Nash equilibria in multi-agent motor interactions.

Social interactions in classic cognitive games like the ultimatum game or the prisoner's dilemma typically lead to Nash equilibria when multiple competitive decision makers with perfect knowledge select optimal strategies. However, in evolutionary game theory it has been shown that Nash equilibria can also arise as attractors in dynamical systems that can describe, for example, the population dynamics of microorganisms. Similar to such evolutionary dynamics, we find that Nash equilibria arise naturally in motor interactions in which players vie for control and try to minimize effort. When confronted with sensorimotor interaction tasks that correspond to the classical prisoner's dilemma and the rope-pulling game, two-player motor interactions led predominantly to Nash solutions. In contrast, when a single player took both roles, playing the sensorimotor game bimanually, cooperative solutions were found. Our methodology opens up a new avenue for the study of human motor interactions within a game theoretic framework, suggesting that the coupling of motor systems can lead to game theoretic solutions.

Synchronization in climate dynamics and other extended systems

Synchronization is now well established as representing coherent behaviour between two or more otherwise autonomous nonlinear systems subject to some degree of coupling. Such behaviour has mainly been studied to date, however, in relatively low-dimensional discrete systems or networks. But the possibility of similar kinds of behaviour in continuous or extended spatiotemporal systems has many potential practical implications, especially in various areas of geophysics. We review here a range of cyclically varying phenomena within the Earth's climate system for which there may be some evidence or indication of the possibility of synchronized behaviour, albeit perhaps imperfect or highly intermittent. The exploitation of this approach is still at a relatively early stage within climate science and dynamics, in which the climate system is regarded as a hierarchy of many coupled sub-systems with complex nonlinear feedbacks and forcings. The possibility of synchronization between climate oscillations (global or local) and a predictable external forcing raises important questions of how models of such phenomena can be validated and verified, since the resulting response may be relatively insensitive to the details of the model being synchronized. The use of laboratory analogues may therefore have an important role to play in the study of natural systems that can only be observed and for which controlled experiments are impossible. We go on to demonstrate that synchronization can be observed in the laboratory, even in weakly coupled fluid dynamical systems that may serve as direct analogues of the behaviour of major components of the Earth's climate system. The potential implications and observability of these effects in the long-term climate variability of the Earth is further discussed. © 2010 Springer-Verlag Berlin Heidelberg.

Representations and algorithms for finite-state bisimulations of linear discrete-time control systems

While a large amount of research over the past two decades has focused on discrete abstractions of infinite-state dynamical systems, many structural and algorithmic details of these abstractions remain unknown. To clarify the computational resources needed to perform discrete abstractions, this paper examines the algorithmic properties of an existing method for deriving finite-state systems that are bisimilar to linear discrete-time control systems. We explicitly find the structure of the finite-state system, show that it can be enormous compared to the original linear system, and give conditions to guarantee that the finite-state system is reasonably sized and efficiently computable. Though constructing the finite-state system is generally impractical, we see that special cases could be amenable to satisfiability based verification techniques. ©2009 IEEE.

Dynamical structure function identifiability conditions enabling signal structure reconstruction

Networks of controlled dynamical systems exhibit a variety of interconnection patterns that could be interpreted as the structure of the system. One such interpretation of system structure is a system's signal structure, characterized as the open-loop causal dependencies among manifest variables and represented by its dynamical structure function. Although this notion of structure is among the weakest available, previous work has shown that if no a priori structural information is known about the system, not even the Boolean structure of the dynamical structure function is identifiable. Consequently, one method previously suggested for obtaining the necessary a priori structural information is to leverage knowledge about target specificity of the controlled inputs. This work extends these results to demonstrate precisely the a priori structural information that is both necessary and sufficient to reconstruct the network from input-output data. This extension is important because it significantly broadens the applicability of the identifiability conditions, enabling the design of network reconstruction experiments that were previously impossible due to practical constraints on the types of actuation mechanisms available to the engineer or scientist. The work is motivated by the proteomics problem of reconstructing the Per-Arnt-Sim Kinase pathway used in the metabolism of sugars. © 2012 IEEE.

Shock transmission and eigenvalue veering within a coupled system.

The operation of dynamical systems in harsh environments requires continuous monitoring. Internal sensors may be used to monitor the conditions in real time. A typical example is the sensor and electronic components used in space structures which, especially during launch, are subject to huge g force. The paper will present an experimental and theoretical study on a simplified model used to analyze the possible cause of high acceleration on the enclosed sensors and equipments due to impulsive loading. The model system consists of two beams coupled using compliant connections. An impulse hammer excites one beam, and vibrations are transmitted to the indirectly driven beam. A theoretical model is developed using a Rayleigh-Ritz approach and validated using experimental results in both the frequency and time domains. Monto Carlo simulation was done with random masses positioned on the indirectly driven beam to determine the worst-case conditions for maximum peak acceleration. Highest acceleration levels were found when mode matching in the two beams led to veering behavior in the coupled modes. The results suggest guidelines for the detailed design of internal components of a structure exposed to shock loading from its environment. [The authors thank Schlumberger Cambridge Research for financial support.].

