Cooperative Dynamical Systems Research Articles

A generalization of linear positive systems with applications to nonlinear systems: Invariant sets and the Poincaré–Bendixson property

The dynamics of linear positive systems maps the positive orthant to itself. In other words, it maps a set of vectors with zero sign variations to itself. What linear systems map the set of vectors with k sign variations to itself? We address this question using tools from the theory of cooperative dynamical systems and the theory of totally positive matrices. This yields a generalization of positive linear systems called k-positive linear systems, that reduces to positive systems for k=1. We describe applications of this new type of systems to the analysis of nonlinear dynamical systems. In particular, we show that such systems admit certain explicit invariant sets, and for the case k=2 establish the Poincaré–Bendixson property for any bounded trajectory.

Dynamical Systems With a Cyclic Sign Variation Diminishing Property

In 1970, Binyamin Schwarz defined and analyzed totally positive differential systems (TPDSs), i.e., linear time-varying systems whose transition matrix is totally positive. He showed that any solution of a TPDS satisfies a sign variation diminishing property with respect to the standard number of sign variations. It has been recently shown that several important results on entrainment [stability] in time-varying [time-invariant] nonlinear tridiagonal cooperative systems follow from the fact that the variational equation associated with these nonlinear systems is a TPDS. Thus, the number of sign variations in the vector of derivatives can be used as an integer-valued Lyapunov function. Here we develop the theory of linear cyclic variation diminishing differential systems (CVDDSs). These are systems whose transition matrix satisfies a variation diminishing property with respect to the cyclic number of sign variations. Thus, the cyclic number of sign variations can be used as an integer-valued Lyapunov function for any vector solution of a CVDDS. We show that several known classes of nonlinear cooperative dynamical systems have a variational equation, which is a CVDDS.

Global and blow-up radial solutions for quasilinear elliptic systems arising in the study of viscous, heat conducting fluids

We study positive radial solutions of quasilinear elliptic systems with a gradient term in the form where is either a ball or the whole space, , , , and . We first classify all the positive radial solutions in case is a ball, according to their behavior at the boundary. Then we obtain that the system has non-constant global solutions if and only if and . Finally, we describe the precise behavior at infinity for such positive global radial solutions by using properties of three component cooperative and irreducible dynamical systems.

On the spectral properties of nonsingular matrices that are strictly sign-regular for some order with applications to totally positive discrete-time systems

A matrix is called strictly sign-regular of order k (denoted by S S R k ) if all its k × k minors are non-zero and have the same sign. For example, totally positive matrices, i.e., matrices with all minors positive, are S S R k for all k . Another important subclass are those that are S S R k for all odd k . Such matrices have interesting sign variation diminishing properties, and it has been recently shown that they play an important role in the analysis of certain nonlinear cooperative dynamical systems. In this paper, the spectral properties of nonsingular matrices that are S S R k for a specific value k are studied. One of the results is that the product of the first k eigenvalues is real and of the same sign as the k × k minors, and that linear combinations of certain eigenvectors have specific sign patterns. It is then shown how known spectral properties for matrices that are S S R k for several values of k can be derived from these results. Using these theoretical results, the notion of a totally positive discrete-time system (TPDTS) is introduced. This may be regarded as the discrete-time analogue of the important notion of a totally positive differential system. It is shown that TPDTSs can be applied to prove that certain time-varying nonlinear dynamical systems entrain to periodic excitations.

Reprogramming cooperative monotone dynamical systems.

Multistable dynamical systems are ubiquitous in nature, especially in the context of regulatory networks controlling cell fate decisions, wherein stable steady states correspond to different cell phenotypes. In the past decade, it has become experimentally possible to "reprogram" the fate of a cell by suitable externally imposed input stimulations. In several of these reprogramming instances, the underlying regulatory network has a known structure and often it falls in the class of cooperative monotone dynamical systems. In this paper, we therefore leverage this structure to provide concrete guidance on the choice of inputs that reprogram a cooperative dynamical system to a desired target steady state. Our results are parameter-independent and therefore can serve as a practical guidance to cell-fate reprogramming experiments.

