A survey on piecewise-linear models of regulatory dynamical systems

Recent developments in understanding the various regulatory systems, especially the developments in biology and genomics, stimulated an interest in modelling such systems. Hybrid systems, originally developed for process control applications, provide advances in modelling such systems. A particular class of hybrid systems which are relatively simpler to analyze mathematically but still capable of demonstrating the essential features of many non-linear dynamical systems is piecewise-linear systems. Implementation of piecewise-linear systems for modelling of regulatory dynamical systems requires different considerations depending on the status of the problem. In this work we considered three different cases. Firstly, we consider the inferential modelling problem based on the empirical observations and study the discrete piecewise-linear system, whose inverse problem is solvable under some assumptions. Secondly, we considered the problem of obtaining some complex regulatory systems by tractable piecewise-linear formulations and study the qualitative dynamic features of the systems and their piecewise-linear models. Finally, we considered Boolean delay equations for building abstract models of regulatory systems, which might be the simplest models demonstrating the essential qualitative features of our interest underlying adaption, learning and memorization.     

Dynamical analysis of chemotherapy models with time-dependent infusion

Classical mathematical models for chemotherapy assume a constant infusion rate of the chemotherapy agent. However in reality the infusion rate usually varies with respect to time, due to the natural (temporal or random) fluctuation of environments or clinical needs. In this work we study a non-autonomous chemotherapy model where the injection rate and injection concentration of the chemotherapy agent are time-dependent. In particular, we prove that the non-autonomous dynamical system generated by solutions to the non-autonomous chemotherapy system possesses a pullback attractor. In addition, we investigate the detailed interior structures of the pullback attractor to provide crucial information on the effectiveness of the treatment. The main analytical tool used is the theory of non-autonomous dynamical systems. Numerical experiments are carried out to supplement the analysis and illustrate the effectiveness of different types of infusions.     

Noncommutative Dynamical Models with Quantum Symmetries

We define dynamical models on the q-Minkowski space algebra (which is a particular case of the Reflection Equation Algebra) as deformations (quantizations) of dynamical models with rotational symmetries, and we find their integrals. In particular, we introduce a q-analog of the Runge-Lenz vector and a q-analog of the dynamics in space-time with a spherically symmetric metric.     

Measuring complexity in a business cycle model of the Kaldor type

The purpose of this paper is to study the dynamical behavior of a family of two-dimensional nonlinear maps associated to an economic model. Our objective is to measure the complexity of the system using techniques of symbolic dynamics in order to compute the topological entropy. The analysis of the variation of this important topological invariant with the parameters of the system, allows us to distinguish different chaotic scenarios. Finally, we use a another topological invariant to distinguish isentropic dynamics and we exhibit numerical results about maps with the same topological entropy. This work provides an illustration of how our understanding of higher dimensional economic models can be enhanced by the theory of dynamical systems.     

The tractability index of memristive circuits: branch‐oriented and tree‐based models

The memory‐resistor or memristor is a new electrical element characterized by a nonlinear charge‐flux relation. This device poses many challenging problems, in particular from the circuit modeling point of view. In this paper, we address the index analysis of certain differential‐algebraic models of memristive circuits; specifically, our attention is focused on so‐called branch‐oriented models, which include in particular tree‐based formulations of the circuit equations. Our approach combines results coming from differential‐algebraic equation theory, matrix analysis and theory of digraphs. This framework should be useful in future studies of dynamical aspects of memristive circuits. Copyright © 2012 John Wiley & Sons, Ltd.     

Role of ergodicity in the transient Fluctuation Relation and a new relation for a dissipative non-chaotic map

Toy model dynamical systems, such as the baker maps, are useful to shed light on some of the conditions verified by deterministic models in non-equilibrium statistical physics. We investigate a 2D dynamical system, enjoying a weak form of reversibility, with peculiar basins of attraction and steady states. In particular, we test the conditions required for the validity of the transient Fluctuation Relation. Our analysis illustrates by means of concrete examples why ergodicity of the equilibrium dynamics (also known as “ergodic consistency”) seems to be a necessary condition for the transient Fluctuation Relation. This investigation then leads to the numerical verification of a kind of transient relation which, differently from the usual transient Fluctuation Relation, holds only asymptotically. At the same time, this relation is not a steady state Fluctuation Relation, because the steady state is a fixed point without fluctuations.     

