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The purpose of this work is to investigate the effect of stochastic perturbations of the white noise type on the stability of capture into autoresonance in oscillating systems with a variable pumping amplitude and frequency such that a center–saddle bifurcation occurs in the corresponding limiting autonomous system. The another purpose is determine the dependence of the intervals of stochastic stability of the autoresonance on the noise intensity. Methods. The existence of autoresonant regimes with increasing amplitude is proved by constructing and justificating asymptotic solutions in the form of power series with constant coefficients. The stability of solutions in terms of probability with respect to noise is substantiated using stochastic Lyapunov functions. Results. The conditions are described under which the autoresonant regime is preserved and disappears when the parameters pass through bifurcation values. The dependence of the intervals of stochastic stability of autoresonance on the degree of damping of the noise intensity is found. It is shown that more stringent restrictions are required to preserve the stability of solutions for the bifurcation values of the parameters. Conclusion. At the level of differential equations describing capture into autoresonance, the effect of damped stochastic perturbations on the center–saddle bifurcation is studied. The results obtained indicate the possibility of using damped oscillating perturbations for stable control of nonlinear systems.

This article investigates the dynamical behavior analysis and soliton solutions of the generalized Whitham–Broer–Kaup–Boussineq–Kupershmidt equations in fluid flow dynamics and plasma waves. Firstly, the generalized Whitham–Broer–Kaup–Boussineq–Kupershmidt equations are simplified into the ordinary differential equations. Secondly, two-dimensional planar dynamical system and its phase portraits are given by utilizing the theory of planar dynamical system analysis. Finally, the soliton solutions of the generalized Whitham–Broer–Kaup–Boussineq–Kupershmidt equations are presented by using the fourth-order complete discriminant system. In order to better reveal the soliton propagation phenomenon of the generalized Whitham–Broer–Kaup–Boussineq–Kupershmidt equations, some three-dimensional, two-dimensional, and contour graphs of the solutions were drawn.

The main purpose of this article is to study the qualitative analysis and explicit solutions of perturbed Chen–Lee–Liu equation with refractive index. By utilizing wave transformation, two-dimensional dynamics in a plane are obtained. Using the qualitative theory of planar dynamical systems, a series of planar phase portraits are presented. By analyzing the bifurcations of these phase portraits and the characteristics of equilibrium points, some explicit solutions such as Jacobian function solutions and hyperbolic function solutions are constructed. These can further understand the dynamic behavior of perturbed Chen–Lee–Liu equation and their wave propagation.

Measles is a highly communicable viral infection that mostly affects children aged 5 years and below. Maternal antibodies in neonates help protect them from infectious diseases, including measles. However, maternal antibodies disappear a few months after birth, necessitating vaccination against measles. A mathematical model of measles, incorporating maternal antibodies and a double-dose vaccination, was proposed. Whenever ℛ0

