The topic of chaotic thresholds for piecewise linear discontinuous (PWLD) systems with multiple-well potentials is a persistent topic in the research of a number of authors. In this article we investigate the chaos of a modified piecewise linear discontinuous (MPWLD) system. The model, containing N free parameters, could be of interest to specialists working in this area. With a specially developed software product, we generate the Melnikov equation M(t)=0 and examine all its zeros. This opens up an opportunity for researchers to correctly understand and formulate the classical Melnikov criterion for the possible occurrence of chaos in dynamical systems. Several simulations are composed. We also demonstrate some specialized modules for investigating the dynamics of the proposed model. Intriguing and new generalizations made through probabilistic constructions are considered.

Efficient retrieval of mathematical and structural similarities in System Dynamics models remains a significant challenge for traditional lexical systems, which often fail to capture the contextual dependencies of simulation processes. This paper presents an architectural approach and implementation of a semantic search module integrated into an existing cloud-based modeling and simulation system. The proposed method employs a strategy for serializing graph structures into textual descriptions, followed by the generation of vector embeddings via local ONNX inference and indexing within a vector database (Qdrant). Experimental validation performed on a diverse corpus of complex dynamic models, compares the proposed approach against traditional information retrieval methods (Full-Text Search, Keyword Search in PostgreSQL, and Apache Lucene with Standard and BM25 scoring). The results demonstrate the distinct advantage of semantic search, achieving high precision (over 90%) within the scope of the evaluated corpus and effectively eliminating information noise. In comparison, keyword search exhibited only 24.8% precision with a significant rate of false positives, while standard full-text analysis failed to identify relevant models for complex conceptual queries (0 results). Despite a recorded increase in latency (~2 s), the study proves that the vector-based approach is a significantly more robust solution for detecting hidden semantic connections in mathematical model databases, providing a foundation for future developments toward multi-vector indexing strategies.

In this article, we investigate the dynamics of new modified third-order Phase-Locked Loops (PLLs). Our goal here is to investigate the effect of the new factor ∑j=1Najsin(jωt) on the dynamics of the proposed model. Using perturbation techniques based on Andronov–Melnikov concepts, we demonstrate that horseshoe chaos exists in three-dimensional nonautonomous systems. Several simulations are performed. Additionally, we present a few specific modules for examining the dynamics of the hypothetical oscillator circuit under consideration. This will be a crucial component of a much broader web-based scientific computing application. We will explicitly note that the proposed model is hypothetical and specialists working in this scientific field have a say. We will consider a numerical example of the possible application of the Melnikov function in the modeling of the radiation Melnikov antenna diagram. In addition, we examine a generalization based on probability distributions.

In this paper we investigate the chaos of a generalized perturbed Lotka–Volterra model based on considerations by other studies used in the literature. The model, containing N number of free parameters, could be of interest to specialists working in the fields of biological applications, chemistry, reaction kinetics, biostatistics, games theory, etc. With a specially developed software product, we generate the Melnikov equation M(t)=0 and examine all its zeros. This opens up an opportunity for the researcher to correctly understand and formulate the classical Melnikov criterion for the possible occurrence of chaos in the dynamical system. Several simulations are composed. We also demonstrate some specialized modules for investigating the dynamics of the proposed model. We further develop our model using the exponential form of the sine function. Thus, the perturbation can be interpreted as a term dependent on the characteristic function of a probability distribution. Although the original formulation leads to a distribution stated on a discrete domain, we can easily generalize the results for arbitrary distributions. Some particular examples are provided.

Many authors consider modified SIQR (susceptible, infected, isolated (quarantined), and recovered individuals) models for childhood diseases. In this paper, we examine a modified differential system with N free parameters that may be of interest to epidemiology experts. We pay special attention to the Melnikov function, which corresponds to the proposed new model. We create the Melnikov equation (M(t)=0) and analyze all of its roots using a specially designed software program. This gives the researcher the chance to accurately comprehend and articulate the classical Melnikov criterion for the potential appearance of chaos in the dynamical system. Additionally, we present a few specific modules for examining the new model’s dynamics. Additionally covered is a potential use of the Melnikov function that corresponds to the differential model under consideration, with particular potential in the modeling and synthesis of antenna diagrams. Last but not least, we consider the proposed generalization from a stochastic point of view.

In this paper, we provide a novel extended mixed differential model that is appealing to users because of its numerous free parameters. The motivation of this research arises from the opportunity for a general investigation of some outstanding classical and novel dynamical models. The higher energy levels known in the literature can be governed by appropriately added correction factors. Furthermore, the different applications of the considered model can be achieved only after a proper parameter calibration. All these necessitate the use of diverse optimization and approximation techniques. The proposed extended model is especially useful in the important field of decision making, namely the antenna array theory. This is due to the possibility of generating high-order Melnikov polynomials. The work is a natural continuation of the authors’ previous research on the topic of chaos generation via the term x|x|a−1. Some specialized modules for investigating the dynamics of the proposed oscillators are provided. Last but not least, the so-defined dynamical model can be of interest for scientists and practitioners in the area of antenna array theory, which is an important part of the decision-making field. The stochastic control of oscillations is also the subject of our consideration. The underlying distributions we use may be symmetric, asymmetric or strongly asymmetric. The same is true for the mass in the tails, too. As a result, the stochastic control of the oscillations we purpose may exhibit a variety of possible behaviors. In the final section, we raise some important issues related to the methodology of teaching Master’s and PhD students.

