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Information patterns in a neuron model describe the possible modes in which information is processed and transmitted within neurons and neural networks. An improved Rulkov neuron with the aim of revealing its unexplored dynamics is introduced and investigated, with possible application to information coding carried out in this work. After introducing the neuron model, its stability around the single equilibrium point is examined, and it is discovered that the system is able to exhibit both stable and unstable dynamics. Using two-parameter charts, the system's global stability dynamics are obtained, and windows of the hidden and self-excited dynamics involving both chaotic and periodic states are clearly separated. For the validation of the result of the mathematical model, an electronic circuit was developed in Pspice simulation environment, and both results were in good accord. Finally, a network of 500 improved Rulkov neurons under the chain configuration is used to explore the phenomenon of the information patterns. From that investigation, it was found that the improved Rulkov neural lattice under modulational instability presents repetitive, regular stripes of bright and dark bands that are almost periodic and localized in space and time related to synchronization. These results could provide guidance in discerning information processing patterns in the nervous system. [ABSTRACT FROM AUTHOR]

The investigated parametrically coupled electromechanical structure is composed of a mechanical Duffing oscillator whose mass sits on a moving belt surface. The driving electrical network is a van der Pol oscillator whose aim is to actuate the attached DC motor to provide some rotatry unbalances and parametric coupling in the vibrating structure. The coupled oscillator is applied to energy harvesting and overcomes the limitation of low energy generation associated with a single oscillator of this kind. The system was solved analytically and validated by numerical methods. The global dynamics of the structure were investigated, and nonlinear phenomena such as Neimark–Sacker bifurcation, discontinuity-induced bifurcation, grazing–sliding, and bifurcation to multiple tori were identified. These nonlinear behaviors affect the harvested energy at bifurcation points, resulting in jumps from one energy level to another. In addition to harnessing the highest energy under hard parametric coupling, the coupling ensures that higher and more useful energy is harvested over a wider range of belt speeds. Finally, the qualitative validation of the numerical concept by experimental setup verifies the workings of the model. [ABSTRACT FROM AUTHOR]

This article addresses the bifurcation characteristics and vibration reduction of a 2 -DOF dynamical system simulating the nonlinear oscillation of an asymmetric rotor model subjected to simultaneous multiparametric and external excitations. To suppress the system's vibrations, two 1 / 2 -DOF active dampers are attached to the system in linear and cubic nonlinear forms via a magnetic coupling actuator. The closed-loop system model is derived as two differential equations with multi-control terms, including cubic, quantic, and septic, coupled nonlinearly to two first-order systems. Applying perturbation theory, the system model is solved, and the autonomous system describing the closed-loop slow-flow dynamics is obtained. Through numerical algorithms, the motion bifurcation is analyzed using various tools such as 2D and 3D bifurcation diagrams, two-parameter stability charts, basins of attraction, orbit plots, and time response profiles. The analytical investigations confirm that the uncontrolled model behaves like a hardening Duffing oscillator with multistability characteristics, displaying simultaneous mono-stable, bi-stable, tri-stable, or quadri-stable periodic oscillations depending on both the asymmetric nonlinearities and angular velocity. Subsequently, the influence of different control parameters is analyzed to determine the threshold between mono and multi-stability conditions. Finally, optimal control parameters are designed to eliminate multistability characteristics and achieve minimum and safe vibration levels. [ABSTRACT FROM AUTHOR]

This work examines the fractional generalized Korteweg-de-Vries-Zakharov-Kuznetsov equation (gKdV-ZKe) by utilizing three well-known analytical methods, the modified G ′ G 2 -expansion method, 1 G ′ -expansion method and the Kudryashov method. The gKdV-ZK equation is a nonlinear model describing the influence of magnetic field on weak ion-acoustic waves in plasma made up of cool and hot electrons. The kink, singular, anti-kink, periodic, and bright soliton solutions are observed. The effect of the fractional parameter on wave shapes have been analyzed by displaying various graphs for fractional-order values of β . In addition, we utilize the Hamiltonian property to observe the stability of the attained solution and Galilean transformation for sensitivity analysis. The suggested methods can also be utilized to evaluate the nonlinear models that are being developed in a variety of scientific and technological fields, such as plasma physics. Findings show the effectiveness simplicity, and generalizability of the chosen computational approach, even when applied to complex models. [ABSTRACT FROM AUTHOR]

Impacting mechanical systems with suitable parameter settings exhibit a large amplitude chaotic oscillation close to the grazing with the impacting surface. The cause behind this uncertainty is the square root singularity and the occurrence of dangerous border collision bifurcation. In the case of one-degree-of-freedom mechanical systems, it has already been shown that this phenomenon occurs under certain conditions. This paper proposes the same uncertainty of a two-degree freedom mechanical impacting system under specific requirements. This paper shows that the phenomena earlier reported in the case of one-degree-of-freedom mechanical systems (like narrow band chaos, finger-shaped attractor, etc.) also occur in the two-degrees-of-freedom mechanical impacting system. We have numerically predicted that the narrowband chaos ensues under specific parameter settings. We have also shown that narrowband chaos can be avoided under some parameter settings. At last, we demonstrate the numerical predictions experimentally by constructing an equivalent electronic circuit of the mechanical rig. [ABSTRACT FROM AUTHOR]

