Review on computational methods for Lyapunov functions

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function

Cultural Contestation in China: Ethnicity, Identity and the State

This chapter explores how the political-administrative design of the Chinese state, characterized as “multi-level governance”, might be the cause of more subtle forms of resistance. By looking at the formulation of heritage policies of Lancang County, Christina Maags illustrates how the administrative fragmentation resulted in both administrative contestation and cultural contestation, with a threatened local identity at its core

Existence of piecewise linear Lyapunov functions in arbitrary dimensions

Lyapunov functions are an important tool to determine the basin of attraction of exponentially stable equilibria in dynamical systems. In Marinósson (2002), a method to construct Lyapunov functions was presented, using finite differences on finite elements and thus transforming the construction problem into a linear programming problem. In Hafstein (2004), it was shown that this method always succeeds in constructing a Lyapunov function, except for a small, given neighbourhood of the equilibrium. For two-dimensional systems, this local problem was overcome by choosing a fan-like triangulation around the equilibrium. In Giesl/Hafstein (2010) the existence of a piecewise linear Lyapunov function was shown, and in Giesl/Hafstein (2012) it was shown that the above method with a fan-like triangulation always succeeds in constructing a Lyapunov function, without any local exception. However, the previous papers only considered two-dimensional systems. This paper generalises the existence of piecewise linear Lyapunov functions to arbitrary dimensions

The Workings of Monsters: Of Monsters and Humans in Icelandic Society

Vestfirðir(Westfjords of Iceland) is the large north-western peninsula of Iceland, which consists of high mountains and deep fjords. It is the most isolated and sparsely populated part of Iceland. Geographically the oldest part of the country, it is also the place which Icelanders see as the perfect environment for trolls and giants. For over 1,100 years, Icelanders have amassed a plethora of diverse monsters, good, bad, and in-between; this diversity can be tested against Cohen's (1996) Seven Theses categorization. These monsters live on the land, within the land (and water); and they are the land. They also live within Icelanders, at times in the form of protectors, at times as adversaries. Such monsters enter the lives and minds of Icelanders in different ways and their place, meaning, and effectiveness are diverse. On the one hand, the powers of Icelandic monsters can be displayed and affected through fear, deceit, and natural catastrophes, while on the other hand their powers can be negotiated and leveled to the benefits of human and “monsterkind.” Like the land, these monsters cannot easily be categorized (Cohen's Thesis III) and at the same time they both represent and reveal Icelandic history, culture, and society (Cohen's Thesis I). As a child growing up in Iceland, I learned about the four Vaettir ­(protectors) of Iceland. They come in the shape of Gridingur (bull), Gammur (griffin), Dreki (dragon), and Bergrisi (rock-giant) and are entrusted with the task of protecting the land—north, south, east, and west respectively—from external forces. Over the years, the symbolic meaning of the Vaettir, and their place in Icelandic mythology, history, and psyche, have settled in my mind, but my earliest feeling for them was awe of their size, for they are all giants

Evoking imaginaries : Art probing, ethnography and more-than-academic practice

I discuss and argue for combinations of artistic practice and cultural analysis, for meta- disciplinary and serendipitous endeavours that can entangle art and ethnographic research. These combinations can be understood as practices that are more-than-academic. I define the artistic side of this combinatory work as art probing. Art probes have a double function. First, they can instil inspiration and be possible points of departure for research, and, second, they can be used to communicate scientific concepts and arguments beyond the scope of academic worlds. According to this point of view, artistic and scientific output should be seen as provisional renditions oriented towards different audiences and as part of an extended open-ended art of inquiry. When working with this more-than-academic practice, a number of stakeholders are involved, ranging from academic professionals to art institutions, museums and visitors of art exhibitions, and performances. I will discuss how I understand ethnography as part of this process and in relation to practices of art probing