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Differentially Positive Systems
The paper introduces and studies differentially positive systems, that is,
systems whose linearization along an arbitrary trajectory is positive. A
generalization of Perron Frobenius theory is developed in this differential
framework to show that the property induces a (conal) order that strongly
constrains the asymptotic behavior of solutions. The results illustrate that
behaviors constrained by local order properties extend beyond the well-studied
class of linear positive systems and monotone systems, which both require a
constant cone field and a linear state space.The research was supported by the Fund for Scientific Research FNRS and by the Engineering and Physical Sciences Research Council under Grant EP/G066477/1.This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/TAC.2015.243752
An operator-theoretic approach to differential positivity
Differentially positive systems are systems whose linearization along
trajectories is positive. Under mild assumptions, their solutions
asymptotically converge to a one-dimensional attractor, which must be a limit
cycle in the absence of fixed points in the limit set. In this paper, we
investigate the general connections between the (geometric) properties of
differentially positive systems and the (spectral) properties of the Koopman
operator. In particular, we obtain converse results for differential
positivity, showing for instance that any hyperbolic limit cycle is
differentially positive in its basin of attraction. We also provide the
construction of a contracting cone field.A. Mauroy holds a BELSPO Return Grant and F. Forni holds a FNRS fellowship. This paper presents research results of the Belgian Network DYSCO, funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office.This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/CDC.2015.740332
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Individual differences in the neuropsychopathology of addiction.
Drug addiction or substance-use disorder is a chronically relapsing disorder that progresses through binge/intoxication, withdrawal/negative affect and preoccupation/anticipation stages. These stages represent diverse neurobiological mechanisms that are differentially involved in the transition from recreational to compulsive drug use and from positive to negative reinforcement. The progression from recreational to compulsive substance use is associated with downregulation of the brain reward systems and upregulation of the brain stress systems. Individual differences in the neurobiological systems that underlie the processing of reward, incentive salience, habits, stress, pain, and executive function may explain (i) the vulnerability to substance-use disorder; (ii) the diversity of emotional, motivational, and cognitive profiles of individuals with substance-use disorders; and (iii) heterogeneous responses to cognitive and pharmacological treatments. Characterization of the neuropsychological mechanisms that underlie individual differences in addiction-like behaviors is the key to understanding the mechanisms of addiction and development of personalized pharmacotherapy
Positivity, monotonicity, and consensus on lie groups
Dynamical systems whose linearizations along trajectories are positive in the sense that they infinitesimally contract a smooth cone field are called differentially positive. The property can be thought of as a generalization of monotonicity, which is differential positivity in a linear space with respect to a constant cone field. Differential positivity places significant constraints on the asymptotic behavior of trajectories under mild technical conditions. This paper studies differentially positive systems defined on Lie groups. The geometry of a Lie group allows for the generation of invariant cone fields over the tangent bundle given a single cone in the Lie algebra. We outline the mathematical framework for studying differential positivity of discrete and continuous-time dynamics on a Lie group with respect to an invariant cone field and motivate the use of this analysis framework in nonlinear control, and, in particular in nonlinear consensus theory. We also introduce a generalized notion of differential positivity of a dynamical system with respect to an extended notion of cone fields generated by cones of rank k. This new property provides the basis for a generalization of differential Perron-Frobenius theory, whereby the Perron-Frobenius vector field which shapes the one-dimensional attractors of a differentially positive system is replaced by a distribution of rank k that results in k-dimensional integral submanifold attractors instead
Differential positivity on compact sets
The paper studies differentially positive systems, that is, systems whose
linearization along an arbitrary trajectory is positive. We illustrate the use
of differential positivity on compact forward invariant sets for the
characterization of bistable and periodic behaviors. Geometric conditions for
differential positivity are provided. The introduction of compact sets
simplifies the use of differential positivity in applications.The research was supported by the Fund for Scientific Research FNRS and by the Engineering and Physical Sciences Research Council under Grant EP/G066477/1.This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/CDC.2015.740322
On the Geometry of Message Passing Algorithms for Gaussian Reciprocal Processes
Reciprocal processes are acausal generalizations of Markov processes
introduced by Bernstein in 1932. In the literature, a significant amount of
attention has been focused on developing dynamical models for reciprocal
processes. Recently, probabilistic graphical models for reciprocal processes
have been provided. This opens the way to the application of efficient
inference algorithms in the machine learning literature to solve the smoothing
problem for reciprocal processes. Such algorithms are known to converge if the
underlying graph is a tree. This is not the case for a reciprocal process,
whose associated graphical model is a single loop network. The contribution of
this paper is twofold. First, we introduce belief propagation for Gaussian
reciprocal processes. Second, we establish a link between convergence analysis
of belief propagation for Gaussian reciprocal processes and stability theory
for differentially positive systems.Comment: 15 pages; Typos corrected; This paper introduces belief propagation
for Gaussian reciprocal processes and extends the convergence analysis in
arXiv:1603.04419 to the Gaussian cas
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