Dynamic Stability Analysis of Size-Dependent Viscoelastic/Piezoelectric Nano-beam

This paper analyzes the dynamic stability of an isotropic viscoelastic Euler–Bernoulli nano-beam using piezoelectric materials. For this purpose, the size-dependent theory was used in the framework of the modified couple stress theory (MCST) for piezoelectric materials. In order to capture the geometrical nonlinearity, the von Karman strain displacement relation was applied. Hamilton’s principle was also employed to obtain the governing equations. Furthermore, the Galerkin method was used in order to convert the governing partial differential equations (PDEs) to a nonlinear second-order ordinary differential one. Dynamic stability analysis was performed and the effects of such parameters as viscoelastic coefficients, size effect, and piezoelectric coefficient were investigated. The results showed that in this system, saddle points, central points, Hopf bifurcation points, and fork bifurcation points could be created, and the phase portraits connecting these equilibrium points exhibit periodic orbits, heteroclinic orbits, and homoclinic orbits.

Diffusion-Wave Type Solutions to the Second-Order Evolutionary Equation with Power Nonlinearities

The paper deals with a nonlinear second-order one-dimensional evolutionary equation related to applications and describes various diffusion, filtration, convection, and other processes. The particular cases of this equation are the well-known porous medium equation and its generalizations. We construct solutions that describe perturbations propagating over a zero background with a finite velocity. Such effects are known to be atypical for parabolic equations and appear as a consequence of the degeneration of the equation at the points where the desired function vanishes. Previously, we have constructed it, but here the case of power nonlinearity is considered. It allows for conducting a more detailed analysis. We prove a new theorem for the existence of solutions of this type in the class of piecewise analytical functions, which generalizes and specifies the earlier statements. We find and study exact solutions having the diffusion wave type, the construction of which is reduced to the second-order Cauchy problem for an ordinary differential equation (ODE) that inherits singularities from the original formulation. Statements that ensure the existence of global continuously differentiable solutions are proved for the Cauchy problems. The properties of the constructed solutions are studied by the methods of the qualitative theory of differential equations. Phase portraits are obtained, and quantitative estimates are determined by constructing and analyzing finite difference schemes. The most significant result is that we have shown that all the special cases for incomplete equations take place for the complete equation, and other configurations of diffusion waves do not arise.

Discretization, Bifurcation and Control for a Class of Predator–Prey Interactions

The present study focuses on the dynamical aspects of a discrete-time Leslie–Gower predator–prey model accompanied by a Holling type III functional response. Discretization is conducted by applying a piecewise constant argument method of differential equations. Moreover, boundedness, existence, uniqueness, and a local stability analysis of biologically feasible equilibria were investigated. By implementing the center manifold theorem and bifurcation theory, our study reveals that the given system undergoes period-doubling and Neimark–Sacker bifurcation around the interior equilibrium point. By contrast, chaotic attractors ensure chaos. To avoid these unpredictable situations, we establish a feedback-control strategy to control the chaos created under the influence of bifurcation. The fractal dimensions of the proposed model are calculated. The maximum Lyapunov exponents and phase portraits are depicted to further confirm the complexity and chaotic behavior. Finally, numerical simulations are presented to confirm the theoretical and analytical findings.

A Study of (2+1)-Dimensional Konopelchenko-Dubrovsky (KD) System: Closed-Form Solutions, Solitary Waves, Bifurcation Analysis and Quasi-Periodic Solution

Nonlinear evolution equations (NLEEs) are extensively used to establish the elementary propositions of natural circumstances. In this work, we study the Konopelchenko-Dubrovsky (KD) equation which depicts non-linear waves in mathematical physics with weak dispersion. The considered model is investigated using the combination of generalized exponential rational function (GERF) method and dynamical system method. The GERF method is utilized to generate closedform invariant solutions to the (2+1)-dimensional KD model in terms of trigonometric, hyperbolic, and exponential forms with the assistance of symbolic computations. Moreover, three-dimensional graphics are displayed to depict the behavior of obtained solitary wave solutions. The model is observed to have single and multiple soliton profiles, kink-wave profiles, and periodic oscillating nonlinear waves. These generated solutions have never been published in the literature. All the newly generated soliton solutions are checked by putting them back into the associated system with the soft computation via Wolfram Mathematica. Moreover, the system is converted into a planer dynamical system using a certain transformation and the analysis of bifurcation is examined. Furthermore, the quasi-periodic solution is investigated numerically for the perturbed system by inserting definite periodic forces into the considered model. With regard to the parameter of the perturbed model, two-dimensional and three-dimensional phase portraits are plotted.