Purpose: In the work there are presented results of the synthesis and additional validation of previously developed mathematical models of two different mechanical oscillators with 1 degree of freedom and harmonic excitation: (i) with magnetically modified elasticity generating a double symmetrical minimum of potential; (ii) with linear mechanical springs and with a one-sided limiter of motion. Methods: In the first case, original mathematical models of non-linear magnetic springs were developed, allowing for effective and fast numerical simulations of the bifurcation dynamics of a real mechanical oscillator with Duffing type stiffness. In the second system, various models of impact were proposed and tested: continuous models based on the generalized Hunt–Crossley model and original discontinuous versions of this model based on the restitution coefficient and with a finite duration of the collision. In the frame of the present work, a system consisting of magnetic springs used in the first system and obstacles from the second oscillator was built and investigated. The system was built as a new configuration of a special universal stand used in the earlier studies mentioned here. Results and Conclusion: In the current study, the parameters of the models identified in previous studies on two different systems were used, the synthesis of which is the current work. A very good agreement was obtained between numerical simulations and experimental data, thus demonstrating the correctness and effectiveness of the adopted mathematical models. [ABSTRACT FROM AUTHOR]

This work aims to investigate the stability, bifurcation, and vibration control of a discontinuous dynamical model simulating the nonlinear oscillation of a horizontally suspended nonlinear rotor system. A novel Proportional-Integral-Resonance-Control (PIRC) algorithm is introduced to dampen the rotor's vibrations. An 8-pole electromagnetic bearing is employed as an active actuator through which the PIRC control signals are applied in the form of eight electrical currents. These currents, in turn, generate controllable attractive forces that counteract the rotor's vibrations. Based on the presented control structure, the mathematical model of the closed-loop system, including the magneto-electromechanical coupling and rotor-actuator rub-impact force, is derived as a 2-DOF discontinuous dynamical system connected to two first-order filters. A closed-form solution using the multiple scales analysis (up to the second-order approximation) of the system model was obtained. Additionally, the autonomous dynamical system governing the evolution of oscillation amplitudes and modified phases was derived when the rub-impact was neglected. Based on this derived autonomous system, the bifurcation characteristics, vibration control, and system stability have been explored. The analytical results obtained are utilized as a heuristic tool to report the conditions under which the system will experience rub and/or impact forces. Then, the entire discontinuous model was numerically analyzed using bifurcation diagrams, zero–one chaotic test, poincaré maps, orbital plots, and instantaneous radial and lateral oscillations at the rub and/or impact conditions reported analytically. The principal findings demonstrate that the introduced analytical investigation can be used to predict accurately when the rotor-actuator will be subject to a rub-impact force. Furthermore, it is proven analytically that the system's damping coefficients are proportional to the cartesian product of the controller's feedback and control gains. Moreover, the numerical analysis also illustrates that the presence of a rotor-actuator rub-impact can lead to period-1, period-2, period-3, period-4, or quasiperiodic oscillations, depending on the rotation speed and/or disc eccentricity. Finally, it is proven that optimizing the controller gains can prevent the rotor-actuator rub-impact effect regardless of the rotor speed and disc eccentricity. [ABSTRACT FROM AUTHOR]

This paper aims to analyze the coupled nonlinear fractional Drinfel'd-Sokolov-Wilson (FDSW) model with beta derivative. The nonlinear FDSW equation plays an important role in describing dispersive water wave structures in mathematical physics and engineering, which is used to describe nonlinear surface gravity waves propagating over horizontal sea bed. We have applied the travelling wave transformation that converts the FDSW model to nonlinear ordinary differential equations. After that, we applied the generalized rational exponential function method (GERFM). Diverse types of soliton solution structures in the form of singular bright, periodic, dark, bell-shaped and trigonometric functions are attained via the proposed method. By selecting a suitable parametric value, the 3D, 2D and contour plots for some solutions are also displayed to visualize their nature in a better way. The modulation instability for the model is also discussed. The results show that the presented method is simple and powerful to get a novel soliton solution for nonlinear PDEs. [ABSTRACT FROM AUTHOR]

Nonlinear chains of atoms (NCA) are complex systems with rich dynamics, that influence various scientific disciplines. The lie symmetry approach is considered to analyze the NCA. The Lie symmetry method is a powerful mathematical tool for analyzing and solving differential equations with symmetries, facilitating the reduction of complexity and obtaining solutions. After getting the entire vector field by using the Lie scheme, we find the optimal system of symmetries. We have converted assumed PDE into nonlinear ODE by using the optimal system. The new auxiliary scheme is used to find the Travelling wave solutions, while graphical behaviour visually represents relationships and patterns in data or mathematical models. The multiplier method enables the identification of conservation laws, and fundamental principles in physics that assert certain quantities remain constant over time. [ABSTRACT FROM AUTHOR]

In this work, with the aim of reducing the cost of the implementation of the traditional 2D FHN neuron circuit, a pair of diodes connected in an anti-parallel direction is used to replace the usual cubic nonlinearity (implemented with two multipliers). Based on the stability of the model, the generation of self-excited firing patterns is justified. Making use of the famous Helmholtz theorem, a Hamilton function is provided for the estimation of the energy released during each electrical activity of the model. From the investigation of the 1D evolution of the maxima of the membrane potential of the model, it was recorded that the considered model is able to experience a period of doubling bifurcation followed by a crisis that enables the increasing of the volume of the attractor. This contribution ends with the realization of a neural circuit without analog multipliers for the validation of the obtained results. [ABSTRACT FROM AUTHOR]

In this work, a memristive FitzHugh–Rinzel (mFHR) neuron prototype is introduced and investigated. The memristive device is exploited to simulate the impact of a magnetic radiation on the FitzHugh–Rinzel (FHR) neuron's behavior. Depending on the strength of the electromagnetic induction and the intensity of the external stimulus, it is found that the model experiences self-excited firing activity. The two-parameter charts of the largest Lyapunov exponent and the bifurcation diagram investigation revealed the model exhibited hysteretic dynamics, which induced the coexistence of bifurcation of sets of parameters not yet revealed in such a model. The energy necessary to provide each firing activity in the proposed model is also estimated based on the Helmholtz theorem. Finally, results of the information patterns with a chain of 100 mFHR neurons are obtained by numerical calculations using Runge–Kutta (fourth-order) calculation method, which approaches solutions of the resulting dynamic equations. The spatiotemporal patterns and time series plots for membrane potential revealed regular localized structures made of alternate bright and dark bands identified as spikes, which are sensitive to external stimulation current, electromagnetic induction coefficients and synaptic coupling strength. [ABSTRACT FROM AUTHOR]