Many authors analyze the chaotic motion of the driven and damped double sine-Gordon equations and compute the Melnikov functions by numerical methods, taking an example to verify good agreement between numerical methods and analytical ones. Unfortunately, due to the lack of an explicit presentation of the Melnikov integral, the reader has difficulty navigating and touching upon Melnikov’s elegant theory and, in particular, the formulation of the Melnikov criterion for the occurrence of chaos in a dynamical system, based solely on the provided illustrations of dependencies between the main parameters of the model under consideration. In this paper we will try to shed additional light on this important problem. A new planar system corresponding to the generalized double sine-Gordon model with many free parameters is considered. We also look at the modeling of radiation diagrams and antenna factors as potential uses for the Melnikov functions. A number of simulations are created. We also show off a few specific modules for examining the model’s behavior. There is also discussion of one use for potential oscillation control.

Many authors analyze the prediction of chaos in a Josephson junction with quadratic damping by the Melnikov technique. Due to the lack of an explicit presentation of the Melnikov integral, the researchers apply numerical methods and illustrative examples to verify a good agreement between the numerical method and the analytical one. The reader has difficulty navigating and touching upon Melnikov’s elegant theory and, in particular, the formulation of the Melnikov criterion for the possible occurrence of chaos in a dynamical system, based solely on the provided illustrations of dependencies between the main parameters of the model under consideration. The statements in a number of publications devoted to this interesting topic, such as “It is easy to see that Melnikov’s integrals are finite and not zero. It is possible to see that the transverse zeros of the Melnikov function”, do not shed enough light on the origin of the “horseshoe”-type chaos. In this paper we will try to shed additional light on this important problem. A new planar system corresponding to the N-generalized Josephson junction with quadratic damping with many free parameters is considered, which may be of interest to specialists in the field of engineering sciences. Prediction of chaos in the proposed model by the Melnikov technique is closely related to the problem of approximately simultaneously finding all roots (simple or multiple) of generalized trigonometric polynomials. Several simulations are composed. We also demonstrate some specialized modules for investigating the dynamics of the model. One application about generating stochastic construction for possible control over oscillations is also discussed.

In this paper, we propose a new system that occurs in modelling “the battle of the sexes” in evolutionary biology (Hofbauer & Sigmund 1988). The existence of a heteroclinic cycle and a continuous family of periodic orbits of the system is established; then the dynamical characteristics of a time-periodic perturbation of the system are investigated. By using the well-known Melnikov’s method, a sufficient condition is obtained for the perturbed system to have a transverse hetero-clinic cycle and hence to possess chaotic behaviour in the sense of Smale. One possible application that Melnikov functions may find in the modelling and synthesis of radiating antenna patterns is considered. We demonstrate some modules for investigating the dynamics of the proposed model. This will be included as an integral part of a planned much more general Web-based application for scientific computing. The proposed new extended model contains many free parameters (the coefficients ai, i = 1, 2, . . . ,N), which makes it attractive for use in the fields of biological applications, chemistry, sociology, lifetime analysis, reaction kinetics, biostatistics, population dynamic, medical research, games theory etc. Finally, a special case of subharmonic solutions is discussed.

The so-call SQUIDs (abbreviated from superconducting quantum interference device) are very sensitive apparatuses especially built for metering very low magnetic fields. These systems have applications in various practical fields—biology, geology, medicine, different engineering areas, etc. Their features are mainly based on superconductors and the Josephson effect. They can be differentiated into two main groups—direct current (DC) and radio frequency (RF) SQUIDs. Both of them were constructed in the 1960s at Ford Research Labs. The main difference between them is that the second ones use only one superconducting tunnel junction. This reduces their sensitivity, but makes them significantly cheaper. We investigate namely the rf-SQUIDs in the present work. A number of authors devote their research to the rf-SQUIDs driven by an oscillating external flux. We aim to enlarge the theoretical base of these systems by adding new factors in their dynamics. Several particular cases are explored and simulated. We demonstrate also some specialized modules for investigating the proposed model. One application for possible control over oscillations is also discussed. It is based on the Fourier transform and, as a consequence, on the characteristic function of some probability distributions.

The literature devoted to the issue of a forced modified Van der Pol–Duffing oscillator with asymmetric potential is a major and varied way to represent nonlinear dissipative chemical dynamics. It is known that this model is based on the real reaction–kinetic scheme. In this paper, we suggest a novel class of oscillators that are appealing to users due to their numerous free parameters and asymmetric potential. The rationale for this is because an expanded model is put out that enables the investigation of both classical and more recent models that have been reported in the literature at a “higher energy level”. We present a few specific modules for examining these oscillators’ behavior. A much broader Web-based application for scientific computing will incorporate this as a key component. Probabilistic construction to offer possible control over the oscillations is also considered.

