Related papers: Differential Positivity on Compact Sets
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…
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…
Positive systems play an important role in systems and control theory and have found many applications in multi-agent systems, neural networks, systems biology, and more. Positive systems map the nonnegative orthant to itself (and also the…
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…
Positive linear systems on arbitrary time scales are studied. The theory developed in the paper unifies and extends concepts and results known for continuous-time and discrete-time systems. A necessary and sufficient condition for a linear…
The dynamics of linear positive systems map the positive orthant to itself. In other words, it maps a set of vectors with zero sign variations to itself. This raises the following question: what linear systems map the set of vectors with…
A class of periodic differential $n$-dimensional systems with patch structure with (possibly infinite) delay and nonlinear impulses is considered. These systems incorporate very general nonlinearities and impulses whose signs may vary.…
Differential analysis aims at inferring global properties of nonlinear behaviors from the local analysis of the linearized dynamics. The paper motivates and illustrates the use of differential analysis on the nonlinear pendulum model, an…
The present document is devoted to structural properties of neural population dynamics and especially their differential flatness. Several applications of differential flatness in the present context can be envisioned, among which:…
The paper shows that normally hyperbolic one-dimensional compact attractors of smooth dynamical systems are characterized by differential positivity, that is, the pointwise infinitesimal contraction of a smooth cone field. The result is…
This note considers the maximal positively invariant set for polynomial discrete time dynamics subject to constraints specified by a basic semialgebraic set. The note utilizes a relatively direct, but apparently overlooked, fact stating…
Stability and bifurcation properties of one-dimensional discrete dynamical systems with positivity, which are derived from continuous ones by tropical discretization, are studied. The discretized time interval is introduced as a bifurcation…
The concept of positively invariant (PI) sets has proven effective in the formal verification of stability and safety properties for autonomous systems. However, the characterization of such sets is challenging for nonlinear systems in…
A time-varying nonlinear dynamical system is called a totally positive differential system (TPDS) if its Jacobian admits a special sign pattern: it is tri-diagonal with positive entries on the super- and sub-diagonals. If the vector field…
We present log-linear dynamical systems, a dynamical system model for positive quantities. We explain the connection to linear dynamical systems and show how convex optimization can be used to identify and control log-linear dynamical…
We develop categorical foundations of discrete dynamical systems, aimed at understanding how the structure of the system affects its dynamics. The key technical innovation is the notion of a cycle set, which provides a formal language in…
In this paper it is shown that the compact linearization approach, that has been previously proposed only for binary quadratic problems with assignment constraints, can be generalized to arbitrary linear equations with positive coefficients…
We investigate the property for an input-output system to map unimodal inputs to unimodal outputs. As a first step, we analyse this property for linear time-invariant (LTI) systems, static nonlinearities, and interconnections of those. In…
Considering second variations about a given minimizer of a causal variational principle, we derive positive functionals in space-time. It is shown that the strict positivity of these functionals ensures that the minimizer is nonlinearly…
It is known that input-output approaches based on scaled small-gain theorems with constant $D$-scalings and integral linear constraints are non-conservative for the analysis of some classes of linear positive systems interconnected with…