Education

This issue contains two articles in the Education section. The first one is “Elementary Number Theory and Rader's FFT,” by Shlomo Engelberg. The article discusses how number theory makes the calculation of the discrete Fourier transform more efficient. The discrete Fourier transform converts a finite sequence of equally spaced samples of a function into another sequence of equally spaced observations of a complex-valued function in the frequency domain. The paper starts with an introduction to the Cooley--Tukey method of the fast Fourier transform (FFT), which can be called a divide-and-conquer algorithm. It uses a substantially smaller number of calculations in comparison to the straightforward definition-based computation. The construction of that method factorizes $N$, the number of terms used to calculate each element in the sequence, i.e., the number of samples taken, and reformulates the summation using the factors. The Cooley--Tukey method works best if the number $N$ is a power of 2. However, if one does not get to choose the number of samples, $N$ might be a prime number, eliminating the advantage of the Cooley--Tukey method. In that case, another approach proves useful. It is based on the cyclic group of integer numbers modulo the number $N$ (the given prime number of samples). Here the reader is reminded also about Fermat's little theorem, which is essential to establish the cyclic nature of the group. That approach carries the name of Rader. Further, the author brings to attention the occurrence of calculation intensity when working with particular prime numbers: Sophie Germain primes $p$, which are such that $2p+1$ are also primes. At the end of the article, several examples are discussed to illustrate how efficient the procedures really are. The authors have created a MATLAB code available for download at http://www.jct.ac.il/~shlomoe/Public/RaderFiles/DownloadRader.html. The second article in the Education section is titled “Instability in Linear Cooperative Systems of Ordinary Differential Equations” and authored by Janusz Mierczyński. The author discusses stability of dynamical systems described by linear ODEs of the form $x'=A(t)x$, where $A(\cdot)$ is a matrix-valued function of time. The main focus is a method for constructing linear systems which are nonautonomous and are strongly cooperative: for each $t \in \mathbb{R}$ the matrix $A(t)$ has positive off-diagonal elements with a certain asymptotic behavior. The systems of interest have the real parts of all eigenvalues of $A(t)$ negative and bounded away from zero, but nonetheless they have at least one (and frequently many) solutions whose magnitude grows to infinity with time. Strongly cooperative dynamical systems are relevant mathematical models for some biological systems. The existence of solutions whose magnitude is bounded away from zero would indicate the survival of the population. The first section of the paper surveys basic properties of autonomous linear systems which are described by a constant matrix $A$ and an initial condition. The matrix $e^{tA}$ is defined, and its role and actions are described. The author then goes on to show some spectral properties of matrices with positive off-diagonal elements. In particular, the principal eigenvalue of such a matrix is a strongly increasing function of any of its elements. Furthermore, it is shown (in the two-dimensional case) that the transition function of strongly cooperative systems is a matrix with all positive entries (Müller--Kamke theorem), and so is the matrix $e^{tA}$. The author returns to the action of the latter matrix on the unit circle in order to provide insight into the behavior of nonautonomous systems. The main part of the paper discusses the construction of a nonautonomous (piecewise constant) planar linear system such that for each time $t$, the larger eigenvalue of $A(t)$ equals $-\frac{1}{2}$, but the system has a solution not converging to zero at infinity. The author provides intuitive explanations, as well as in-depth analysis, pictures, and animations. In section 4, we find a discussion on how to construct systems with continuous or smooth matrix functions instead of piecewise constant ones. Finally, the author points out that the constructions could target quasi-periodic, almost periodic, and also more general time dependence (discussed in the penultimate section). The paper concludes with a discussion and ideas for project work: if random switching times were allowed, how would the systems need to be described in order to observe similar phenomena?

Discrimination of time-dependent inflow properties with a cooperative dynamical system.

Many physical, chemical, and biological systems exhibit a cooperative or sigmoidal response with respect to the input. In biochemistry, such behavior is called an allosteric effect. Here, we demonstrate that a system with such properties can be used to discriminate the amplitude or frequency of an external periodic perturbation. Numerical simulations performed for a model sigmoidal kinetics illustrate that there exists a narrow range of frequencies and amplitudes within which the system evolves toward significantly different states. Therefore, observation of system evolution should provide information about the characteristics of the perturbation. The discrimination properties for periodic perturbation are generic. They can be observed in various dynamical systems and for different types of periodic perturbation.

Global analysis for spread of infectious diseases via transportation networks.

We formulate an epidemic model for the spread of an infectious disease along with population dispersal over an arbitrary number of distinct regions. Structuring the population by the time elapsed since the start of travel, we describe the infectious disease dynamics during transportation as well as in the regions. As a result, we obtain a system of delay differential equations. We define the basic reproduction number {mathcal {R}}_0 as the spectral radius of a next generation matrix. For multi-regional systems with strongly connected transportation networks, we prove that if {mathcal {R}}_0le 1 then the disease will be eradicated from each region, while if {mathcal {R}}_0> 1 there is a globally asymptotically stable equilibrium, which is endemic in every region. If the transportation network is not strongly connected, then the model analysis shows that numerous endemic patterns can exist by admitting a globally asymptotically stable equilibrium, which may be disease free in some regions while endemic in other regions. We provide a procedure to detect the disease free and the endemic regions according to the network topology and local reproduction numbers. The main ingredients of the mathematical proofs are the inductive applications of the theory of asymptotically autonomous semiflows and cooperative dynamical systems. We visualise stability boundaries of equilibria in a parameter plane to illustrate the influence of the transportation network on the disease dynamics. For a system consisting of two regions, we find that due to spatial heterogeneity characterised by different local reproduction numbers, {mathcal {R}}_0 may depend non-monotonically on the dispersal rates, thus travel restrictions are not always beneficial.