This paper estimates a new ultimate bound set (UBS) for the chaotic system caused by the interaction of the whistler and ion-acoustic waves with the plasma oscillation. The intrinsic Hamilton energy is estimated by using the Helmholtz theorem, and this kind of energy function is the most suitable Lyapunov function to discern its dynamic stability. It is found that the Hamilton energy is relative to the firing states of the dynamical system. In a stable state, the energy is also a constant, while a chaotic state is resulting from an oscillation in the energy. We use the Lagrange coefficient method to solve an optimization problem analytically so that we can find an accurate UBS for the plasma chaotic system. Further, we present the competitive modes (CMs) for the plasma system to investigate its different dynamical behaviors for different parameters. Simulation results for CMs confirm the theories presented about the Hamilton energy function and UBS. [ABSTRACT FROM AUTHOR]

Interaction between parametric excitation and self-excited vibration has been subjected to numerous investigations in continuous systems. The ability of parametric excitation to quench self-excited vibrations in such systems has also been well documented. But such effects in discontinuous systems do not seem to have received comparable attention. In this article, we investigate the interaction between parametric excitation and self-excited vibration in a four degree of freedom discontinuous mechanical system. Unlike majority of studies in which oscillatory nature of stiffness accounts for parametric excitation, we consider a much more practical case in which parametric excitation is provided by a massless rotor of rectangular cross section with a cylinder-like mass concentrated at the center. The rotor arrangement is placed on a friction-induced self-excited support in the form of a frame placed on a belt moving with constant velocity. This frame is connected to a supplementary mass. A Stribeck friction model is considered for the mass in contact with the belt. The frictional force between the mass and the belt is oscillatory in nature because of the variation of normal force due to parametric excitation from the rotor. Our investigations reveal mutual synchronization of parametric excitation and self-excited vibration in the system for specific parameter values. The existence of a stable limit cycle with constant synchronized fundamental frequency, for a range of parametric excitation frequencies, is established numerically. Investigation based on frequency spectra and Lissajous curves reveals complex synchronization patterns owing to the presence of higher harmonics. The system is also shown to exhibit Neimark–Sacker bifurcations under the variation of belt velocity. Furthermore, variation in belt velocity and coupling stiffness is seen to cause a breakup of quasi-periodic torus with small-amplitude oscillations to form large amplitude chaotic orbits. This points toward the possibility of vibration suppression in the system by tuning the parameters for stabilizing the small-amplitude quasi-periodic response. An example of co-existence of different attractors in the system is also presented. [ABSTRACT FROM AUTHOR]

Free vibrations of the orthotropic micro/nanoplate with nonclassical shape are investigated. The considered model is based on the nonlocal elasticity theory. The developed method uses the Ritz method as well as R-function theory for the construction of the system of coordinate functions. The linear frequencies are obtained for a rectangular plate with two cutouts on opposite sides, while the boundary conditions are considered of several types, including simply supported and clamped edges. The small-scale effects for various sizes of cutouts are discussed. [ABSTRACT FROM AUTHOR]

This paper introduces and investigates the dynamics of an improved Wilson neuron model with a memristive autapse. The study of the stability of the equilibrium points of the proposed model revealed that it can move from a rest state to a firing state through a Hopf bifurcation. Therefore, the improved model experiences self-excited dynamics. The study of the dynamics of the proposed model revealed neuronal behaviors such as quiescent, bursting, spiking, and hysteretic dynamics characterized by the coexistence of two and three firing patterns for the same set of parameters. In addition, the dynamics of modulated information patterns via the membrane potential related to potential information coding in a network of 500 neurons are presented and discussed. Planar impulse wave solutions of the generic model as initial conditions result in a localized unstable wave pattern under autaptic coupling and magnetic flux time delay. High memristive autaptic coupling and magnetic flux time delay have antagonistic effects on the spatiotemporal patterns. Our results suggest that high autaptic coupling could be a possible route to chaotic and chimera-like behaviors in the network, while high magnetic flux time delay could be an efficient bifurcation parameter in controlling the chaotic and chimera-like behaviors. Finally, a microcontroller implementation of the introduced memristive Wilson neuron has been addressed based on the 32-bit STM32F407ZE development board. Results obtained from the experimental investigations of the model agreed with the numerical ones based on the physically captured phase trajectories and time-domain waveforms. [ABSTRACT FROM AUTHOR]

The geometrically nonlinear vibrations of simply supported double-layer graphene sheet systems under in-plane magnetic field are considered in the presented manuscript. The interaction between layers is taken into account due to van der Waals forces. The investigation is based on the nonlocal elasticity theory, Kirchhoff plate theory and von Kármán theory. The effect of the magnetic field is due to the Lorentz force based on Maxwell's equations. The governing equations are used in mixed form by introducing the stress Airy function. The analytical presentation of the nonlinear frequency ratio for in-phase vibration and anti-phase vibration modes is presented. It is shown that the nonlocal parameter in the compatibility equation can significantly change the vibration characteristics. [ABSTRACT FROM AUTHOR]

