Homoclinic bifurcations in reversible systems

Zusammenfassung zur Dissertation: Homoclinic Bifurcations in Reversible Systems von Thomas Wagenknecht eingereicht bei der Fakultät für Mathematik und Naturwissenschaften der Technischen Universität Ilmenau am 17. Juni 2003 öffentlich verteidigt am 12. Dezember 2003 Gutachter: Prof. Dr. A. R. Champneys (University of Bristol) Prof. Dr. B. Fiedler (Freie Universität Berlin) Prof. Dr. B. Marx (Technische Universität Ilmenau) Zusammenfassung Die vorliegende Arbeit untersucht Bifurkationen homokliner Lösungen in gewöhnlichen Differentialgleichungen. Homokline Lösungen sind in positiver und negativer Zeit asymptotisch zu einer Gleichgewichtslage, d.h. zu einer konstanten Lösung der Differentialgleichung. Die Arbeit betrachtet solche homokline Bifurkationen, die von einer Veränderung des Typs dieser assoziierten Gleichgewichtslage herrühren. Verschiedene Szenarien werden in der Klasse der reversiblen Differentialgleichungen analysiert. Der Hauptteil der Arbeit beschäftigt sich mit Homoklinen an Gleichgewichtslagen, welche selbst in einer lokalen Bifurkation verzweigen. Dabei verändert sich der Typ der Gleichgewichtslage vom reellen Sattel (mit führenden reellen Eigenwerten) zum Sattel-Zentrum (mit einem Paar rein imaginärer Eigenwerte). Das Miteinander lokaler und globaler Bifurkationseffekte erfordert eine neuartige Behandlung: Durch eine Kombination analytischer und geometrischer Techniken wird eine Beschreibung verzweigender Homoklinen gewonnen. Dabei werden sowohl rein reversible Systeme als auch Systeme mit zusätzlicher Symmetrie und Hamilton-Struktur betrachtet. Im zweiten Teil der Arbeit werden homokline Bifurkationsphänomene untersucht, die von einer Typveränderung der Gleichgewichtslage von rellem Sattel zu komplexem Sattel- Fokus (mit komplexen führenden Eigenwerten) herrühren. Dabei wird die Existenz von zwei Ausgangshomoklinen in sogenannter Blasebalg-Konfiguration (homoclinic bellows configuration) vorausgesetzt. Unter Verwendung einer auf Lin zurückgehenden analytischen Methode werden Bifurkationsresultate für verzweigende N-Homoklinen erzielt. Die allgemeinen Bifurkationsresultate werden auf physikalische Probleme der nichtlinearen Optik und Wasserwellentheorie, sowie auf zwei mathematische Modellgleichungen angewendet und in numerischen Untersuchungen bestätigt.The thesis investigates bifurcations from homoclinic solutions of ordinary differential equations. Homoclinic solutions are characterised by approaching an equilibrium, i.e. a constant solution of a differential equation, in both positive and negative time. The thesis is devoted to the analysis of homoclinic bifurcations that originate from a change in the type of the associated equilibrium. Several scenarios are considered in the class of reversible ordinary differential equations. The main part of the thesis deals with solutions homoclinic to equilibria that themselves undergo a local bifurcation. In this process the type of the equilibrium changes from a real saddle (with real leading eigenvalues) to a saddle-centre (with a pair of imaginary eigenvalues). The interplay of local and global bifurcation effects requires a new analytical approach. By a combination of analytical and geometric techniques a description of bifurcating homoclinic solutions is derived. Thereby both purely reversible systems and systems with additional symmetry or Hamiltonian structure are considered. The second part of the thesis discusses a homoclinic bifurcation in which the associated equilibrium undergoes a transition from real saddle to complex saddle-focus (with complex leading eigenvalues). The existence of two primary homoclinic solutions forming a so-called bellows structure is assumed. Using an analytical technique known as Lin?s method results about the bifurcation of N-homoclinic orbits are derived. The theory is applied to physical problems from nonlinear optics and water wave theory as well as to two mathematical model systems. Numerical investigations confirm the general bifurcation results