Complex systems, as interwoven miscellaneous interacting entities that emerge and evolve through self-organization in a myriad of spiraling contexts, exhibit subtleties on global scale besides steering the way to understand complexity which has been under evolutionary processes with unfolding cumulative nature wherein order is viewed as the unifying framework. Indicating the striking feature of non-separability in components, a complex system cannot be understood in terms of the individual isolated constituents' properties per se, it can rather be comprehended as a way to multilevel approach systems behavior with systems whose emergent behavior and pattern transcend the characteristics of ubiquitous units composing the system itself. This observation specifies a change of scientific paradigm, presenting that a reductionist perspective does not by any means imply a constructionist view; and in that vein, complex systems science, associated with multiscale problems, is regarded as ascendancy of emergence over reductionism and level of mechanistic insight evolving into complex system. While evolvability being related to the species and humans owing their existence to their ancestors' capability with regards to adapting, emerging and evolving besides the relation between complexity of models, designs, visualization and optimality, a horizon that can take into account the subtleties making their own means of solutions applicable is to be entailed by complexity. Such views attach their germane importance to the future science of complexity which may probably be best regarded as a minimal history congruent with observable variations, namely the most parallelizable or symmetric process which can turn random inputs into regular outputs. Interestingly enough, chaos and nonlinear systems come into this picture as cousins of complexity which with tons of its components are involved in a hectic interaction with one another in a nonlinear fashion amongst the other related systems and fields. Relation, in mathematics, is a way of connecting two or more things, which is to say numbers, sets or other mathematical objects, and it is a relation that describes the way the things are interrelated to facilitate making sense of complex mathematical systems. Accordingly, mathematical modeling and scientific computing are proven principal tools toward the solution of problems arising in complex systems' exploration with sound, stimulating and innovative aspects attributed to data science as a tailored-made discipline to enable making sense out of voluminous (-big) data. Regarding the computation of the complexity of any mathematical model, conducting the analyses over the run time is related to the sort of data determined and employed along with the methods. This enables the possibility of examining the data applied in the study, which is dependent on the capacity of the computer at work. Besides these, varying capacities of the computers have impact on the results; nevertheless, the application of the method on the code step by step must be taken into consideration. In this sense, the definition of complexity evaluated over different data lends a broader applicability range with more realism and convenience since the process is dependent on concrete mathematical foundations. All of these indicate that the methods need to be investigated based on their mathematical foundation together with the methods. In that way, it can become foreseeable what level of complexity will emerge for any data desired to be employed. With relation to fractals, fractal theory and analysis are geared toward assessing the fractal characteristics of data, several methods being at stake to assign fractal dimensions to the datasets, and within that perspective, fractal analysis provides expansion of knowledge regarding the functions and structures of complex systems while acting as a potential means to evaluate the novel areas of research and to capture the roughness of objects, their nonlinearity, randomness, and so on. The idea of fractional-order integration and differentiation as well as the inverse relationship between them lends fractional calculus applications in various fields spanning across science, medicine and engineering, amongst the others. The approach of fractional calculus, within mathematics-informed frameworks employed to enable reliable comprehension into complex processes which encompass an array of temporal and spatial scales notably provides the novel applicable models through fractional-order calculus to optimization methods. Computational science and modeling, notwithstanding, are oriented toward the simulation and investigation of complex systems through the use of computers by making use of domains ranging from mathematics to physics as well as computer science. A computational model consisting of numerous variables that characterize the system under consideration allows the performing of many simulated experiments via computerized means. Furthermore, Artificial Intelligence (AI) techniques whether combined or not with fractal, fractional analysis as well as mathematical models have enabled various applications including the prediction of mechanisms ranging extensively from living organisms to other interactions across incredible spectra besides providing solutions to real-world complex problems both on local and global scale. While enabling model accuracy maximization, AI can also ensure the minimization of functions such as computational burden. Relatedly, level of complexity, often employed in computer science for decision-making and problem-solving processes, aims to evaluate the difficulty of algorithms, and by so doing, it helps to determine the number of required resources and time for task completion. Computational (-algorithmic) complexity, referring to the measure of the amount of computing resources (memory and storage) which a specific algorithm consumes when it is run, essentially signifies the complexity of an algorithm, yielding an approximate sense of the volume of computing resources and seeking to prove the input data with different values and sizes. Computational complexity, with search algorithms and solution landscapes, eventually points toward reductions vis à vis universality to explore varying degrees of problems with different ranges of predictability. Taken together, this line of sophisticated and computer-assisted proof approach can fulfill the requirements of accuracy, interpretability, predictability and reliance on mathematical sciences with the assistance of AI and machine learning being at the plinth of and at the intersection with different domains among many other related points in line with the concurrent technical analyses, computing processes, computational foundations and mathematical modeling. Consequently, as distinctive from the other ones, our special issue series provides a novel direction for stimulating, refreshing and innovative interdisciplinary, multidisciplinary and transdisciplinary understanding and research in model-based, data-driven modes to be able to obtain feasible accurate solutions, designed simulations, optimization processes, among many more. Hence, we address the theoretical reflections on how all these processes are modeled, merging all together the advanced methods, mathematical analyses, computational technologies, quantum means elaborating and exhibiting the implications of applicable approaches in real-world systems and other related domains. [ABSTRACT FROM AUTHOR]

This research is focused into the exploration of the inherent nonlinear dynamics related to the negative-order (3+1)-dimensional KdV–Calogero–Bogoyavlenskii–Schiff equation. The simplified Hirota method is employed to construct and visualize the multiple solitons through phase portraits. The underlying methodology is advantageous as the model is not needed to convert in bilinear form. The symmetry analysis and the corresponding invariant solutions for the analyzed model are calculated using Lie groups. The underlying model is converted into second-order ordinary differential equation, which is further converted into the dynamical system of first-order differential equations through the utilization of Galilean transformation. Additionally, the qualitative examination of the acquired dynamical system is executed by employing the concept of bifurcation analysis. Following this, an external force is introduced to the model in order to generate a disruption, consequently leading to the formation of a perturbed dynamical system. The confirmation of chaotic behavior in the perturbed dynamical system is observed by employing a range of tools designed to detect chaos. The findings of this piece of research are highly captivating and provide a significant contribution to the field of soliton dynamics in particular and mathematical physics in general.