In this paper, planar forced oscillations of a particle connected to the support via two nonlinear springs linked in series and two viscous dampers are investigated. The constitutive relationships for elastic forces of both springs are postulated in the form of the third-order power law. The geometric nonlinearity caused by the transverse motion of the pendulum is approximated by three terms of the Taylor series, which limits the range of applicability of the obtained results to swings with maximum amplitudes of about 0.6 rad. The system has two degrees of freedom, but its motion is described by two differential equations and one algebraic equation which have been derived using the Lagrange equations of the second kind. The classical multiple scales method (MSM) in the time domain was employed. However, the MSM variant with three scales of the time variable has been modified by developing new and dedicated algorithms to adapt the technique to solving problems described by the differential and algebraic equations (DAEs). The paper investigates the cases of forced and damped oscillation in non-resonant conditions, three cases of external resonances, and the internal 1: 2 resonance in the system. Moreover, the analysis of the stationary periodic states with external resonances was carried out, and investigations into the system's stability were concluded in each case. Two methods of assessing asymptotic solutions have been proposed. The first is based on the determination of the error satisfying the equations of the mathematical model. The second one is a relative measure in the sense of the L 2 -norm, which compares the asymptotic solution with the numerical one determined using the NDSolve procedure of Mathematica software. These measures show that the applied MSM solves the system to a high degree of accuracy and exposes the key dynamical features of the system. It was observed that the system exhibits jump phenomena at some points in the resonance cases, with stable and unstable periodic orbits. This feature predicts chaotic vibration in the system and defines the regions for its applications. [ABSTRACT FROM AUTHOR]

This presented work investigate the bio-convections effects of the magnetized time dependent axisymmetric flow of Carreau-nanomaterial performances with multiple slip effects over a stretching sheet. The momentum, heat, concentration and density of motile micro-organism are renovated into the system of equation via using well known similarity revolution. Well known Mathematical computational techniques and software (i.e. bvp4c and MATLAB) are used to draw graphical and tabular results. Velocity profile equation f ′ (ζ) , energy equation θ (ζ) , volumetric nanoparticles ϕ (ζ) , density motile microorganism χ (ζ) .The Carreau viscosity model is use to reduce the viscosity of fluid when W e = 0 and n = 0 . Besides we moderate this into power law index with n = 0.5 and n = 1.5 partial slip condition of velocity is also instigated at the surface. Gravity dependent gyrotactic nanoparticles are utilized for well observing axisymmetric flow with convective boundary layer condition and comparatively better heat transfer rate result and applicable to maximum realistic approach. [ABSTRACT FROM AUTHOR]

The reliability of the usage of a splitter plate (passive control device) downstream of the obstacle, in suppressing the fluid forces on a circular obstacle of diameter D = 0.1 m is studied in this paper. The first parameter of the current study is the attachment of a splitter plate of various lengths (L i) with the obstacle, whereas the gap separation (G i) between the splitter plate and the obstacle, is used as a second parameter. The control elements of the first and second parameters are varied from 0.1 to 0.3 . For the attached splitter plates of lengths 0.2 and 0.3 , the oscillatory behavior of transient flow at R e = 100 is successfully controlled. For the gap separation, 0.1 and 0.2 similar results are obtained. However, it is observed that a splitter plate of too short length and a plate located at the inappropriate gap from the obstacle, are worthless. A computational strategy based on the finite element method is utilized due to the complicated representative equations. For a clear physical depiction of the problem, velocity and pressure plots have been provided. Drag and lift coefficients the hydrodynamic benchmark values are also evaluated in a graphical representation surrounding the obstacle's peripheral surface as well as the splitter plate. In a conclusion, a splitter plate can function to control fluid forces whether it is attached or detached, based on plate length and gap separation between obstacle and plate, respectively. [ABSTRACT FROM AUTHOR]

The communication describes a theoretical framework for tangent hyperbolic fluid of nano-biofilm due to an extending or shrinking sheet that comprises a stagnation point flow, chemical reaction with activation energy, and bioconvection of gyrotactic microorganisms. The varying transport features due to dynamic viscosity, thermal conductivity, nano-particle mass permeability and microbe organisms diffusivity are taken into account for the novelty of this work. The inspiration is developed to enhance heat transfer. A set of leading partial differential equations is formed along with appropriate boundary constraints. Using similarity transformations, the basic formulation is transitioned into non-linear differential equations. To produce observational data, the shooting technique and Runge-Kutta fourth order method are employed. The coding of numerical scheme is developed in Matlab script. The visual representation of the effects of diverse fluid transport properties and distinctive parameters on speed, temperature, concentration and motile density are evaluated. The velocity become faster when the parameters ω , λ , ϵ and V 0 are enhanced. Brownian motion, thermal conductivity, heat generation as well as thermophoresis factors all strengthen the temperature distribution, however the nano-particle concentration profile is enhanced as the nano-particle mass conductivity variable, activation energy as well as the thermophoresis variable are boosted. The microorganism density improves significantly when the microorganism diffusivity factor increases. The skin friction, Sherwood number, Nusselt number and motile density number decline against the incremented transport parameters. [ABSTRACT FROM AUTHOR]

The unsteady energy and mass transport of magnetohydrodynamics (MHD) second grade nanofluid via an exponentially extending surface with Dufour and Soret effects are investigated in this study. Variable thermal conductivity and mixed convection effects are used to investigate the heat transfer mechanism. There are also new characteristics such as slip flow, viscous dissipation, Brownian motion, nonlinear thermal radiation, and thermophoresis. In the problem formulation, the boundary-layer approximation is used. Using the suitable transformations, the energy, momentum, and concentration equations are generated into non-linear ordinary differential equations (ODEs). The solution to the resultant problems was calculated via the Homotopy analysis method (HAM). The effects of environmental parameters on velocity, temperature, and concentration profiles are graphically depicted. When comparing the current results to the previous literature, there was also a satisfactory level of agreement. In comparison to a flow based on constant characteristics, the flow with variable thermal conductivity is shown to be significantly different and realistic. The temperature of the fluid grew in direct proportion to the thermophoresis motion, buoyancy ratio, and Brownian motion parameters. According to the findings, the slippery porous surface may be employed efficiently in chemical and mechanical sectors that deal with a variety of very viscous flows. [ABSTRACT FROM AUTHOR]