The complete classification for global dynamics of a model for the persistence of HTLV-I infection

Li and Lim (2011) [8] proposed a three-dimensional system of ordinary differential equations modeling the role of Tax expression in the persistence of Human T-lymphotropic virus Type I and presented a mathematical analysis and its immunovirological explanation. The previous paper shows that three-dimensional system is a cooperative system if a sharp condition is satisfied. By means of the theory of cooperative dynamical systems, they studied the global dynamics. If the sharp condition is not satisfied, they leave it an open problem. The present paper is to provide a complete classification of global dynamics for this ODE model. In particular, our global analysis establishes that the dynamics of the HTLV-I infection are completely determined by a basic reproductive number, which completely solves the open problem above.

Evolution in Bayesian Supermodular Population Games

We study evolution in Bayesian supermodular population games. We define such games in the context of large populations of agents and establish the existence of a maximum and a minimum pure strategy Nash equilibria which are monotone in types. To study evolution in such games, we introduce Bayesian perturbed best response dynamics and the corresponding aggregate perturbed best response dynamic. Using the theory of cooperative dynamical systems, we show that solution trajectories of the aggregate perturbed best response dynamic converge to the set of perturbed equilibrium distributions. Using results from Ely and Sandholm (2005), we then conclude that the L1 solution trajectories of the Bayesian perturbed best response dynamic also converge to the set of Bayesian perturbed equilibria.

Stability Analysis of Transportation Networks with Multiscale Driver Decisions

Stability of Wardrop equilibria is analyzed for dynamical transportation networks in which the drivers' route choices are influenced by information at multiple temporal and spatial scales. The considered model involves a continuum of nonatomic indistinguishable drivers commuting between a common origin-destination pair in an acyclic transportation network. The drivers' route choices are affected by their relatively infrequent perturbed best responses to global information about the current network congestion levels, as well as their instantaneous local observation of the immediate surroundings as they transit through the network. A novel model is proposed for driver route choice behavior, exhibiting local consistency with their preference toward globally less congested paths as well as myopic decisions in favor of locally less congested paths. The simultaneous evolution of the traffic congestion on the network and of the aggregate path preference is modeled by a system of coupled ordinary differential equations. The main result shows that if the frequency of updates of path preferences is sufficiently small as compared to the frequency of the traffic flow dynamics, then the state of the transportation network ultimately approaches a neighborhood of the Wardrop equilibrium. The presented results may be read as further evidence in support of Wardrop's postulate of equilibrium, showing robustness of it with respect to nonpersistent perturbations. The proposed analysis combines techniques from singular perturbation theory, evolutionary game theory, and cooperative dynamical systems.

Hierarchical control of cooperative nonlinear dynamical systems

Cooperative control methods that are scalable with low computational cost are crucial for networked dynamical systems to respond quickly in unknown or cluttered environments. In an attempt to make the problem tractable, many existing cooperative controls are designed with oversimplified assumptions and/or without the capabilities of rapidly handling different environmental and dynamical constraints. In this article, proposed is a two-level hierarchical, cooperative control framework using a divide-and-conquer strategy so that challenges can be separately handled at different levels. It is scalable and has low computational cost. Based on a simplified homogeneous double-integrator dynamic model, the top-level planner first computes cooperative trajectories satisfying obstacle avoidance requirements. Then at the lower level, state and control constraints, nonlinear dynamics and self-collision/obstacle avoidance as related to the real system are addressed through a bio-inspired fast trajectory planning algorithm. The stability of the overall hierarchical structure is proven. Two examples, a differential-drive ground vehicle formation control and an unmanned aerial vehicle formation flight, are used to illustrate the advantages of the proposed hierarchical framework.

Cone invariance and rendezvous of multiple agents

In this article is presented a dynamical systems framework for analysing multi-agent rendezvous problems and characterize the dynamical behaviour of the collective system. Recently, the problem of rendezvous has been addressed considerably in the graph theoretic framework, which is strongly based on the communication aspects of the problem. The proposed approach is based on the set invariance theory and focusses on how to generate feedback between the vehicles, a key part of the rendezvous problem. The rendezvous problem is defined on the positions of the agents and the dynamics is modelled as linear first-order systems. These algorithms have also been applied to non-linear first-order systems. The rendezvous problem in the framework of cooperative and competitive dynamical systems is analysed that has had some remarkable applications to biological sciences. Cooperative and competitive dynamical systems are shown to generate monotone flows by the classical Muller—Kamke theorem, which is analysed using the set invariance theory. In this article, equivalence between the rendezvous problem and invariance of an appropriately defined cone is established. The problem of rendezvous is cast as a stabilization problem, with a the set of constraints on the trajectories of the agents defined on the phase plane. The n-agent rendezvous problem is formulated as an ellipsoidal cone invariance problem in the n-dimensional phase space. Theoretical results based on set invariance theory and monotone dynamical systems are developed. The necessary and sufficient conditions for rendezvous of linear systems are presented in the form of linear matrix inequalities. These conditions are also interpreted in the Lyapunov framework using multiple Lyapunov functions. Numerical examples that demonstrate application are also presented.