Crowdsourcing in a Public Organization: Transformation and Culture

Employees increasingly want to participate in the decision-making processes of their organization. Approaches that facilitate organizational participation, such as internal crowdsourcing platforms, can achieve this. However, although organizations of various backgrounds should be able to reap benefits from such systems, in practice, implementation might fail due to an organizational culture built on hierarchical structures, low experience with employee participation and fixed processes. Taking a design science approach, we explore this relation in a five-month case study of a transformational process at a public organization. We design, implement and evaluate an information system for internal crowdsourcing. Based on the results from user-generated content analysis of the proposals and comments, survey data and several in-depth interviews, we observe an IT culture conflict that significantly affects system adoption and use. Our findings contribute to the discussion on theoretical and practical implications for design and implementation of information systems in organizational contexts

PEER RATINGS AND ASSESSMENT QUALITY IN CROWD-BASED INNOVATION PROCESSES

Social networks – whether public or in enterprises – regularly ask users to rate their peers’ content using different voting techniques. When employed in innovation challenges, these rating procedures are part of an open, interactive, and continuous engagement among customers, employees, or citizens. In this regard, assessment accuracy (i.e., correctly identifying good and bad ideas) in crowdsourced eval-uation processes may be influenced by the display of peer ratings. While it could sometimes be useful for users to follow their peers, it is not entirely clear under which circumstances this actually holds true. Thus, in this research-in-progress article, we propose a study design to systematically investigate the effect of peer ratings on assessment accuracy in crowdsourced idea evaluation processes. Based on the elaboration likelihood model and social psychology, we develop a research model that incorporates the mediating factors extraversion, locus of control, as well as peer rating quality (i.e., the ratings’ corre-lation with the evaluated content’s actual quality). We suggest that the availability of peer ratings de-creases assessment accuracy and that rating quality, extraversion, as well as an internal locus of control mitigate this effect

Ideate. Collaborate. Repeat. A Research Agenda for Idea Generation, Collaboration and Evaluation in Open Innovation

Open innovation has been and remains to be a rapidly changing field of research in Information Systems and various other disciplines. With the rise of professional open innovation platforms and the emergence of crowdsourcing as well as employee-driven innovation, studies on the front-end of open innovation – namely idea generation, collaboration and evaluation – are facing new challenges. In this structured literature review, we analyze a large body of prior research in order to derive a framework, which is able to classify and reflect the lively debate on open innovation. In addition, we identify important implications for practitioners with advise on the design of open innovation systems. Moreover, our study identifies several promising areas for future research

When Life Gives You Lemons: How rating scales affect user activity and frustration in collaborative evaluation processes

Initiators of open innovation processes involving customers or employees often face vast amounts of idea proposals. These proposals vary greatly in terms of quality, which is why organizers often engage the users themselves in the evaluation process. Building on the concept of information overload, we evaluate the effects of three distinct rating scales on users’ activity and frustration measures. On the basis of an open innovation campaign for employees of a public-private institution in Germany, we systematically compare the novel “bag of lemons” method with conventional Likert scales and up-down-voting schemes. Our results demonstrate that the “bag of lemons”-approach yields higher levels of user activity, but is also perceived as significantly more frustrating. We find this effect to be fully mediated by perceived information overload, which points to potential avenues for the design of stimulating yet tolerably complex Information Systems for open innovation and rating techniques

Pattern Formation on Networks:from Localised Activity to Turing Patterns

Systems of dynamical interactions between competing species can be used to model many complex systems, and can be mathematically described by {\em random} networks. Understanding how patterns of activity arise in such systems is important for understanding many natural phenomena. The emergence of patterns of activity on complex networks with reaction-diffusion dynamics on the nodes is studied here. The connection between solutions with a single activated node, which can bifurcate from an undifferentiated state, and the fully developed system-scale patterns are investigated computationally. The different coexisting patterns of activity the network can exhibit are shown to be connected via a snaking type bifurcation structure, similar to those responsible for organising localised pattern formation in regular lattices. These results reveal the origin of the multistable patterns found in systems organised on complex networks. A key role is found to be played by nodes with so called {\em optimal degree}, on which the interaction between the reaction kinetics and the network structure organise the behaviour of the system. A statistical representation of the density of solutions over the parameter space is used as a means to answer important questions about the number of accessible states that can be exhibited in systems with such a high degree of complexity