The original article can be found online at https://doi.org/10.1007/s11071-024-10020-3.Correction to: Nonlinear Dynhttps://doi.org/10.1007/s11071-024-10020-3The article was originally published with many equation errors missed during proofing. The author and publisher regret the errors and the original article is now updated. The original article has been corrected.Publisher's NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.By Sasan Rahmanian and Jan AwrejcewiczReported by Author; Author [Extracted from the article]

In this contribution, an image encryption technique jointly based on the contribution of chaotic behavior resulting from an extreme multistable system and DNA coding is proposed. The chaotic system used is based on a memristive model with quadratic memductance. This memductance is used to build the nonlinear term of a Jerk system. The investigation of the obtained system reveals that it is dissipative and deals with a line of equilibria. Using nonlinear analysis tools, including the two-parameter Lyapunov exponent, the graph of the Lyapunov spectrum, the bifurcation diagram, and phase portraits, to study the considered model, the results show that the considered system is able to exhibit the coexistence of an infinite number of stable states. Besides, the results of this work have been supported by using theoretical, numerical, and experimental approaches. Furthermore, the generated chaotic sequence by the proposed memristive Jerk system is used in the confusion and diffusion steps combined with DNA coding, thus enhancing the efficiency of the cryptosystem. The experimental results and the analyses (statistic, differential, and noise and data loss attacks) sufficiently show that the proposed cryptosystem is robust by the quality of the obtained results. [ABSTRACT FROM AUTHOR]

Free vibrations of shallow sandwich shells resting on elastic foundations are investigated. It is assumed that the shell consists of three layers of defined thickness. The core is made of ceramics or metal, while the upper and lower layers are made of functionally graded material (FGM). The volume fractions of metal and ceramics are described by the power law. Hence, estimation methods for higher accuracy remain a challenge. The elastic foundation is described by two-parameter Pasternak's model. Both higher-order shear deformation shell theory (HSDT), which includes interactions with elastic foundation, and the R-functions theory combined with the variational Ritz method are used to study shells with arbitrary planforms. The numerical study is carried out in the framework of the refined third-order theory. The proposed method and developed numerical techniques have been validated on test problems for FG shell with rectangular planform. Furthermore, new results for shallow shells with a cut-out of the complex form are obtained. The effects of the gradient index, boundary conditions, thickness of core and face sheet layers, as well as elastic foundations on fundamental frequencies, are investigated. In addition, we demonstrate how the divergence of the results varies with the gradient index. The considered studied cases show that the natural frequencies depend on the foundation parameters, the thickness of the layers, boundary conditions, and the thickness arrangement. [ABSTRACT FROM AUTHOR]

In the present exploration, some new wave profiles are constructed via Lie symmetry of transformations scheme by the one-dimensional optimal system in a pre-compressed one-dimensional granular crystal. Lie group scheme is used to construct the infinitesimal generators, entire vector field, commutation relations, and adjoint representations. Infinitesimal generators are used to construct one optimal system of the one-dimensional sub-algebras. Employing the optimal system, similarity reductions are obtained for the considered nonlinear model. Translational vectors and their linear combination are used to covert the assumed model in a nonlinear ordinary differential equation. These methods give us some new and interesting solutions which contain the trigonometric, hyperbolic trigonometric, and exponential type functions. To represent the physical significance of the assumed model, some 3D and 2D diagrams of acquired results are plotted by using Mathematica under the suitable choice of involving parameters values. The multiplier scheme is employed to describe the conserved vectors of the assumed problem. [ABSTRACT FROM AUTHOR]

Countries affected by the coronavirus epidemic have reported many infected cases and deaths based on world health statistics. The crowding factor, which we named "crowding effects," plays a significant role in spreading the diseases. However, the introduction of vaccines marks a turning point in the rate of spread of coronavirus infections. Modeling both effects is vastly essential as it directly impacts the overall population of the studied region. To determine the peak of the infection curve by considering the third strain, we develop a mathematical model (susceptible–infected–vaccinated–recovered) with reported cases from August 01, 2021, till August 29, 2021. The nonlinear incidence rate with the inclusion of both effects is the best approach to analyze the dynamics. The model's positivity, boundedness, existence, uniqueness, and stability (local and global) are addressed with the help of a reproduction number. In addition, the strength number and second derivative Lyapunov analysis are examined, and the model was found to be asymptotically stable. The suggested parameters efficiently control the active cases of the third strain in Pakistan. It was shown that a systematic vaccination program regulates the infection rate. However, the crowding effect reduces the impact of vaccination. The present results show that the model can be applied to other countries' data to predict the infection rate. [ABSTRACT FROM AUTHOR]

Brain functions are sometimes emulated using some analog integrated circuits based on the organizational principle of natural neural networks. Neuromorphic engineering is the research branch devoted to the study and realization of such circuits with striking features. In this contribution, a novel small network of three neurons is introduced and investigated. The model is built from the coupling between two 2D Hindmarsh–Rose neurons through a 2D FitzHugh–Nagumo neuron. Thus, a heterogeneous coupled network is obtained. The biophysical energy released by the network during each electrical activity is evaluated. In addition, nonlinear analysis tools such as two-parameter Lyapunov exponent, bifurcation diagrams, the graph of the largest Lyapunov exponent, phase portraits, time series, as well as the basin of attractions are used to numerically investigate the network. It is found that the model can experience hysteresis justified by the simultaneous existence of three distinct electrical activities using the same set of parameters. Finally, the circuit implementation of the network is addressed in PSPICE to further support the obtained results. [ABSTRACT FROM AUTHOR]

