Analysis Of Discrete Dynamic Systems Research Articles

An evolutive model of a boundedly rational consumer with changing preferences and reference group consumption

In this paper, two dynamic models, recently proposed to describe the adaptive repeated choices of a boundedly rational consumer, are joined together. One considers a consumer adjusting the consumption level of a given good over time according to the observed discrepancy between expected and realized utility gain and modifies the utility function according to past consumption experience, also including saturation effects when past consumption is excessive. The other one considers the same adjustment mechanism with constant preferences but with a behavioral effect that introduces a tendency (or bias) to imitate a reference group of consumers. Merging these two models, a two-dimensional nonlinear dynamical system is obtained which describes consumers that decide their next period consumption of a given good by following two different (sometimes contrasting) criteria: their own utility maximization on the one side and imitation of a reference group of consumers on the other side. This leads to a greater uncertainty with respect to the model without the behavioral bias. Such uncertainty is studied through a numerical exploration of the long-run dynamics, guided by some global dynamical features of the nonlinear model, such as the folding action of the critical curves that characterize the behavior of the iterated noninvertible map and the singularities related to the presence of a vanishing denominator, namely focal points and prefocal curves. So, the aim of the paper is twofold: on the one side, it tries to contribute to the literature on the economic theory of boundedly rational consumers represented by evolutionary and behavioral approaches; On the other side, it tries to contribute to the recent literature about the global analysis of discrete dynamical systems characterized by contact bifurcations leading to the creation of complex topological structures of the attractors and their basins of attraction.

Forward invariant set preservation in discrete dynamical systems and numerical schemes for ODEs: application in biosciences

We present two results on the analysis of discrete dynamical systems and finite difference discretizations of continuous dynamical systems, which preserve their dynamics and essential properties. The first result provides a sufficient condition for forward invariance of a set under discrete dynamical systems of specific type, namely time-reversible ones. The condition involves only the boundary of the set. It is a discrete analog of the widely used tangent condition for continuous systems (viz. the vector field points either inwards or is tangent to the boundary of the set). The second result is nonstandard finite difference (NSFD) scheme for dynamical systems defined by systems of ordinary differential equations. The NSFD scheme preserves the hyperbolic equilibria of the continuous system as well as their stability. Further, the scheme is time reversible and, through the first result, inherits from the continuous model the forward invariance of the domain. We show that the scheme is of second order, thereby solving a pending problem on the construction of higher-order nonstandard schemes without spurious solutions. It is shown that the new scheme applies directly for mass action-based models of biological and chemical processes. The application of these results, including some numerical simulations for invariant sets, is exemplified on a general Susceptible-Infective-Recovered/Removed (SIR)-type epidemiological model, which may have arbitrary large number of infective or recovered/removed compartments.

Novel Techniques for Classifying Exotic Spheres in High Dimensions

Discrete calculus deals with developing the concepts and techniques of differential and integral calculus in a discrete setting, often using difference equations and discrete function spaces. This paper explores how differential-difference algebra can provide an algebraic framework for advancing discrete calculus. Differential-difference algebra studies algebraic structures equipped with both differential and difference operators. These hybrid algebraic systems unify continuous and discrete analogues of derivatives and shifts. This allows the development of general theorems and properties that cover both settings. In particular, we construct differential-difference polynomial rings and fields over discrete function spaces. We define discrete derivatives and shifts algebraically using these operators. We then study integration, summation formulas, fundamental theorems, and discrete analogues of multivariate calculus concepts from an algebraic perspective. A key benefit is being able to state unified theorems in differential-difference algebra that simultaneously yield results for both the continuous and discrete cases. This provides new tools and insights for discrete calculus using modern algebraic techniques. We also discuss applications of representing discrete calculus problems in differential-difference algebras. This allows bringing to bear algebraic methods and software tools for their solution. Specific examples are provided in areas such as numerical analysis of discrete dynamical systems defined through difference equations. The paper aims to demonstrate the capabilities of differential-difference algebra as a unifying framework for further developing the foundations and applications of discrete calculus. Broader connections to algebraic modeling of discrete physical systems are also discussed.

Stability and Bifurcation Analysis of Discrete Dynamical Systems 2020

Stability and Bifurcation Analysis of Discrete Dynamical Systems 2020

Computing Integrated Information (Φ) in Discrete Dynamical Systems with Multi-Valued Elements

Integrated information theory (IIT) provides a mathematical framework to characterize the cause-effect structure of a physical system and its amount of integrated information (). An accompanying Python software package (“PyPhi”) was recently introduced to implement this framework for the causal analysis of discrete dynamical systems of binary elements. Here, we present an update to PyPhi that extends its applicability to systems constituted of discrete, but multi-valued elements. This allows us to analyze and compare general causal properties of random networks made up of binary, ternary, quaternary, and mixed nodes. Moreover, we apply the developed tools for causal analysis to a simple non-binary regulatory network model (p53-Mdm2) and discuss commonly used binarization methods in light of their capacity to preserve the causal structure of the original system with multi-valued elements.

Stability and Bifurcations Analysis of Discrete Dynamical Systems

Stability and Bifurcations Analysis of Discrete Dynamical Systems

Theoretical Study of the One Self-Regulating Gene in the Modified Wagner Model

Predicting how a genetic change affects a given character is a major challenge in biology, and being able to tackle this problem relies on our ability to develop realistic models of gene networks. However, such models are rarely tractable mathematically. In this paper, we propose a mathematical analysis of the sigmoid variant of the Wagner gene-network model. By considering the simplest case, that is, one unique self-regulating gene, we show that numerical simulations are not the only tool available to study such models: theoretical studies can be done too, by mathematical analysis of discrete dynamical systems. It is first shown that the particular sigmoid function can be theoretically investigated. Secondly, we provide an illustration of how to apply such investigations in the case of the dynamical system representing the one self-regulating gene. In this context, we focused on the composite function f a ( m . x ) where f a is the parametric sigmoid function and m is a scalar not in { 0 , 1 } and we have proven that the number of fixed-point can be deduced theoretically, according to the values of a and m.

BIFURCATIONS IN AN ITERATIVE OPTIMIZATION PROCESS FOR INTENSITY MODULATED RADIATION THERAPY

We investigate bifurcations in an iterative method for optimizing an objective function derived from the intensity modulated radiation therapy (IMRT). Through the bifurcation analysis of discrete dynamical systems for small IMRT plans, we obtained bifurcation diagrams showing suitable parameter values for the convergence of fixed points corresponding to global or local minima of the objectives, and we found that rich nonlinear phenomena including a chaotic state can occur. We also illustrate that a similar bifurcation phenomenon occurs in a large IMRT system. The results on bifurcation structure are useful for suggesting the parameter design of IMRT plans with normal operation.