The post-processed Galerkin method applied to non-linear shell vibrations

We present the results of a computational study of the post-processed Galerkin methods put forward by Garcia-Archilla et al. applied to the non-linear von Karman equations governing the dynamic response of a thin cylindrical panel periodically forced by a transverse point load. We spatially discretize the shell using finite differences to produce a large system of ordinary differential equations (ODEs). By analogy with spectral non-linear Galerkin methods we split this large system into a 'slowly' contracting subsystem and a 'quickly' contracting subsystem. We then compare the accuracy and efficiency of (i) ignoring the dynamics of the 'quick' system (analogous to a traditional spectral Galerkin truncation and sometimes referred to as 'subspace dynamics' in the finite element community when applied to numerical eigenvectors), (ii) slaving the dynamics of the quick system to the slow system during numerical integration (analogous to a non-linear Galerkin method), and (iii) ignoring the influence of the dynamics of the quick system on the evolution of the slow system until we require some output, when we 'lift' the variables from the slow system to the quick using the same slaving rule as in (ii). This corresponds to the post-processing of Garcia-Archilla et al. We find that method (iii) produces essentially the same accuracy as method (ii) but requires only the computational power of method (i) and is thus more efficient than either. In contrast with spectral methods, this type of finite-difference technique can be applied to irregularly shaped domains. We feel that post-processing of this form is a valuable method that can be implemented in computational schemes for a wide variety of partial differential equations (PDEs) of practical importance.

An organizing center in a planar model of neuronal excitability

This paper studies the excitability properties of a generalized FitzHugh-Nagumo model. The model differs from the classical FitzHugh-Nagumo model in that it accounts for the effect of cooperative gating variables such as activation of calcium currents. Excitability is explored by unfolding a pitchfork bifurcation that is shown to organize five different types of excitability. In addition to the three classical types of neuronal excitability, two novel types are described and distinctly associated to the presence of cooperative variables. © 2012 Society for Industrial and Applied Mathematics.

Nonlinear drillstring dynamics analysis

This paper studies the dynamical response of a rotary drilling system with a drag bit, using a lumped parameter model that takes into consideration the axial and torsional vibration modes of the bit. These vibrations are coupled through a bit-rock interaction law. At the bit-rock interface, the cutting process introduces a state-dependent delay, while the frictional process is responsible for discontinuous right-hand sides in the equations governing the motion of the bit. This complex system is characterized by a fast axial dynamics compared to the slow torsional dynamics. A dimensionless formulation exhibits a large parameter in the axial equation, enabling a two-time-scales analysis that uses a combination of averaging methods and a singular perturbation approach. An approximate model of the decoupled axial dynamics permits us to derive a pseudoanalytical expression of the solution of the axial equation. Its averaged behavior influences the slow torsional dynamics by generating an apparent velocity weakening friction law that has been proposed empirically in earlier work. The analytical expression of the solution of the axial dynamics is used to derive an approximate analytical expression of the velocity weakening friction law related to the physical parameters of the system. This expression can be used to provide recommendations on the operating parameters and the drillstring or the bit design in order to reduce the amplitude of the torsional vibrations. Moreover, it is an appropriate candidate model to replace empirical friction laws encountered in torsional models used for control. © 2009 Society for Industrial and Applied Mathematics.

Lock-in and quasiperiodicity in a forced hydrodynamically self-excited jet

The ability of hydrodynamically self-excited jets to lock into strong external forcing is well known. Their dynamics before lock-in and the specific bifurcations through which they lock in, however, are less well known. In this experimental study, we acoustically force a low-density jet around its natural global frequency. We examine its response leading up to lock-in and compare this to that of a forced van der Pol oscillator. We find that, when forced at increasing amplitudes, the jet undergoes a sequence of two nonlinear transitions: (i) from periodicity to T{double-struck}2 quasiperiodicity via a torus-birth bifurcation; and then (ii) from T{double-struck}2 quasiperiodicity to 1:1 lock-in via either a saddle-node bifurcation with frequency pulling, if the forcing and natural frequencies are close together, or a torus-death bifurcation without frequency pulling, but with a gradual suppression of the natural mode, if the two frequencies are far apart. We also find that the jet locks in most readily when forced close to its natural frequency, but that the details contain two asymmetries: the jet (i) locks in more readily and (ii) oscillates more strongly when it is forced below its natural frequency than when it is forced above it. Except for the second asymmetry, all of these transitions, bifurcations and dynamics are accurately reproduced by the forced van der Pol oscillator. This shows that this complex (infinite-dimensional) forced self-excited jet can be modelled reasonably well as a simple (three-dimensional) forced self-excited oscillator. This result adds to the growing evidence that open self-excited flows behave essentially like low-dimensional nonlinear dynamical systems. It also strengthens the universality of such flows, raising the possibility that more of them, including some industrially relevant flames, can be similarly modelled. © 2013 Cambridge University Press.

Bayesian inference and learning in Gaussian process state-space models with Particle MCMC

State-space models are successfully used in many areas of science, engineering and economics to model time series and dynamical systems. We present a fully Bayesian approach to inference and learning (i.e. state estimation and system identification) in nonlinear nonparametric state-space models. We place a Gaussian process prior over the state transition dynamics, resulting in a flexible model able to capture complex dynamical phenomena. To enable efficient inference, we marginalize over the transition dynamics function and, instead, infer directly the joint smoothing distribution using specially tailored Particle Markov Chain Monte Carlo samplers. Once a sample from the smoothing distribution is computed, the state transition predictive distribution can be formulated analytically. Our approach preserves the full nonparametric expressivity of the model and can make use of sparse Gaussian processes to greatly reduce computational complexity.