit en chaos, saam, kan mens dus van a nuttige raamwerk vir die verstaan van die wese en werkinge van nie-lineêre stelsels, voorsien. Daar moet egter onthou word dat elke waarnemer van kennis dit doen uit sy/haar persoonlike raamwerk van oortuiginge, omstandighede en geskiedenis, en dat kennis dus nooit 100 persent objektief kan wees nie. Kennis en waarheid kan egter ook nooit heeltemaal relatief wees nie, want dit sou beteken dat alle kennis (en hiermee ook alle teenstrydige aansprake en stellings) gelyk korrek of waar is. Hierdie is duidelik onmoontlik. Wat wel moontlik is, is die vervullende en suksesvolle strewe na kennis ter wille van die reis van leer en verstaan.

published_or_final_version / abstract / Education / Doctoral / Doctor of Philosophy

A renormalization analysis is presented for a generalized Harper equation (1 + α cos(2π(ω(i + 1/2) + φ)))ψi+1 + (1 + α cos(2π(ω(i − 1/2) + φ)))ψi−1 +2λ cos(2π(iω + φ))ψi = Eψi. (0.1) For values of the parameter ω having periodic continued-fraction expansion, we construct the periodic orbits of the renormalization strange sets in function space that govern the wave function fluctuations of the solutions of the generalized Harper equation in the strong-coupling limit λ→∞. For values of ω with non-periodic continued fraction expansions, we make some conjectures based on work of Mestel and Osbaldestin on the likely structure of the renormalization strange set.

Sustained resonance in a linear oscillator is achievable with a drive whose constant frequency matches the resonant frequency of the oscillator. In oscillators with nonlinear restoring forces, i.e., Dung-type oscillators, resonant frequency changes with amplitude, so a constant frequency drive generates a beat oscillation instead of sustained resonance. Dung-type oscillators can be driven into sustained resonance, called autoresonance (AR), when drive frequency is swept in time to match the changing resonant frequency of the oscillator. It is found that near-optimal drive linear sweep rates for autoresonance can be estimated from the beat oscillation resulting from constant frequency excitation. Specically, a least squares estimate of the slope of the Teager-Kaiser instantaneous frequency versus time plot for the rising half-cycle of the beat response to a stationary drive provides a near-optimal estimate of the linear drive sweep rate that sustains resonance in the pendulum, Dung and Dung-Van der Pol oscillators. These predictions are confirmed with model-based numerical simulations. A closed-form approximation to the AM-FM nonlinear resonance beat response of a Dung oscillator driven at its low-amplitude oscillator frequency is obtained from a solution to an associated Mathieu equation. AR time responses are found to evolve along a Mathieu equation primary resonance stability boundary. AR breakdown occurs at sweep rates just past optimal and map to a single stable point just off the Mathieu equation primary resonance stability boundary. Optimal AR sweep rates produce oscillating phase dierences with extrema near 90 degrees, allowing extended time in resonance. AR breakdown occurs when phase difference equals 180 degrees. Nonlinear resonance of the van der Pol type may play a role in the extraordinary sensitivity of the human ear. / The mechanism for maintaining the cochlear amplifier at its critical point is currently unknown. The possibility of open-loop control of cochlear operating point, maintaining criticality on average through periodically varying damping (super-regeneration) motivates a study of spontaneous otoacoustic emission (SOAE) amplitude modulation on a short (msec) time scale. An example of periodic amplitude modulation within a wide filter bandwidth is found that appears to be a beat oscillation of two SOAEs. / by Carey Witkov. / Thesis (Ph.D.)--Florida Atlantic University, 2011. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2011. Mode of access: World Wide Web.