This manuscript is a captivating exploration of the (3+1)-D nonlinear extended Quantum Zakharov–Kuznetsov (NLEQZK) equation, revealing its complex dynamics and solitary wave solutions. We unveil the general method behind this equation and transform it into an ordinary differential equation (ODE), then venture further by using the Galilean transformation to create a system of ODEs. Further, we journey through bifurcations, chaos, and other fascinating dynamical properties, culminating in the visualization and analysis of solitary wave solutions. From the elegant W-shaped solitons to singular solitons with unconventional characteristics and hybrid singular and periodic solitons, each is discussed in vivid detail. This work presents a significant advancement in understanding the unpredictable and intricate behavior of the model, inviting readers to explore the captivating world of non-linear waves and dynamical systems.

This study focuses on the variant Boussinesq equation, which is used to model waves in shallow water and electrical signals in telegraph lines based on tunnel diodes. The aim of this study is to find closed-form wave solutions using the extended direct algebraic method. By employing this method, a range of wave solutions with distinct shapes, including shock, mixed-complex solitary-shock, singular,mixed-singular, mixed trigonometric, periodic, mixed-shock singular, mixed-periodic, and mixed-hyperbolic solutions, are attained. To illustrate the propagation of selected exact solutions, graphical representations in 2D, contour, and 3D are provided with various parametric values. The equation is transformed into a planar dynamical structure through the Galilean transformation. By utilizing bifurcation theory, the potential phase portraits of nonlinear and super-nonlinear traveling wave solutions are investigated. The Hamiltonian function of the dynamical system of differential equations is established, revealing the system’s conservative nature over time. The graphical representation of energy levels offers valuable insights and demonstrates that the model has closed-form solutions.

In this paper, the (1+1)-dimensional Biswas-Milovic equation with parabolic law and nonlocal nonlinearity is studied. Firstly, the Biswas-Milovic equation with parabolic law and nonlocal nonlinearity is transformed into ordinary differential equation through traveling wave transformation. Secondly, two-dimensional planar dynamic system is given by using the trial method of polynomial for rank homogeneous equations together with the principle of homogeneous balance. Moreover, the phase portrait of the dynamical system is given. Meanwhile, the phase portrait, Poincaré section and sensitivity analysis of its perturbation system are drawn by mathematical software. Finally, the optical soliton solutions of the Biswas-Milovic equation with parabolic law and nonlocal nonlinearity are constructed by using the complete discrimination system method.

This study delves deep into the complexities of the modified nonlinear Schrödinger equation. Through the Galilean transformation, we derive a dynamic system linked to the equation. Using planar dynamical systems theory, we investigate bifurcation phenomena and introduce perturbations to reveal chaotic behaviors. Phase portraits offer visual insights, while sensitivity analysis using the Runge–Kutta method emphasizes solution stability against initial condition variations. Leveraging the planar dynamical system method, we generate diverse solitons, including periodic, bright, and dark solitons. This work enhances our grasp of intricate dynamics and their broader implications.

Ecotourism is a form of tourism involving responsible travel to natural areas, conserving the environment, and improving the well-being of the local people. Its purpose may be to educate the traveler, to provide funds for ecological censervation, to directly benifit the economic development, and political empowerment of local communities. Ecotourism has come up as an important conservation strategy in the tropical areas where diversity of species and habitats are threatened because of the traditional forms of development. This study deals with a non-linear mathematical model with a novel idea for sustainable development of biomass with ecotourism which is imperative in the present scenario. Stability and bifurcation analysis of the model is done and it is observed from our study that the system predicts unstability and exhibits bifurcation if ecotourism goes beyond a threshold value.

