A non-linear discrete-time dynamical system related to epidemic SISI model

We consider SISI epidemic model with discrete-time. The crucial point of this model is that an individual can be infected twice. This non-linear evolution operator depends on seven parameters and we assume that the population size under consideration is constant, so death rate is the same with birth rate per unit time. Reducing to quadratic stochastic operator (QSO) we study the dynamical system of the SISI model.

Optimal measurement budget allocation for Kalman prediction over a finite time horizon by genetic algorithms

In this paper, we address the problem of optimal measurement budget allocation to estimate the state of a linear discrete-time dynamical system over a finite horizon. More precisely, our aim is to select the measurement times in order to minimize the variance of the estimation error over a finite horizon. In addition, we investigate the closely related problem of finding a trade-off between number of measurements and signal to noise ratio.First, the optimal measurement budget allocation problem is reduced to a deterministic combinatorial program. Then, we propose a genetic algorithm implementing a count preserving crossover to solve it. On the theoretical side, we provide a one-dimensional analysis that indicates that the benefit of using irregular measurements grows when the system is unstable or when the process noise becomes important. Then, using the duality between estimation and control, we show that the problem of selecting optimal control times for a linear quadratic regulator can be reduced to our initial problem.Finally, numerical implementations demonstrate that using measurement times optimized by our genetic algorithm gives better estimate than regularly spaced measurements. Our method is applied to a discrete version of a continuous-time system and the impact of the discretization time step is studied. It reveals good convergence properties, showing that our method is well suited to both continuous-time and discrete-time setups.

One-dimensional wave equation with set-valued boundary damping: well-posedness, asymptotic stability, and decay rates

This paper is concerned with the analysis of a 1D wave equation $z_{tt}-z_{xx}=0$ on $[0,1]$ with a Dirichlet condition at $x=0$ and a damping acting at $x=1$ of the form $(z_t(t,1),-z_x(t,1))\in\Sigma$ for $t\geq 0$, where $\Sigma$ is a given subset of $\mathbb R^2$. The study is performed within an $L^p$ functional framework, $p\in [1, +\infty]$. We determine conditions on $\Sigma$ ensuring existence and uniqueness of solutions of that wave equation, its strong stability and uniform global asymptotic stability of the solutions. In the latter case, we study the corresponding decay rates and their optimality. We first establish a correspondence between the solutions of that wave equation and the iterated sequences of a discrete-time dynamical system in terms of which we investigate the above mentioned issues. This enables us to provide a necessary and sufficient condition on $\Sigma$ ensuring existence and uniqueness of solutions of the wave equation and an efficient strategy for determining optimal decay rates when $\Sigma$ verifies a generalized sector condition. In case the boundary damping is subject to perturbations, we derive sharp results regarding asymptotic perturbation rejection and input-to-state issues.

On Sufficient Conditions for Chaotic Behavior of Multidimensional Discrete Time Dynamical System

In this work, we look at the extension of classical discrete dynamical system to multidimensional discrete-time dynamical system by characterizing chaos notions on $${\mathbb {Z}}^d$$ Z d -action. The $${\mathbb {Z}}^d$$ Z d -action on a space X has been defined in a very general manner, and therefore we introduce a $${\mathbb {Z}}^d$$ Z d -action on X which is induced by a continuous map, $$f:{\mathbb {Z}}\times X \rightarrow X$$ f : Z × X → X and denotes it as $$T_f:{\mathbb {Z}}^d \times X \rightarrow X$$ T f : Z d × X → X . Basically, we wish to relate the behavior of origin discrete dynamical systems (X, f) and its induced multidimensional discrete-time $$(X,T_f)$$ ( X , T f ) . The chaotic behaviors that we emphasized are the transitivity and dense periodicity property. Analogues to these chaos notions, we consider k-type transitivity and k-type dense periodicity property in the multidimensional discrete-time dynamical system. In the process, we obtain some conditions on $$(X,T_f)$$ ( X , T f ) under which the chaotic behavior of $$(X,T_f)$$ ( X , T f ) is inherited from the original dynamical system (X, f). The conditions varies whenever f is open, totally transitive or mixing. Some examples are given to illustrate these conditions.