Num problema de espalhamento temos partículas incidentes sobre uma região de espalhamento que, depois de interagir por algum tempo nessa região, escapam para o infinito. Quando o espalhamento é caótico, a função de espalhamento (que é a relação entre uma variável antes do espalhamento e outra variável depois do espalhamento), apresenta singularidades sobre um conjunto de Cantor de condições iniciais. O espalhamento caótico pode ser dividido em dois tipos: espalhamento não-hiperbólico e hiperbólico. No espalhamento não-hiperbólico, o conjunto invariante contém órbitas estáveis. O decaimento das partículas que escapam do conjunto invariante é regido por uma lei de potência com relação ao tempo. No caso do espalhamento hiperbólico, a sela caótica é hiperbólica e todas as órbitas que a compõem são instáveis. O decaimento das partículas na região de espalhamento segue uma exponencial decrescente. Investigamos a transição do espalhamento não-hiperbólico para o hiperbólico quando ruído é adicionado à dinâmica do sistema. Isto porque prevíamos que o ruído reduzisse o efeito de aprisionamento (stickness) dos conjuntos de órbitas estáveis, provocando um decaimento exponencial. Introduzimos perturbações randômicas a fim de simular flutuações reais que ocorrem em sistemas físicos, como por exemplo, um vórtex que depende irregularmente do tempo no estudo de fluidos. Assim, usamos o conceito de mapas randômicos, que são mapas onde um ou mais parâmetros são variados aleatoriamente a cada iteração. Estudamos então, os efeitos provocados por perturbações randômicas em um sistema com espalhamento caótico não-hiperbólico. / In a scattering problem we have particles inciding on a scattering region and these particles, after spending some time in this region, escape towards infinity. When the scattering is chaotic, the scattering function (a function that relates an input variable with an output variable), is singular on a Cantor set of initial conditions. The chaotic scattering can be either non-hyperbolic or hyperbolic. In the non-hyperbolic scattering, the invariant set has stable orbits. This decay is governed by a power law in time. In the hyperbolic case, the chaotic saddle is hyperbolic and all the orbits are unstable. The decay of the particles is a decreasing exponential in the time. We investigate the transition from non-hyperbolic to hyperbolic scattering as noise is added to the system. One expects that noise will reduce the stickness of the regular regions, resulting in an exponential decay law, typical of hyperbolic systems. We apply random perturbations in order to simulate the real fluctuations that occur in physical systems, for example, an aperiodic vortex in a fluid flow. So, we work with random maps, where we change randomly one or more parameters on each iteration. We study thus, the effects of the random perturbations on a system having non-hyperbolic scattering.

acase@tulane.edu / I review my three papers on ray stretching statistics, hot spot formation, and universality in motion through weak random media. In the first paper, we study the connection between stretching exponents and ray densities in weak ray scattering through a random medium. The stretching exponent is a quantitative measure that describes the degree of exponential convergence or divergence among nearby ray trajectories. In the context of non-relativistic particle motion through a correlated random potential, we show how particle densities are strongly related to the stretching exponents, where the `hot spots' in the intensity profile correspond to minima in the stretching exponents. This strong connection is expected to be valid for different random potential distributions, and is also expected to apply to other physical contexts, such as deep ocean waves. The surprising minimum in the average stretching exponent is of great interest due to the associated appearance of the first generation of hot spots, and a detailed discussion will be found in the third paper. In the second paper, we study the stretching statistics of weak ray scattering in various physical contexts and for different types of correlated disorder. The stretching exponent is mathematically linked to the monodromy matrix that evolves the phase space vector over time. From this point of view, we demonstrate analytically and numerically that the stretching statistics along the forward direction follow universal scaling relationships for different dispersion relations and in disorders of differing correlation structures. Predictions about the location of first caustics can be made using the universal evolution pattern of stretching exponents. Furthermore, we observe that the distribution of stretching exponents in 2D ray dynamics with small angular spread is equivalent to the same distribution in a simple 1D kicked model, which allows us to further explore the relation between stretching statistics and the form of the disorder. Finally, the third paper focuses on the 1D kicked model with stretching statistics that resemble 2D small-angle ray scattering. While the long time behavior of the stretching exponent displays a simple linear growth, the behavior on the scale of the Lyapunov time is mathematically nontrivial. From an analysis of the evolving monodromy matrices, we demonstrate how the stretching exponent depends on the statistics of the second derivative of the random disorder, especially the mean and standard deviation. Furthermore, the maximal Lyapunov exponent or the Lyapunov length can be expressed as nontrivial functions of the mean and standard deviation of the kicks. Lastly, we show that the higher moments of the second derivative of the disorder have small or negligible effect on the evolution of the stretching exponents or the maximal Lyapunov exponents. / 1 / SicongChen