In this work, the heat transfer features and stagnation point flow of Magnetohydrodynamics (MHD) hybrid second-grade nanofluid through a convectively heated permeable shrinking/stretching sheet is reported. The purpose of the present investigation is to consider hybrid nanofluids comprising of Alumina Al 2 O 3 and Copper Cu nanoparticles within the Sodium Alginate (SA) as a host fluid for boosting the heat transfer rate. Also, the effects of free convection, viscous dissipation, heat source/sink, and nonlinear thermal radiation are considered. The converted nonlinear coupled fuzzy differential equations (FDEs) with the help of triangular fuzzy numbers (TFNs) are solved using the numerical scheme bvp4c. The numerical results are acquired for various engineering parameters to study the Nusselt number, skin friction coefficient, velocity, and temperature distribution through figures and tables. For the validation, the current numerical results were found to be good as compared to existing results in limiting cases. It is also inspected by this work that with the enhancement of the volume fraction of nanoparticles, the heat transfer rate also increases. So, it may be taken as a fuzzy parameter for a better understanding of fuzzy variables. For the comparison, the volume fraction of nanofluids and hybrid nanofluid are said to be TFN [0, 0.1, 0.2]. In the end, we can see that fuzzy triangular membership functions (MFs) have not only helped to overcome the computational cost but also given better accuracy than the existent results. Finding from fuzzy MFs, the performance of hybrid nanofluids is better than nanofluids. [ABSTRACT FROM AUTHOR]

In this paper a new approach is taken to find the exact solutions for generalized unsteady magnetohydrodynamic transport of a rate-type fluid near an unbounded upright plate and is analyzed for ramped wall temperature and velocity with constant concentration. The vertical plate is suspended in a porous medium and encounters radiation effects. Solutions based on special functions are obtained using an integral transform for an unsteady MHD Maxwell fluid in the presence of ramped velocity, temperature and constant concentration. The relations for Nusselt number and skin-friction coefficient are efficiently computed to precisely estimate the rate of heat transfer at the boundary and the shear stress. Results are also discussed in detail and demonstrated graphically using software to comprehensively analyze the dynamics of the proposed problem, and the physical impact of several system parameters, such as magnetic field M, Prandtl number Pr, the relaxation time λ, dimensionless time τ, Schmidt number Sc, Mass and Thermal Grashof numbers Gm and Gr, respectively, is studied. Furthermore, solutions for some recently published work are compared with the current study that endorses the authenticity of our derived results and proves that those investigations are limiting or special cases of the current problem. [ABSTRACT FROM AUTHOR]

A mechanical system composed of two weakly coupled oscillators under harmonic excitation is considered. Its main part is a vibro-impact unit composed of a linear oscillator with an internally colliding small block. This block is coupled with the secondary part being a damped linear oscillator. The mathematical model of the system has been presented in a non-dimensional form. The analytical studies are restricted to the case of a periodic steady-state motion with two symmetric impacts per cycle near 1:1 resonance. The multiple scales method combined with the sawtooth-function-based modelling of the non-smooth dynamics is employed. A conception of the stability analysis of the periodic motions suited for this theoretical approach is presented. The frequency–response curves and force–response curves with stable and unstable branches are determined, and the interplay between various model parameters is investigated. The theoretical predictions related to the motion amplitude and the range of stability of the periodic steady-state response are verified via a series of numerical experiments and computation of Lyapunov exponents. Finally, the limitations and extensibility of the approach are discussed. [ABSTRACT FROM AUTHOR]

Parametric vibrations of the single-layered graphene sheet (SLGS) are studied in the presented work. The equations of motion govern geometrically nonlinear oscillations. The appearance of small effects is analysed due to the application of the nonlocal elasticity theory. The approach is developed for rectangular simply supported small-scale plate and it employs the Bubnov–Galerkin method with a double mode model, which reduces the problem to investigation of the system of the second-order ordinary differential equations (ODEs). The dynamic behaviour of the micro/nanoplate with varying excitation parameter is analysed to determine the chaotic regimes. As well the influence of small-scale effects to change the nature of vibrations is studied. The bifurcation diagrams, phase plots, Poincaré sections and the largest Lyapunov exponent are constructed and analysed. It is established that the use of nonlocal equations in the dynamic analysis of graphene sheets leads to a significant alteration in the character of oscillations, including the appearance of chaotic attractors. [ABSTRACT FROM AUTHOR]

In this paper vibrations of the isotropic micro/nanoplates subjected to transverse and in-plane excitation are investigated. The governing equations of the problem are based on the von Kármán plate theory and Kirchhoff–Love hypothesis. The small-size effect is taken into account due to the nonlocal elasticity theory. The formulation of the problem is mixed and employs the Airy stress function. The two-mode approximation of the deflection and application of the Bubnov–Galerkin method reduces the governing system of equations to the system of ordinary differential equations. Varying the load parameters and the nonlocal parameter, the bifurcation analysis is performed. The bifurcations diagrams, the maximum Lyapunov exponents, phase portraits as well as Poincare maps are constructed based on the numerical simulations. It is shown that for some excitation conditions the chaotic motion may occur in the system. Also, the small-scale effects on the character of vibrating regimes are illustrated and discussed. [ABSTRACT FROM AUTHOR]

The paper deals with the dynamics of a lumped mass mechanical system containing two nonlinear springs connected in series. The external harmonic excitation, linear and nonlinear damping are included into considerations. The mathematical model contains both differential and algebraic equations, so it belongs to the class of dynamical systems governed by the differential–algebraic system of equations (DAEs). An approximate analytical approach is used to solve the initial value problem for the DAEs. We adopt the multiple scales method (MSM) that allows one to obtain the sufficiently correct approximate solutions both far from the resonance and at the resonance conditions. The steady and non-steady resonant vibrations are analyzed by employing the modulation equations of the amplitudes and phases which are yielded by the MSM procedure. [ABSTRACT FROM AUTHOR]