Stochastic Theory of Coarse-Grained Deterministic Systems: Martingales and Markov Approximations

Many works have been devoted to show that Thermodynamics and Statistical Physics can be rigorously deduced from an exact underlying classical Hamiltonian dynamics, and to resolve the related paradoxes. In particular, the concept of equilibrium state and the derivation of Master Equations should result from purely Hamiltonian considerations. In this chapter, we reexamine this problem, following the point of view developed by Kolmogorov more than 60 years ago, in great part known from the work published by Arnold and Avez in 1967. Our setting is a discrete time dynamical system, namely the successive iterations of a measure-preserving mapping on a measure space, generalizing Hamiltonian dynamics in phase space. Using the notion of Kolmogorov entropy and martingale theory, we prove that a coarse-grained description both in space and in time leads to an approximate Master Equation satisfied by the probability distribution of partial histories of the coarse-grained state.

Poncelet property and quasi-periodicity of the integrable Boltzmann system

We study the motion of a particle in a plane subject to an attractive central force with inverse-square law on one side of a wall at which it is reflected elastically. This model is a special case of a class of systems considered by Boltzmann which was recently shown by Gallavotti and Jauslin to admit a second integral of motion additionally to the energy. By recording the subsequent positions and momenta of the particle as it hits the wall, we obtain a three-dimensional discrete-time dynamical system. We show that this system has the Poncelet property: If for given generic values of the integrals one orbit is periodic, then all orbits for these values are periodic and have the same period. The reason for this is the same as in the case of the Poncelet theorem: The generic level set of the integrals of motion is an elliptic curve, and the Poincaré map is the composition of two involutions with fixed points and is thus the translation by a fixed element. Another consequence of our result is the proof of a conjecture of Gallavotti and Jauslin on the quasi-periodicity of the integrable Boltzmann system, implying the applicability of KAM perturbation theory to the Boltzmann system with weak centrifugal force.

Inspecting debt servicing mechanism in Nigeria using ARMAX model of the Koyck-kind

The burden of external debt affects the wellbeing of an economy (or a country) by making the economy vulnerable to external shocks and crowding out investment. When dealing with debt management in indebted poor countries like Nigeria, the rational approach is to allocate a portion of export earnings for debt service payments. Along this line, there is a need to identify the link between debt servicing and export earnings. Hence, the current and long-run effects of export earnings on debt service payments is modelled as a single-input-single-output discrete-time dynamical system within the framework of Autoregressive Moving Average Explanatory Input model of the Koyck-kind (KARMAX). The KARMAX model is identified for Nigeria, using data from the World Bank database from 1970 to 2018 based on the maximum likelihood (ML) method, and the obtained results are compared to the prediction error and the instrumental variable methods. From a theoretical perspective, the KARMAX specification identified by the ML method is more ideal and inspiring. By doing so, this article contributes to the literature on the econometrics of public debt management.

Intercontinental prediction of soybean phenology via hybrid ensemble of knowledge-based and data-driven models

The timing of crop development has significant impacts on management decisions and subsequent yield formation. A large intercontinental dataset recording the timing of soybean developmental stages was used to establish ensembling approaches that leverage both discrete-time dynamical system models of soybean phenology and data-driven, machine-learned models to achieve accurate and interpretable predictions. We demonstrate that the knowledge-based, dynamical models can improve machine learning by generating expert-engineered features. Combining the predictions of the diverse component models via super learning resulted in a mean absolute error of 4.12 and 4.55 days to flowering (R1) and physiological maturity (R7), providing an improvement relative to the best benchmark model error of 6.90 and 15.47 days, respectively. The hybrid intercontinental model applies to a much wider range of management and temperature conditions than previous mechanistic models, enabling improved decision support as alternative cropping systems arise, farm sizes increase, and changes in the global climate continue to accelerate.