The definition of semi-hyperbolic dynamical systems generated by Lipschitz continuous and not necessarily invertible mappings in Banach spaces is presented in this thesis. Like hyperbolic mappings, they involve a splitting into stable and unstable spaces, but a slight leakage from the strict invariance of the spaces is possible and the unstable subspaces are assumed to be finite dimensional. Bi-shadowing is a combination of the concepts of shadowing and inverse shadowing and is usually used to compare pseudo-trajectories calculated by a computer with the true trajectories. In this thesis, the concept of bi-shadowing in a Banach space is defined and proved for semi-hyperbolic dynamical systems generated by Lipschitz mappings. As an application to the concept of bishadowing, linear delay differential equations are shown to be bi-shadowing with respect to pseudo-trajectories generated by nonlinear small perturbations of the linear delay equation. This shows robustness of solutions of the linear delay equation with respect to small nonlinear perturbations. Complicated dynamical behaviour is often a consequence of the expansivity of a dynamical system. Semi-hyperbolic dynamical systems generated by Lipschitz mappings on a Banach space are shown to be exponentially expansive, and explicit rates of expansion are determined. The result is applied to a nonsmooth noninvertible system generated by delay differential equation. It is shown that semi-hyperbolic mappings are locally φ-contracting, where -0 is the Hausdorff measure of noncompactness, and that a linear operator is semi-hyperbolic if and only if it is φ-contracting and has no spectral values on the unit circle. The definition of φ-bi-shadowing is given and it is shown that semi-hyperbolic mappings in Banach spaces are φ-bi-shadowing with respect to locally condensing continuous comparison mappings. The result is applied to linear delay differential equations of neutral type with nonsmooth perturbations. Finally, it is shown that a small delay perturbation of an ordinary differential equation with a homoclinic trajectory is ‘chaotic’.

Anti-control of chaos has attracted a lot of attention recently due to its potential applications in science and engineering. How to generate useful chaos that is also practically implementable and useful is a current focus of research. This research aims at developing new chaos generation schemes which demonstrate complex dynamical behaviours using simple linear systems with hysteresis function series. A continuous-time linear unstable second-order system with a feedback of hysteresis function is first proposed for generating chaos. The design for chaos generation is studied theoretically. A Poincaré map is used to demonstrate the dynamical behaviour of the system. The existence and the analytic solution of the limit cycle that bounds the basin of attraction of the chaotic attractor are derived. Conditions for the existence of chaotic attractors are studied. A hysteresis based system with a maximum chaotic stability margin is designed. Second, systematic methods for generating 1D n-scroll chaotic attractors in the directions of the state variables and 2D nxm-grid scroll chaotic attractors in the phase plane via continuous-time linear unstable second-order systems with a feedback of hysteresis function series are proposed. Furthermore, systematic methods for generating 1D n-scroll, 2D nxm-grid scroll and 3D nxmxl-space scroll chaotic attractors via continuous-time linear unstable third-order systems using hysteresis function series feedback are also presented in this thesis. Simulation results are presented to demonstrate effectiveness of the schemes. It is shown that the multi-scroll chaos generation systems can be represented in Lur'e form, and as a result it may be used within synchronization schemes for secure communication. Third, the limit cycle that bounds the basin of attraction in the multi-scroll chaos generation with second-order systems case is studied. The relationship of the size of the basin of attraction with the numbers of hysteresis function series is studied. The multi-scroll chaos generation mechanism is then further explored by analyzing the system trajectories; the switching boundaries, switching rules and the trajectories on each subspace. The chaotic behaviours are confirmed theoretically and it is proved that a non-ordinary attractor exists in the multi-scroll chaotic attractor of the second-order systems case. The abundant dynamical behaviour of the multi-scroll chaos generation systems using different hysteresis feedback are demonstrated. A double-hysteresis function, which is the superimposition of two basic hysteresis functions, is proposed for the implementation of the hysteresis based chaotic system. In this design, the double-hysteresis block and its series are constructed via a systematic method. The ideal hysteresis function series can be implemented easily with the proposed double-hysteresis function. The number of scroll attractors can be designed arbitrarily, and the multi-scroll chaotic attractors can be located anywhere and cover any chosen area of the phase plane. The circuitry implementation for generating 1D n-scroll, 2D nxm-grid scroll chaotic attractors with linear second-order systems and hysteresis function series is given. And the oscilloscope illustrated waveforms which included as many as 9x9=81 scrolls chaotic attractor are presented. The experimental results confirmed the theoretical analysis very well and validated the effectiveness as well as the feasibility of the proposed multi-scroll chaos generation schemes. This research may find potential engineering applications in areas such as digital coding and image processing, etc.