The nonlinearities of geometric nature that is characteristic for pendulum-type systems are expressed by the trigonometric functions. In order to apply the method of multiple scales in time domain to solve problems concerning such systems, the trigonometric functions of the generalised coordinates are usually approximated by a few terms of their Taylor series. In the paper we apply the polynomial approximation in quadratic means. In contrast to the approximation by Taylor series, the proposed manner approximates the trigonometric functions not around a given point but on the given interval. Quality and accuracy of the solutions obtained using the multiple scales method based on such approach have been tested. The steady state responses in the main resonance have been also examined and compared with their counterparts obtained using the method of multiple scales based on the Taylor series. [ABSTRACT FROM AUTHOR]

In this article, the authors simulate and study dark and bright soliton solutions of 1D and 2D regularized long wave (RLW) models. The RLW model occurred in various fields such as shallow-water waves, plasma drift waves, longitudinal dispersive waves in elastic rods, rotating flow down a tube, and the anharmonic lattice and pressure waves in liquid–gas bubble mixtures. First of all, the tanh–coth method is applied to obtain the soliton solutions of RLW equations, and thereafter, the approximation of finite domain interval is done by truncating the infinite domain interval. For computational modeling of the problems, a meshfree method based on local radial basis functions and differential quadrature technique is developed. The meshfree method converts the RLW model into a system of nonlinear ordinary differential equations (ODEs), then the obtained system of ODEs is simulated by the Runge–Kutta method. Further, the stability of the proposed method is discussed by the matrix technique. Finally, in numerical experiments, some problems are considered to check the competence and chastity of the developed method. [ABSTRACT FROM AUTHOR]

Our aim in this article is to solve the composite fractional relaxation differential equation by using different definitions of the non-integer order derivative operator D t α , more specifically we employ the definitions of Caputo, Caputo–Fabrizio and Atangana–Baleanu of non-integer order derivative operators. We apply the Laplace transform method to solve the problem and express our solutions in terms of Lorenzo and Hartley's generalised G function. Furthermore, the effects of the parameters involved in the model are graphically highlighted. [ABSTRACT FROM AUTHOR]

The effective properties of the fiber-reinforced composite materials with fibers of circular cross section are investigated. The novel estimation for the effective coefficient of thermal conductivity refining the classical Maxwell formula is derived. The method of asymptotic homogenization is used. For analytical solution of the periodically repeated cell problem, the Schwarz alternating process is employed. The principal term of the refined formula coincides with the classical Maxwell formula. On the other hand, the refined formula can be used far beyond the area of applicability of the Maxwell formula. It can be used for dilute and non-dilute composites. It is confirmed by comparison with known numerical and asymptotic results. [ABSTRACT FROM AUTHOR]

The effective properties of the fiber-reinforced composite materials with fibers of circle cross section are investigated. The novel estimation for the effective coefficient of thermal conductivity refining the classical Maxwell formula is derived. The method of asymptotic homogenization is used. For an analytical solution of the periodically repeated cell problem the Schwarz alternating process (SAP) was employed. Convergence of this method was proved by S. Mikhlin, S. Sobolev, V. Mityushev. Unfortunately, the rate of the convergence is often slow, especially for nondilute high-contrast composite materials. For improving this drawback we used Padé approximations for various forms of SAP solutions with the following additive matching of obtained expressions. As a result, the solutions in our paper are obtained in a fairly simple and convenient form. They can be used even for a volume fraction of inclusion very near the physically possible maximum value as well as for high-contrast composite constituents. The results are confirmed by comparison with known numerical and asymptotic results. [ABSTRACT FROM AUTHOR]

The nonlinear dynamics of the six-pole rotor active magnetic bearings system is studied in this article for the first time. Two control configurations based on a proportional-derivative feedback current controller are proposed to mitigate lateral vibrations of the considered system. The first configuration is designed in such a way that only four electromagnetic poles are responsible for the system vibration control in the Y -direction, while all six poles control the lateral vibration in the X -direction. A second configuration is proposed in which the same four poles of the first configuration control the system vibration in the Y -direction, while the other two poles are responsible for controlling the system vibration in the X -direction. According to the suggested control methods, a mathematical model is derived that simulates lateral oscillations of the system. Using the perturbation analysis, four autonomous and coupled first-order nonlinear differential equations that govern the system oscillation amplitudes and the corresponding phase angles in both X and Y -directions are extracted. Various bifurcation diagrams are obtained using rotor spinning-speed and disk eccentricity as a bifurcation control parameter. The conditions at which the system can whirl either forward or backward are investigated. Numerical validations for the obtained bifurcation diagrams are introduced which illustrate excellent agreement with the analytical results. Based on the acquired analytical and numerical results, it is found that the first control configuration has the best dynamical behavior for controlling the vibration of such systems. [ABSTRACT FROM AUTHOR]

Biological models are usually described using difference equations. As a result, we are - in this work - interested in studying a general difference model which includes two biological models as special cases. In detail, we study the qualitative behaviors (local and global stability, boundedness and periodicity character) of a general difference model. Furthermore, we apply our general results to the population model with two age classes and the flour beetle model. [ABSTRACT FROM AUTHOR]

The paper presents a new method for the identification of joint friction, reductor self-locking and gear backlash parameters. The method was developed using an existing hexapod robot. During the process, the motor driving currents given by the model and measured on the real device were observed and compared. Initially, the current curves showed specific differences. After the analysis of these deviations, their causes were identified as joint friction, reductor self-locking and gear backlash. The parameters of these phenomena were determined using swarm optimization. As a result of this process, a validated model was obtained. The mentioned mechanical properties were identified using a step-by-step method, in which one property aided in the discovery of another. The robot model was created in a MATLAB/Simulink environment, using the Simscape Multibody Toolbox. [ABSTRACT FROM AUTHOR]