The complex service environment of railway vehicles leads to changes in the wheel–rail adhesion coefficient, and the decrease in critical speed may lead to hunting instability. This paper aims to reveal the diversity of periodic hunting motion patterns and the internal correlation relationship with wheel–rail impact velocities after the hunting instability of a bogie system. A nonlinear, non-smooth lateral dynamic model of a bogie system with 7 degrees of freedom is constructed. The wheel–rail contact relations and the piecewise smooth flange forces are the main nonlinear, non-smooth factors in the system. Based on Poincaré mapping and the two-parameter co-simulation theory, hunting motion modes and existence regions are obtained in the parameter plane consisting of running speed v and the wheel–rail adhesion coefficient μ. Three-dimensional cloud maps of the maximum lateral wheel–rail impact velocity are obtained, and the correlation with the hunting motion pattern is analyzed. The coexistence of periodic hunting motions is further revealed based on combined bifurcation diagrams and multi-initial value phase diagrams. The results show that grazing bifurcation causes the number of wheel–rail impacts to increase at a low-speed range. Periodic hunting motion with period number n = 1 has smaller lateral wheel–rail impact velocities, whereas chaotic motion induces more severe wheel–rail impacts. Subharmonic periodic hunting motion windows within the speed range of chaotic motion, pitchfork bifurcation, and jump bifurcation are the primary forms that induce the coexistence of periodic motion.

This study focuses on the extended (3+1)-dimensional Sakovich equation, which is utilized for the description of nonlinear wave motion and manifests increased dispersion and nonlinear effects within the realm of nonlinear dynamics. The Lie symmetries are initially identified, followed by the construction of the Abelian algebra corresponding to these symmetries. Through the utilization of this Abelian algebra, the governing equation is transformed into an ordinary differential equation. Precise solutions are obtained using the unified technique, and the results are visually represented through 3D, 2D, and contour plots. Subsequently, the qualitative behavior of the equation is explored from multiple perspectives, encompassing bifurcation, chaos, and sensitivity analysis. Bifurcation is examined at critical points, while the dynamical system is subjected to an external force, leading to the detection of chaotic behavior employing diverse tools such as 3D and 2D phase portraits, time series, Poincaré maps, and multistability analysis. Furthermore, sensitivity analysis is conducted for various initial values, revealing the considerable sensitivity of the presented model, wherein even slight changes in the initial conditions result in significant variations. The reported outcomes are intriguing, showcasing the efficacy and applicability of the suggested methodologies for the assessment of soliton solutions and phase patterns in a diverse range of nonlinear models.

One of the major causes of the intermittent nature of long-term climate changes is the interaction between the surface energy balance and the mass balance of the Cryosphere. In this paper, the pre-existing surface energy balance-mass balance model of the Cryosphere is modified by incorporating the radiative forcing of CO2 to observe the effect of global warming on the nature of the previous model. The modified model is generalized using Caputo’s non-local operator. The stability of the new model is analyzed at all the equilibrium points and also it is shown that there exists a unique and bounded solution for the modified system. Using bifurcation analysis and calculating the values of Lyapunov exponents for different fractional orders of the modified system, it is found that the system exhibits chaos for certain values of the radiating forcing of CO2 To observe and visualize the changes in the nature of the new model, the system is solved using the highly efficient 7th-order Runge–Kutta method. It is observed that with the inclusion of the radiative forcing of CO2, the nature of the system changes from asymptotically stable to chaotic as we decrease the order of the system. The Poincare map shows that the modified model even has characteristics of a strange attractor which is highly chaotic. The system becomes unstable when the value of the radiative forcing of CO2 is increased. As a result, the predictive power of the surface energy balance-mass balance model of the Cryosphere decreases. In addition to providing insights into the transition from stability to chaos in the Cryosphere model due to CO2 radiative forcing, this research offers a deeper understanding of the intricate interplay between climate dynamics and complex systems behavior.

Real mechanisms that are advancing in a complex network exhibit chaotic behaviour. This behaviour is crucial in physical and complex systems involving numerical modelling frameworks because it essentially determines the framework’s evolutionary process. In this context, notwithstanding its difficulty, the potential of intentional oversight of the phenomenon has feasible effects; this is why theoretical approaches are advantageous in such scenarios. This study investigates the functioning of a Willamowski–Rössler (W–R) mechanism, including the synchronization of two minimal W–R structures depending on the responsive suggestion technique for regulation, with the purpose of achieving chaos influence in chemical interactions. We investigate the reliability of the steady state at various fractional order (FO) factors. Employing maximum Lyapunov exponents (MLEs), phase depictions, bifurcation schematics, the 0–1 evaluation and approximated entropy, it is demonstrated that adjusting the FOs causes a system’s behavioural pattern to undergo a transition from steady to chaotic. In addition to demonstrating that the proposed scheme fits chaotically under certain circumstances, simulation outcomes demonstrate that mathematical modelling is used to illustrate theoretical debates. To verify that the community detects chaos, the MLE and bifurcation illustrations, whose hallmark factors are plotted, display erratic behaviour while effectively attempting to control the chaos.