The Duffing–van der Pol oscillator is a very prominent and interesting standard model. There is a substantial body of varied literature on this topic. In this article, we propose a new class of oscillators by adding new factors to its dynamics. Investigations in light of Melnikov’s approach are considered. Several simulations are composed. A few specialized modules for testing the dynamics of the hypothetical oscillator under consideration are also given. This will be an essential component of a much broader Web-based scientific computing application that is planned. Possible control over oscillations: approximation with restrictions is also discussed; some probabilistic constructions are also presented.

In this article, we propose a new hypothetical differential model with many free parameters, which makes it attractive to users. The motivation is as follows: an extended model is proposed that allows us to investigate classical and newer models appearing in the literature at a “higher energy level”, as well as the generation of high–order Melnikov polynomials (corresponding to the proposed extended model) with possible applications in the field of antenna feeder technology. We present a few specific modules for examining these oscillators’ behavior. A much broader Web-based application for scientific computing will incorporate this as a key component.

We suggest a few kinds of extended classical oscillators in this study. We present a few specific modules for examining these oscillators’ behavior. This will be an essential component of a broader web-based scientific computing platform that is in the works. The modeling and synthesis of radiating antenna designs is also taken into consideration as a potential use case for Melnikov functions. Additionally, we discuss strategies for achieving probabilistic control over system perturbations.

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.

The purpose of this paper is to investigate some Morse-type oscillators. In its original form, it is a model for describing the vibrations of a diatomic molecule. The Morse potential generalizes the harmonic oscillator by introducing deviations from the classical theoretical model. In the present study, we perturbed the Morse differential equation by several periodic terms based on the cosine function and by a damping term. The frequency is driven by different coefficients. The size of the deviations is controlled by another constant. We provide two modifications w.r.t. the damping term. The Melnikov approach is applied as an indicator of the possible chaotic opportunities. We also propose a novel approach for stochastic control of the perturbations. It is based on the assumption that the coefficients of the periodic terms are the probabilities of underlying distribution. As a result, the dynamics are driven by its characteristic function. Several applications are considered. We demonstrate some specialized modules for investigating the dynamics of the proposed models, along with the synthesis of radiating antenna patterns.

In this paper, we propose a new class of micro-electromechanical oscillators. Some investigations based on Melnikov’s approach are applied for identifying some chaotic possibilities. We demonstrate also some specialized modules for investigating the dynamics of these oscillators. This will be included as an integral part of a planned much more general Web-based application for scientific computing. It turns out that the theoretical apparatus for studying the circuit implementation (design, fabricating, etc.) of the considered differential model for large parameter values is extremely complex and requires a serious investigation. This is the reason to offer this model to the attention of specialists working in this scientific direction. Some open problems related to the use of existing computer algebraic systems for the study of this class of oscillators for large values of n,m and N are also posed. In general, the entire article is subordinated to this frank conversation with the readers with the sole purpose being the professional upgrading of the specialized modules provided for this purpose in subsequent licensed versions of CAS.

The Kies model can be regarded as an efficient model in terms of goodness–of–fit in the field of Software Reliability Analysis. In the books. Pavlov et al., (Some software reliability models: Approximation and modeling aspects. LAP LAMBERT Academic Publishing, Chisinau, 2018) [20], Pavlov et al., (Nontrivial Models in Debugging Theory (Part 2). LAP LAMBERT Academic Publishing, Chisinau, 2018) [21], we explore some models which correspond to debugging theory. In this article we consider the three–parameter Kies model modified as “deterministic” model. We give how a modification of the model with “polynomial variable transfer” can be applied. The usage of such new model for approximation of key data from many scientific fields which concern growth theory. CAS Mathematica gives us possibility for successful implementation of our results.

In this paper we consider the Dickson polynomials D_m+1,n(x,a) of the sixth and seven kind (i.e. m=5 and m=6 , respectively). First, a model with Dickson polynomials as corrections in the Lienard system is presented and “level curves” are studied. In the second place, we will note that some specifics of the amplitudes of these high-degree polynomials open up the possibility of modeling signals from the field of antenna-feeder technology. So, for example, changing the variable t by t=bcosθ +c ( θ is the azimuthal angle and c is the phase difference) in the y(t)-component of the solution of Lienard differential system results in the generation of radiation diagrams. Numerical examples, illustrating our results using CAS MATHEMATICA are given.

In this paper, we propose a new modified planar Kelvin–Stuart model. We demonstrate some modules for investigating the dynamics of the proposed model. This will be included as an integral part of a planned, much more general Web-based application for scientific computing. Investigations in light of Melnikov’s approach are considered. Some simulations and applications are also presented. The proposed new modifications of planar Kelvin–Stuart models contain many free parameters (the coefficients gi,i=1,2,…,N), which makes them attractive for use in engineering applications such as the antenna feeder technique (a possible generating and simulating of antenna factors) and the theory of approximations (a possible good approximation of a given electrical stage). The probabilistic control of the perturbations is discussed.