We analyze the dynamics of a nonlinear mechanical system under the influence of an external harmonic force. The system consists of a linear oscillator (primary mass) and attached nonlinear dynamic absorber. It is supposed that the frequency of the external force is close to the natural frequency of the main mass. Assuming that the parameters of the system are uncertain, the stability conditions of the stationary regimes of the averaged equations are obtained analytically; these regimes correspond to the quasi-periodic motions of the original input system. An analytical approach to the problem of selecting the parameters of a dynamic absorber is proposed in order to reduce the amplitude of oscillations of the main system. The results obtained are compared with the results of the numerical integration of the equations of the motion with different initial conditions and parameter values. [ABSTRACT FROM AUTHOR]

In this article, the authors proposed a meshfree approach for simulation of non-linear Schrödinger equation with constant and variable coefficients. Schrödinger equation is a classical field equation whose principal applications are to the propagation of light in non-linear optical fibers and planar waveguides and in quantum mechanics. First of all, spatial derivatives are discretized by using local radial basis functions based on differential quadrature method (LRBF-DQM) and, subsequently, the obtained system of non-linear ordinary differential equations (ODEs) is solved by fourth-order Runge–Kutta (RK-4). The stability analysis of the proposed approach is discussed by the matrix method. Numerical experiments ensure that the proposed approach is accurate and computationally efficient. [ABSTRACT FROM AUTHOR]

The pendulum is excited horizontally by a system of a DC motor and a slider–crank mechanism. Mathematical modeling is realistic and based on experimental rig, taking into account details concerning friction in the joints as well as realistic mass distribution in particular elements of the system. Using basic nonlinear tools as phase portraits, Poincaré maps, and Fourier spectra, we report various solutions including periodic, quasi-periodic and non-periodic ones. To identify chaotic solutions, we used the 0–1 test. The simulation results were qualitatively confirmed by experiments. [ABSTRACT FROM AUTHOR]

Three-DOF system with a limited power supply (or non-ideal system) having the Mises girder as absorber is considered. Stationary resonance regimes of vibrations near stable equilibrium positions of the system are constructed in two approximations of the multiple scales method. Namely, vibrations near the resonance 1:1 between the motor rotation frequency and the linear sub-system frequency and vibrations near the resonance 1:1 between the motor and absorber frequencies are analyzed. This analysis near the first resonance is made as in supposition that elastic springs of the Mises girder are weak and have an order of the small parameter, as well as in supposition that these springs are not weak. It is obtained that the effective absorption of the elastic vibrations can be obtained in the second case. Additional numerical simulation permits to find regimes of absorption when it is possible to guarantee also the fast outcome from the first resonance region. The most appropriate for absorption and such outcome from the first resonance are vibrations near the second resonance and the snap-through motions. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of harmonically forced impacting oscillator. The physical model for oscillator is a cart on a guide connected to the support with springs and excited by the stepper motor. The support also is provided with limiter of motion. The mathematical model for this system is defined with the second-order piecewise smooth differential equation. Model's nonlinearity is connected with the incorporation of dry friction and generalized Hertz contact law. Analyzing the classical Poincare sections and inter-impact sequences obtained experimentally and numerically, the bifurcations and statistical properties of periodic, multi-periodic, and chaotic regimes were examined. The development of impact-adding regime as a new nonlinear phenomenon when the forcing frequency varies was observed. [ABSTRACT FROM AUTHOR]

Developing new designs and optimization of the cancer treatment is extremely important task. In this work, the nonlinear multi-scale diffusion cancer invasion model that describes the interactions of the tumor cells, matrix-metalloproteinases, matrix-degradative enzymes and oxygen is studied. The conditions under which the cancerous biological system exhibits chaotic behavior were obtained by means of the method based on wandering trajectories analysis. Regions of parameters leading to carcinogenesis in the biological system studied were found in control parameter planes 'number of tumor cells versus diffusion saturation level.' Significant influence of the biological system initial state to carcinogenesis was ascertained and illustrated by regions in phase planes of initial conditions. Evolution of all regions obtained is presented depending on glucose level. [ABSTRACT FROM AUTHOR]

Dynamics behavior of the micromechanical gyroscope designed for measuring one component of the angular velocity is studied in the paper. The Cardan suspension is applied to connect the sensing plate with the substrate whose angular velocity is measured. The gimbal and the plate with sensors are connected via torsional joints. Vibrating motion of the sensing plate is excited mainly by a torque resulting from the Coriolis effect. The mathematical model equations have been derived using the Lagrange equation of the second kind. Both nonlinear effects of the geometrical nature and the nonlinear characteristics of the torsional joints are taken into account. The governing equations are solved with help of the method of multiple scales in time domain that belongs to the broad class of asymptotic methods. The approximate solution of analytical form has been obtained for non-resonant vibration as well as for the case of the main and internal resonances that occur simultaneously. Analytical form of solution allows for extensive analysis of the behavior of the system. The desirable state of the gyroscope work is steady-state vibration in resonance that is discussed in detail. [ABSTRACT FROM AUTHOR]

In the work there is presented a mathematical model of a rattleback lying on a horizontal plane, with special attention paid to modelling of the contact forces. Friction force modelling concerns a certain class of problems, where it is possible to assume fully developed sliding on the contact zone. The models are based on the integral expressions assuming the classical Coulomb friction law valid at each point of the contact. In order to obtain a higher simulation speed, a special class of approximations if integral functions is used, being some kind of generalization of Padé expansion. The rolling resistance is modelled as a result of special distortion of contact pressure distribution and takes into account the rolling direction and the elliptic shape of the contact. The work uses previous results of the authors, but more attention is paid to the rolling resistance model, which is extended and analysed in more detail. The models are validated experimentally and their various elements are tested. [ABSTRACT FROM AUTHOR]