Dynamical systems for solving variational inclusion and fixed point problems on Hadamard manifolds

In this paper, we investigate a new class of dynamical systems for solving variational inclusion and fixed point problems on Hadamard manifolds. Then we prove that the dynamical system has a unique solution under some suitable assumptions. Moreover, the global exponential stability and invariance property of the dynamical systems are also established. Our main results in this work are new and extend the existing ones in the literature.     

Distribution of frequencies of digits via multifractal analysis

We study the Hausdorff dimension of a large class of sets in the real line defined in terms of the distribution of frequencies of digits for the representation in some integer base. In particular, our results unify and extend classical work of Borel, Besicovitch, Eggleston, and Billingsley in several directions. Our methods are based on recent results concerning the multifractal analysis of dynamical systems and often allow us to obtain explicit expressions for the Hausdorff dimension. This work is still another illustration of the role that the theory of dynamical systems can play in number theory.     

Models of p-adic mechanics

We segregate the class of ultrametric (p-adic) systems within the standard models of classical and quantum mechanics. We show that ultrametric models can be described in the language of standard models but also have several distinguishing properties. In particular, we show that a stronger Poincaré recurrence theorem holds for classical ultrametric dynamical systems. As an example of a quantum p-adic system, we consider the algebra of commutation relations of the one-dimensional quantum mechanics. We show that this algebra, as in the real case, is isomorphic to the algebra of compact operators.     

Stabilization in a 3D eco-epidemiological model: From the complete extinction of a predator population to their self-healing

In this paper, using the localization method of compact invariant sets, we examine the ultimate dynamics of the 3D prey–predator model containing two subpopulations of susceptible and infected predators. Our attention is focused to finding ultimate sizes of interacting populations, and, in addition, we show the existence of a global attracting set. Then, we derive various global conditions of ultimate extinction of at least one of the predators subpopulations and describe conditions under which all types of internal bounded dynamics are ruled out. In particular, we describe convergence conditions to omega-limit sets located (1) in the intersection of the prey-free plane with the infected predators-free plane and (2) in the infected predators-free plane. Based on the dynamical analysis of the 2D infection-free subsystem, we obtain conditions of global attraction to (i) the prey-only disease-free equilibrium point, (ii) the disease-free prey-predator equilibrium point (self-healing of the predator population), and (iii) the omega-limit set containing an equilibrium point or a periodic orbit. Main theoretical results are illustrated by numerical simulation. Tools and techniques developed in this work can be appropriated in the studies within predictive population ecology of more complex eco-epidemiological models.     

When Lanchester met Richardson,the outcome was stalemate: A parable for mathematical models of insurgency

Many authors have used dynamical systems to model asymmetric war. We explore this approach more broadly, first returning to the prototypical models such as Richardson’s arms race, Lanchester’s attrition models and Deitchman’s guerrilla model. We investigate combinations of these and their generalizations, understanding how they relate to assumptions about asymmetric conflict. Our main result is that the typical long-term outcome is neither annihilation nor escalation but a stable fixed point, a stalemate. The state cannot defeat the insurgency by force alone, but must alter the underlying parameters. We show how our models relate to or subsume other recent models. This paper is a self-contained introduction to 2D continuous dynamical models of war, and we intend that, by laying bare their assumptions, it should enable the reader to critically evaluate such models and serve as a reminder of their limitations.     

A note on random walks with absorbing barriers and sequential Monte Carlo methods

In this article, we consider importance sampling (IS) and sequential Monte Carlo (SMC) methods in the context of one-dimensional random walks with absorbing barriers. In particular, we develop a very precise variance analysis for several IS and SMC procedures. We take advantage of some explicit spectral formulae available for these models to derive sharp and explicit estimates; this provides stability properties of the associated normalized Feynman–Kac semigroups. Our analysis allows one to compare the variance of SMC and IS techniques for these models. The work in this article is one of the few to consider an in-depth analysis of an SMC method for a particular model-type as well as variance comparison of SMC algorithms.     

Dynamics of a stage-structured predator–prey model with prey impulsively diffusing between two patches

In this work, we propose a stage-structured predator–prey model, with prey impulsively diffusing between two patches. Using the discrete dynamical system determined by the stroboscopic map, we obtain a predator-extinction periodic solution. Further, the predator-extinction periodic solution is globally attractive. By the theory on the delay and impulsive differential equation, we prove that the investigated system is permanent. Our results indicate that the discrete time delay has influence to the dynamical behaviors of the investigated system.     

