We study the uniform approximation of the sigmoid cut function by smooth sigmoid functions such as the logistic and the Gompertz functions. The limiting case of the interval-valued В step function is В discussed В using В Hausdorff metric. В Various expressions for the error estimates of the corresponding uniform and Hausdorff approximations are obtained. Numerical examples are presented using CAS MATHEMATICA.

In this paper, we present a new class of extended oscillators in light of chaos theory. It is based on dynamical complex systems built on the concept of self-describing with a stopping criterion process. We offer an effective studying approach with a specific focus on learning, provoking students’ thinking through the triad of enigmatics–creativity–acmeology. Dynamic processes are the basis of mathematical modeling; thus, we can reach the goal of the above-mentioned triad by the proposed differential systems. The results we derive strongly confirm the presence of symmetry in the outcomes of the proposed models. We suggest a stochastic approach to structuring the proposed dynamical systems by modeling the coefficients that drive them by some discrete probability distribution that exhibits symmetry or asymmetry. We propose specific tools for researching the behavior of these systems.

In this article, we propose some extended oscillator models. Various experiments are performed. The models are studied using the Melnikov approach. We show some integral units for researching the behavior of these hypothetical oscillators. These will be implemented as add-on sections of a thoughtful main web-based application for researching computations. One of the main goals of the study is to share the difficulties that researchers (who are not necessarily professional mathematicians) encounter in using contemporary computer algebraic systems (CASs) for scientific research to examine in detail the dynamics of modifications of classical and newer models that are emerging in the literature (for the large values of the parameters of the models). The present article is a natural continuation of the research in the direction that has been indicated and discussed in our previous investigations. One possible application that the Melnikov function may find in the modeling of a radiating antenna diagram is also discussed. Some probability-based constructions are also presented. We hope that some of these notes will be reflected in upcoming registered rectifications of the CAS. The aim of studying the design realization (scheme, manufacture, output, etc.) of the explored differential models can be viewed as not yet being met.

In this paper, we propose new generalized oscillator model. Considerations in the light of the Melnikov's approach are also given. We focus on some interesting simulations based on the proposed new model. This is an integral part of a planned much more general Web-based application for scientific computing.