Solvability of a nonlinear three-dimensional system of difference equations with constant coefficients

In this paper, we show that the following three-dimensional system of difference equations x n + 1 = y n x n − 2 a x n − 2 + b z n − 1 , y n + 1 = z n y n − 2 c y n − 2 + d x n − 1 , z n + 1 = x n z n − 2 e z n − 2 + f y n − 1 , n ∈ N 0 , $$\begin{equation*} x_{n+1}=\frac{y_{n}x_{n-2}}{ax_{n-2}+bz_{n-1}}, \quad y_{n+1}=\frac{z_{n}y_{n-2}}{cy_{n-2}+dx_{n-1}}, \quad z_{n+1}=\frac{x_{n}z_{n-2}}{ez_{n-2}+fy_{n-1}}, \quad n\in \mathbb{N}_{0}, \end{equation*}$$ where the parameters a, b, c, d, e, f and the initial values x −i , y −i , z −i , i ∈ {0, 1, 2}, are complex numbers, can be solved, extending further some results in the literature. Also, we determine the forbidden set of the initial values by using the obtained formulas. Finally, an application concerning a three-dimensional system of difference equations are given.

Motivation and information in motor performance: modelling of self-efficacy and knowledge of results interaction in a timing task

BACKGROUND: Despite the literature positing a strong relation between motor performance and self-efficacy, few studies address the phenomenon formally. In this sense, how self-efficacy modulates corrections in a trial-to-trial basis and how the performance that individuals consider to be satisfactory modulate both corrections and self-efficacy are not well understood. AIM: The aim of this study is to develop and evaluate a model that relates self-efficacy and performance through a system of difference equations. METHOD: First, we demonstrate the model’s capabilities through constrained simulations. The, to evaluate the model’s grasp of empirical data, we parameterized the model to capture the constant, variable error, self-efficacy and believed satisfactory performance for each individual. RESULTS: The model demonstrates capacity to reproduce these summary results when initial conditions are fed to the system of difference equations. However, we observe features that must be improved and qualitative deviations when individuals demonstrate highly variable behavior. CONCLUSION: The initial results support the current assumptions and included variables in this model.

ON A SOLVABLE SYSTEM OF NON-LINEAR DIFFERENCE EQUATIONS WITH VARIABLE COEFFICIENTS

In this paper, we show that the system of difference equations can be solved in the closed form. Also, we determine the forbidden set of the initial values by using the obtained formulas. Finally, we obtain periodic solutions of aforementioned system.

Characterization of Homogeneous System of Difference Equations

In this paper, we introduced new definitions of the system of homogenous difference equations of order two; namely homogenous and semi homogenous system, where we focused on finding the equivalents for these definitions of order one as well as of order greater than one for the system of difference equations of the second order and given some examples. We also a given formula to find the power of the matrix that we used in this research.

Neimark-Sacker, flip and transcritical bifurcation in a symmetric system of difference equations with exponential terms

In this paper, we study the conditions under which the following symmetric system of difference equations with exponential terms: \[ x_{n+1} =a_1\frac{y_n}{b_1+y_n} +c_1\frac{x_ne^{k_1-d_1x_n}}{1+e^{k_1-d_1x_n}},\] \[ y_{n+1} =a_2\frac{x_n}{b_2+x_n} +c_2\frac{y_ne^{k_2-d_2y_n}}{1+e^{k_2-d_2y_n}}\] where $a_i$, $b_i$, $c_i$, $d_i$, $k_i$, for $i=1,2$, are real constants and the initial values $x_0$, $y_0$ are real numbers, undergoes Neimark-Sacker, flip and transcritical bifurcation. The analysis is conducted applying center manifold theory and the normal form bifurcation analysis.

Inference on the dynamics of the COVID pandemic from observational data

We describe a time dependent stochastic dynamic model in discrete time for the evolution of the COVID-19 pandemic in various states of USA. The proposed multi-compartment model is expressed through a system of difference equations that describe their temporal dynamics. Various compartments in our model is connected to the social distancing measures and diagnostic testing rates. A nonparametric estimation strategy is employed for obtaining estimates of interpretable temporally static and dynamic epidemiological rate parameters. The confidence bands of the parameters are obtained using a residual bootstrap procedure. A key feature of the methodology is its ability to estimate latent compartments such as the trajectory of the number of asymptomatic but infected individuals which are the key vectors of COVID-19 spread. The nature of the disease dynamics is further quantified by the proposed epidemiological markers, which use estimates of such key latent compartments.