A method for reduction of human ventricular action potential model

ABSTRACT

Mathematical modelling and computer simulations are important tools in the field of cardiac electrophysiology. High computational costs of complex models make them difficult to apply in large-scale simulations like tissue. Therefore, model reduction are of particular importance in heart studies. In this paper, we introduce a technique for simplification of ventricular cell(VC) complex models. By using this technique, starting with a complex model of human VC including 17state variables, we reduce the number of state variables to two. Our simplified model is compared with the original one via several electrophysiological features and computational efficiency. Results show that the reduced model has acceptable behaviours in single cell and one-dimensional simulation, moreover, is 55 times faster than the original one. As the presented method does not depend on the reference model, it may be applied to every cardiac cell models or each complex excitable dynamical systems with the same dynamics as VC.     

Learning the Structure of Mixed Graphical Models

We consider the problem of learning the structure of a pairwise graphical model over continuous and discrete variables. We present a new pairwise model for graphical models with both continuous and discrete variables that is amenable to structure learning. In previous work, authors have considered structure learning of Gaussian graphical models and structure learning of discrete models. Our approach is a natural generalization of these two lines of work to the mixed case. The penalization scheme involves a novel symmetric use of the group-lasso norm and follows naturally from a particular parameterization of the model. Supplementary materials for this article are available online.     

Competitive exclusion in a nonlocal reaction–diffusion–advection model of phytoplankton populations

We continue our study on the global dynamics of a nonlocal reaction–diffusion–advection system modeling the population dynamics of two competing phytoplankton species in a eutrophic environment, where both populations depend solely on light for their metabolism. In our previous work, we proved that system (1.1) is a strongly monotone dynamical system with respect to a non-standard cone related to the cumulative distribution functions, and further determined the global dynamics when the species have either identical diffusion rate or identical advection rate. In this paper, we study the trade-off of diffusion and advection and their joint influence on the outcome of competition. Two critical curves for the local stability of two semi-trivial equilibria are analyzed, and some new competitive exclusion results are obtained. Our main tools, besides the theory of monotone dynamical system, include some new monotonicity results for the principal eigenvalues of elliptic operators in one-dimensional domains.     

Random attractors for stochastic differential equations driven by two-sided Lévy processes

Abstract

In this paper, the asymptotic behavior of solutions for a nonlinear Marcus stochastic differential equation with multiplicative two-sided Lévy noise is studied. We plan to consider this equation as a random dynamical system. Thus, we have to interpret a Lévy noise as a two-sided metric dynamical system. For that, we have to introduce some fundamental properties of such a noise. So far most studies have only discussed two-sided Lévy processes which are defined by combining two-independent Lévy processes. In this paper, we use another definition of two-sided Lévy process by expanding the probability space. Having this metric dynamical system we will show that the Marcus stochastic differential equation with a particular drift coefficient and multiplicative noise generates a random dynamical system which has a random attractor.     

Tilings,substitution systems and dynamical systems generated by them

The object of this work is to study the properties of dynamical systems defined by tilings. A connection to symbolic dynamical systems defined by one- and two-dimensional substitution systems is shown. This is used in particular to show the existence of a tiling system such that its corresponding dynamical system is minimal and topological weakly mixing. We remark that for one-dimensional tilings the dynamical system always contains periodic points.     

A dynamical systems framework for intermittent data assimilation

We consider the problem of discrete time filtering (intermittent data assimilation) for differential equation models and discuss methods for its numerical approximation. The focus is on methods based on ensemble/particle techniques and on the ensemble Kalman filter technique in particular. We summarize as well as extend recent work on continuous ensemble Kalman filter formulations, which provide a concise dynamical systems formulation of the combined dynamics-assimilation problem. Possible extensions to fully nonlinear ensemble/particle based filters are also outlined using the framework of optimal transportation theory.     

New approach in dynamics of regenerative chatter research of turning

In this paper, regenerative chatter phenomena in a turning process is discussed from impulsive dynamical point of view. By introducing the instantaneous pulse when vibration occurs and the vibratory condition set, we optimize the models and present a certain kind of second-order impulsive differential systems, which is a specific discontinuous dynamical system. Then we search for the general results of the nonoccurrence of chatter phenomena by discussing the number of the vibration pulse times, utilizing the method of flow theory in discontinuous systems and transversal property at the boundary. Our results give a convenient way to estimate the available parameters to keep the turning process stable.