Synchronization with partial state coupling on SO(n)

This paper studies autonomous synchronization of k agents whose states evolve on SO(n), but which are only coupled through the action of their states on one "reference vector" in Rn for each link. Thus each link conveys only partial state information at each time, and to reach synchronization agents must combine this information over time or throughout the network. A natural gradient coupling law for synchronization is proposed. Extensive convergence analysis of the coupled agents is provided, both for fixed and time-varying reference vectors. The case of SO(3) with fixed reference vectors is discussed in more detail. For comparison, we also treat the equivalent setting in Rn, i.e. with states in Rn and connected agents comparing scalar product of their states with a reference vector.Comment: to be submitted to SIAM Journal on Control and Optimizatio

Intrinsic Reduced Attitude Formation with Ring Inter-Agent Graph

This paper investigates the reduced attitude formation control problem for a group of rigid-body agents using feedback based on relative attitude information. Under both undirected and directed cycle graph topologies, it is shown that reversing the sign of a classic consensus protocol yields asymptotical convergence to formations whose shape depends on the parity of the group size. Specifically, in the case of even parity the reduced attitudes converge asymptotically to a pair of antipodal points and distribute equidistantly on a great circle in the case of odd parity. Moreover, when the inter-agent graph is an undirected ring, the desired formation is shown to be achieved from almost all initial states

Dynamics of Oscillators Coupled by a Medium with Adaptive Impact

In this article we study the dynamics of coupled oscillators. We use mechanical metronomes that are placed over a rigid base. The base moves by a motor in a one-dimensional direction and the movements of the base follow some functions of the phases of the metronomes (in other words, it is controlled to move according to a provided function). Because of the motor and the feedback, the phases of the metronomes affect the movements of the base while on the other hand, when the base moves, it affects the phases of the metronomes in return. For a simple function for the base movement (such as $y = \gamma_{x} [r \theta_1 + (1 - r) \theta_2]$ in which $y$ is the velocity of the base, $\gamma_{x}$ is a multiplier, $r$ is a proportion and $\theta_1$ and $\theta_2$ are phases of the metronomes), we show the effects on the dynamics of the oscillators. Then we study how this function changes in time when its parameters adapt by a feedback. By numerical simulations and experimental tests, we show that the dynamic of the set of oscillators and the base tends to evolve towards a certain region. This region is close to a transition in dynamics of the oscillators; where more frequencies start to appear in the frequency spectra of the phases of the metronomes

Disordered proteins and network disorder in network descriptions of protein structure, dynamics and function. Hypotheses and a comprehensive review

During the last decade, network approaches became a powerful tool to describe protein structure and dynamics. Here we review the links between disordered proteins and the associated networks, and describe the consequences of local, mesoscopic and global network disorder on changes in protein structure and dynamics. We introduce a new classification of protein networks into ‘cumulus-type’, i.e., those similar to puffy (white) clouds, and ‘stratus-type’, i.e., those similar to flat, dense (dark) low-lying clouds, and relate these network types to protein disorder dynamics and to differences in energy transmission processes. In the first class, there is limited overlap between the modules, which implies higher rigidity of the individual units; there the conformational changes can be described by an ‘energy transfer’ mechanism. In the second class, the topology presents a compact structure with significant overlap between the modules; there the conformational changes can be described by ‘multi-trajectories’; that is, multiple highly populated pathways. We further propose that disordered protein regions evolved to help other protein segments reach ‘rarely visited’ but functionally-related states. We also show the role of disorder in ‘spatial games’ of amino acids; highlight the effects of intrinsically disordered proteins (IDPs) on cellular networks and list some possible studies linking protein disorder and protein structure networks