This work investigates the nonlinear Rangwala–Rao equation, which stems from the mixed derivative nonlinear Schrödinger equation. For retrieving new exact solutions to the equation, the complete discriminant system for polynomial method is employed. In results, some novel traveling wave solutions, including solitary wave solutions, triangular function solutions, periodic solutions and Jacobi elliptic function solutions are obtained and demonstrated through numerical simulations. The bifurcations of phase velocity fields are depicted to reveal the dynamic behavior of the Rangwala-Rao equation using the qualitative theory of dynamical systems. Furthermore, considering external perturbation, the chaotic motions of the perturbed Rangwala-Rao equation are investigated.

The main objective of this paper is to study the soliton solutions and dynamics analysis of the fractional Radhakrishnan–Kundu–Lakshmanan equation with multiplicative noise in the Stratonovich sense. Firstly, the wave transformation is used to obtain the nonlinear ordinary differential equation, and then the nonlinear ordinary differential equation is transformed into a two-dimensional plane dynamic system with a Hamiltonian system. Secondly, the phase portrait and sensitivity of the plane dynamic system and its perturbed system are studied using Maple software. Thirdly, the soliton solutions of the stochastic fractional Radhakrishnan–Kundu–Lakshmanan equation can be constructed, and the Jacobian function solutions and hyperbolic function solutions are obtained. Finally, some three-dimensional and two-dimensional diagrams of the obtained solutions are also drawn. Moreover, the modulation stability of the equation under consideration is also given.

Roll-to-roll manufacturing is the primary means of flexible electronics to facilitate scale-up production, and the web as the printed substrate is guided by the intermediate roller in this process. This paper aims to investigate the nonlinear superharmonic resonance and chaotic motion of a moving web under an intermediate nonlinear support. This support is modeled as a nonlinear elastic spring with linear and nonlinear stiffness, and the equation of motion of the web attached with nonlinear elastic support is derived from the D’Alembert principle and von Karman theory. The resulting equation is reduced to the two-degree ordinary differential equations via the Galerkin truncation, the superharmonic resonance responses of the web system are obtained by the multi-scale method, and the bifurcation analysis and stability are analyzed by the Runge-Kutta numerical method. The results indicate that the linear and nonlinear stiffnesses have a significant effect on amplitude-frequency responses and chaotic motion. This study provides an exploration of vibration behaviors of the web in flexible manufacturing, thereby laying the foundation for the improvement of fabrication productivity.

This study analyzed a modified conservative 4D chaotic system is investigated using both integer and non-integer order derivatives. Several dynamical aspects of the said model are explored, such as stable equilibrium points, Lyapunov spectra (LS), attractor projection, Poincare, bifurcations and phase portrait. The system is also analyzed using a singular fractional operator, and the theory of the existence of solutions is established through functional analysis. To obtain numerical results of the fractional order system, a numerical method based on Newton polynomial is applied. The study reveals the presence of hidden and fixed point chaotic attractors for certain fractional order values.

This paper presents several novel contributions in the field of nonlinear fractional low-pass electrical transmission line model (NFLETLM). Firstly, using the modified (G′G2)-expansion method and the extended modified Jacobi elliptic expansion method, we discovered new and accurate solutions for NFLETLM, which have not been reported in the existing literature (Tala-Tcbuc et al. 2014; Nuruzzaman et al. 2021). These solutions, denoted as U3,U4,U7,U8,U9,U10,U13,U14 and UJ1,UJ2,UJ3,UJ4 represent novel contributions to the fields. Secondly, by utilizing computer simulations, we observed various intriguing phenomena in the wave solution graphs, such as anti-kink waves, periodic waves, intense singular periodic waves, bright singular wave solutions, multi-periodic waves, intense double periodic waves, and alternating patterns of light and shade waves. These findings shed light on previously unexplored aspects of the problem. Thirdly, through an extensive study on the newly discovered solutions, we provided a comprehensive understanding of the solitons inherent in the NFLETLM, including comparison with other derivative solutions. Our conformable fractional derivative solutions showed similarities to beta derivative solutions, distinguishing them from Riemann–Liouville derivative solutions. Lastly, we explored the phase portrait, bifurcation analysis, sensitivity, and potential chaotic behaviors of the NFLETLM, which have not been addressed in previous works (Tala-Tcbuc et al. 2014; Nuruzzaman et al. 2021). This innovative contribution expands our understanding of the NFLETLM model and uncovers new dynamics and phenomena.

