Stochastic perturbations to dynamical systems: a response theory approach

Using the formalism of the Ruelle response theory, we study how the invariant measure of an Axiom A dynamical system changes as a result of adding noise, and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics. We first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the additive and in the multiplicative case. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. This provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings. Using Kramers-Kronig theory, it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case analysed. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide an example of application of our results by considering the spatially extended chaotic Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations. Taking into account the chaotic hypothesis, we expect that our results have practical relevance for a more general class of system than those belonging to Axiom A.

Field manual 3-24 and the heritage of counterinsurgency theory

The Counterinsurgency Manual FM 3-24 has been accused of being over-dependent on the counterinsurgency 'classics' Galula and Thompson. But comparison reveals that it is different in spirit. Galula and Thompson seek practical control; the Manual seeks to build 'legitimacy'. Its concept of legitimacy is superficially Weberian, but owes more to the writings of the American Max Manwaring. The Manual presupposes that a rights-based legal order can (other things being equal) be made to be cross-culturally attractive; 'effective governance' by itself can build legitimacy. The fusion of its methods with an ideology creates unrealistic criteria for success. Its weaknesses suggest a level of incapacity to think politically that will, in time, result in further failures.

The heliospheric magnetic field

The heliospheric magnetic field (HMF) is the extension of the coronal magnetic field carried out into the solar system by the solar wind. It is the means by which the Sun interacts with planetary magnetospheres and channels charged particles propagating through the heliosphere. As the HMF remains rooted at the solar photosphere as the Sun rotates, the large-scale HMF traces out an Archimedean spiral. This pattern is distorted by the interaction of fast and slow solar wind streams, as well as the interplanetary manifestations of transient solar eruptions called coronal mass ejections. On the smaller scale, the HMF exhibits an array of waves, discontinuities, and turbulence, which give hints to the solar wind formation process. This review aims to summarise observations and theory of the small- and large-scale structure of the HMF. Solar-cycle and cycle-to-cycle evolution of the HMF is discussed in terms of recent spacecraft observations and pre-spaceage proxies for the HMF in geomagnetic and galactic cosmic ray records.

Multilevel research in the field of organizational behavior: an empirical look at 10 years of theory and research

During the last 30 years, significant debate has taken place regarding multilevel research. However, the extent to which multilevel research is overtly practiced remains to be examined. This article analyzes 10 years of organizational research within a multilevel framework (from 2001 to 2011). The goals of this article are (a) to understand what has been done, during this decade, in the field of organizational multilevel research and (b) to suggest new arenas of research for the next decade. A total of 132 articles were selected for analysis through ISI Web of Knowledge. Through a broad-based literature review, results suggest that there is equilibrium between the amount of empirical and conceptual papers regarding multilevel research, with most studies addressing the cross-level dynamics between teams and individuals. In addition, this study also found that the time still has little presence in organizational multilevel research. Implications, limitations, and future directions are addressed in the end. Organizations are made of interacting layers. That is, between layers (such as divisions, departments, teams, and individuals) there is often some degree of interdependence that leads to bottom-up and top-down influence mechanisms. Teams and organizations are contexts for the development of individual cognitions, attitudes, and behaviors (top-down effects; Kozlowski & Klein, 2000). Conversely, individual cognitions, attitudes, and behaviors can also influence the functioning and outcomes of teams and organizations (bottom-up effects; Arrow, McGrath, & Berdahl, 2000). One example is when the rewards system of one organization may influence employees’ intention to quit and the existence or absence of extra role behaviors. At the same time, many studies have showed the importance of bottom-up emergent processes that yield higher level phenomena (Bashshur, Hernández, & González-Romá, 2011; Katz-Navon & Erez, 2005; Marques-Quinteiro, Curral, Passos, & Lewis, in press). For example, the affectivity of individual employees may influence their team’s interactions and outcomes (Costa, Passos, & Bakker, 2012). Several authors agree that organizations must be understood as multilevel systems, meaning that adopting a multilevel perspective is fundamental to understand real-world phenomena (Kozlowski & Klein, 2000). However, whether this agreement is reflected in practicing multilevel research seems to be less clear. In fact, how much is known about the quantity and quality of multilevel research done in the last decade? The aim of this study is to compare what has been proposed theoretically, concerning the importance of multilevel research, with what has really been empirically studied and published. First, this article outlines a review of the multilevel theory, followed by what has been theoretically “put forward” by researchers. Second, this article presents what has really been “practiced” based on the results of a review of multilevel studies published from 2001 to 2011 in business and management journals. Finally, some barriers and challenges to true multilevel research are suggested. This study contributes to multilevel research as it describes the last 10 years of research. It quantitatively depicts the type of articles being written, and where we can find the majority of the publications on empirical and conceptual work related to multilevel thinking.

Ion populations on open field lines within the dayside low-latitude boundary layer: theory and observations during a transient event

We analyze of ion populations observed by the NOAA-12 satellite within dayside auroral transients. The data are matched with an open magnetopause model which allows for the transmission of magnetosheath ions across one or both of the two Alfvén waves which emanate from the magnetopause reconnection site. It also allows for reflection and acceleration of ions of magnetospheric origin by these waves. From the good agreement found between the model and the observations, we propose that the events and the low-latitude boundary precipitation are both on open field lines.