Concept of searching for anchor goods based on the study of commodity indices

The article is devoted to the search for anchor goods — goods whose prices are forming for the prices of groups of goods. The role of the desired anchor product is similar to the role of the anchor currency. The identification of such anchors will allow us to understand the pricing structure, which is presumably of a self-affinity nature. The authors believe that there are several options for why a particular product becomes an anchor: historical, cultural or economic. To identify the main requirements for the anchor product, the behavior and dynamics of price changes of goods included in the CRB index were analyzed. Attention to this index is due, firstly, to its complex but significant role in setting exchange rates, and secondly, its role in analyzing world markets. The CRB index is an indicator of trends in the global commodity market. It displays all the movements and changes in the structure of the world commodity market. The characteristic patterns in the behavior of the goods in question are revealed. Typical phase portraits of their relational series for CRB index products have been revealed. The goods included in the CRB index were regrouped according to the similarity of their behavior relative to each other. Four groups were obtained, within each of which phase portraits of relational time series based on a pair of goods have a similar appearance. Reference phase portraits are described. The basic rules of the anchor product search are formulated based on the study of goods included in the selected commodity index. The conceptual proposals for the search for anchor goods using commodity indexes are formulated Based on the proposed approach, some products were investigated and their ability to play the role of an anchor was evaluated.

Complexity of a Discrete-Time Predator-Prey Model Involving Prey Refuge Proportional to Predator

In this paper, a discrete-time predator-prey model involving prey refuge proportional to predator density is studied. It is assumed that the rate at which prey moves to the refuge is proportional to the predator density. The fixed points, their local stability, and the existence of Neimark-Sacker bifurcation are investigated. At last, the numerical simulations consisting of bifurcation diagrams, phase portraits, and time-series are given to support analytical findings. The occurrence of chaotic solutions are also presented by showing the Lyapunov exponent while some parameters are varied.

Dynamical systems approach in automobiles technological transition from environmental drivers

The effects of climate change have led us to identify the need to minimize emissions, particularly in the transport sector. However, the demand for cars has increased and, the only alternative is the technological transition towards zero-emission technologies such as electric cars. This document proposes a mathematical model based on system dynamics to evaluate with dynamical systems the technological transition of automobiles from environmental drives. The results show bifurcations of codimension 2 between 4 different types of phase portraits, each of which allowed to identify a different type of transition. It is concluded that the demand for each technology and its useful life will determine its permanence in the market, but that each of these gives rise to environmental impacts that will guide the preferences of consumers, with the possibility that there is a loss in the motivation of consumers generated by the emergence of some other type of alternative, perhaps such as active mobility.

Computational Analysis and Bifurcation of Regular and Chaotic Ca2+ Oscillations

This study investigated the stability and bifurcation of a nonlinear system model developed by Marhl et al. based on the total Ca2+ concentration among three different Ca2+ stores. In this study, qualitative theories of center manifold and bifurcation were used to analyze the stability of equilibria. The bifurcation parameter drove the system to undergo two supercritical bifurcations. It was hypothesized that the appearance and disappearance of Ca2+ oscillations are driven by them. At the same time, saddle-node bifurcation and torus bifurcation were also found in the process of exploring bifurcation. Finally, numerical simulation was carried out to determine the validity of the proposed approach by drawing bifurcation diagrams, time series, phase portraits, etc.

Dynamical Effects of Offset Terms on a Modified Chua’s Oscillator and Its Circuit Implementation

This paper addresses the effects of offset terms on the dynamics of a modified Chua’s oscillator. The mathematical model is derived using Kirchhoff’s laws. The model is analyzed with the help of the maximal Lyapunov exponent, bifurcation diagrams, phase portraits, and basins of attraction. The investigations show that the offset terms break the symmetry of the system, generating more complex nonlinear phenomena like coexisting asymmetric bifurcations, coexisting asymmetric attractors, asymmetric double-scroll chaotic attractors and asymmetric attraction basins. Also, a hidden attractor (period-1 limit cycle) is found when varying the initial conditions. More interestingly, this latter attractor coexists with all other self-excited ones. A microcontroller-based implementation of the circuit is carried out to verify the numerical investigations.