Attractors in coherent systems of differential equations

Attractors of cooperative dynamical systems are particularly simple; for example, a nontrivial periodic orbit cannot be an attractor, and orbits are nowhere dense. This paper provides characterizations of attractors for the wider class of coherent systems, defined by the property that all directed feedback loops are positive. Several new results for cooperative systems are obtained in the process. Connections with biological models are discussed.

On a Smale theorem and nonhomogeneous equilibria in cooperative systems

A standard result by Smale states that n dimensional strongly cooperative dynamical systems can have arbitrary dynamics when restricted to unordered invariant hyperspaces. In this paper this result is extended to the case when all solutions of the strongly cooperative system are bounded and converge towards one of only two equilibria outside of the hyperplane. An application is given in the context of strongly cooperative systems of reaction diffusion equations. It is shown that such a system can have a continuum of spatially inhomogeneous steady states, even when all solutions of the underlying reaction system converge to one of only three equilibria.

Fingerprint classification using a feedback-based line detector.

We present a fingerprint classification algorithm in this paper. This algorithm classifies a fingerprint image into one of the five classes: Arch, Left loop, Right loop, Whorl, and Tented arch. We use a new low-dimensional feature vector obtained from the output of a novel oriented line detector presented here. Our line detector is a co-operative dynamical system that gives oriented lines and preserves multiple orientations at points where differently oriented lines meet. Our feature extraction process is based on characterizing the distribution of orientations around the fingerprint. We discuss three different classifiers: support vector machines, nearest-neighbor classifier, and neural network classifier. We present results obtained on a National Institute of Standards and Technology (NIST) fingerprint database and compare with other published results on NIST databases. All our classifiers perform equally well, and this suggests that our novel line detection and feature extraction process indeed captures all the crucial information needed for classification in this problem.

Convergence in discrete-time neural networks with specific performance.

We analyze convergence in discrete-time neural networks with specific performance such as decay rate and trajectory bounds in terms of componentwise absolute (exponential) stability. Simple necessary and sufficient stability and positive invariance conditions are presented, which allow us to design a convergent network with prescribed performance. Our approach is based on a decomposition of competitive-cooperative connectivity or inhibitory-excitatory interaction that abounds in neural networks, without assuming symmetry of the connection matrices. The key idea is that through the decomposition, we can always relate a competitive-cooperative network with a cooperative dynamical system. The latter possesses significant order-preserving properties that are basic to our analysis. The explicit division of connection weights into inhibitory and excitatory types offers a higher potential for relating formal neural network models to neurophysiology.

Convergence of the one-dimensional Kohonen algorithm

We show in a very general framework the a.s. convergence of the one-dimensional Kohonen algorithm–after self-organization–to its unique equilibrium when the learning rate decreases to 0 in a suitable way. The main requirement is a log-concavity assumption on the stimuli distribution which includes all the usual (truncated) probability distributions (uniform, exponential, gamma distribution with parameter ≥ 1, etc.). For the constant step algorithm, the weak convergence of the invariant distributions towards equilibrium as the step goes to 0 is established too. The main ingredients of the proof are the Poincaré-Hopf Theorem and a result of Hirsch on the convergence of cooperative dynamical systems.

Convergence of the one-dimensional Kohonen algorithm

We show in a very general framework the a.s. convergence of the one-dimensional Kohonen algorithm–after self-organization–to its unique equilibrium when the learning rate decreases to 0 in a suitable way. The main requirement is a log-concavity assumption on the stimuli distribution which includes all the usual (truncated) probability distributions (uniform, exponential, gamma distribution with parameter ≥ 1, etc.). For the constant step algorithm, the weak convergence of the invariant distributions towards equilibrium as the step goes to 0 is established too. The main ingredients of the proof are the Poincaré-Hopf Theorem and a result of Hirsch on the convergence of cooperative dynamical systems.

Convergence presque sûre de l'algorithme de Kohonen unidimensionnel

We prove the a.s. convergence of the one dimensional Kohonen algorithm after selforganization and with decreasing step. Under some mild asumption of log-concavity on the input probability distribution, we first show the uniqueness of the limit and then the a.s. convergence. The proof relies on the index formula for a vector field on a manifold and a result by M. Hirsh about the convergence of cooperative dynamical systems applied to the ODE.