Memristors, which are inevitably dynamical and memory components, could replace capacitors in the next-generation dynamical supercomputing converters that can exhibit behavioural characteristics, such as chaos. The nonlocal and nonsingular kernel Mittag-Leffler function (MLF) in the context of Caputo is researched in a new memristor framework via variable order. The existence and uniqueness of the fractional depiction of the specific process are confirmed using the Banach fixed-point approach for the contraction criterion. A newly developed numerical scheme for fractional-order systems introduced by Toufik and Atangana is utilized to obtain the phase portraits of the suggested system for various variable-order values of the system. The memristor-based model and many chaotic patterns of the Newton–Leipnik, Rabinovich–Fabrikant, Dadras, Aizawa, Thomas’ and four-wing types were the core issues of our study. The sensitivity of the chaotic systems of the considered model will be focused with precision using the bifurcation diagrams and the Lyapunov exponent. The stability of the equilibrium points of the commensurable fractional-order chaotic system will be addressed in the context of fractional calculus. In other words, we will use the standard Matignon criterion to address the problem of stability. The main attraction and novelty of this paper will be the use of the Lyapunov exponent (LE˜)s. It is demonstrated that the attempted framework is chaotic as well as extremely responsive to modifications in the parameters and the small initial conditions.

In this paper, the stochastic space-fractional long-short-wave interaction system (SF-LSWIS) with multiplicative white noise is considered. The stochastic exact solutions, including triangular function solutions, hyperbolic function solutions, and Jacobi elliptic function solutions, are obtained via the complete discriminant system for polynomial method. For this purpose, the traveling wave transformation is applied to carry out the ordinary differential equation of this model, which is also converted to a dynamical system. Thereafter, qualitative behavior and bifurcation of the phase portraits of the dynamical system are investigated using the qualitative theory of dynamical systems. The chaotic behaviors of the system considering external periodic perturbation are studied. Furthermore, the 3D surface, graphs of contour plots, and level curves of some exact solutions are depicted by choosing proper parameter values. The influence and effect of fractional derivatives and noise strength on the solutions are also studied.

As per the World Health Organization’s (WHO’s) suggestions, personal protection via adopting precautionary measures is one of the most effective control aspects to avoid Zika infection in the absence of suitable medical treatment. This personal protection further can be enhanced and explored by propagating information about disease prevalence. Therefore, in this study, we wish to see the effect of information on Zika transmission by formulating a compartmental mathematical model that quantifies the effect of an individual’s behavioral response as self-protection due to information. Furthermore, the basic reproduction number was calculated using the next-generation matrix technique. The model analysis was carried out to determine the local and global stability properties of equilibrium points. In addition, the model shows the occurrence of forward bifurcation when the reproduction number crosses unity. To understand the impact of various model parameters, we conducted a sensitivity analysis using both the normalized sensitivity index and the partial rank correlation coefficient methods. Moreover, we performed numerical simulations to assess the influence of important parameters on the model’s behavior for Zika prevalence. Our study accentuates that as information-induced self-protection increases, the prevalence of Zika infection will be at a very minimum level, and this observation is in line with WHO suggestions.

The insect predator–prey system mediates several feedback mechanisms which regulate species abundance and spatial distribution. However, the spatiotemporal dynamics of such discrete systems with the refuge effect remain elusive. In this study, we analyzed a discrete Holling type II model incorporating the refuge effect using theoretical calculations and numerical simulations, and selected moths with high and low growth rates as two exemplifications. The result indicates that only the flip bifurcation opens the routes to chaos, and the system undergoes four spatiotemporally behavioral patterns (from the frozen random pattern to the defect chaotic diffusion pattern, then the competition intermittency pattern, and finally to the fully developed turbulence pattern). Furthermore, as the refuge effect increases, moths with relatively slower growth rates tend to maintain stability at relatively low densities, whereas moths with relatively faster growth rates can induce chaos and unpredictability on the population. According to the theoretical guidance of this study, the refuge effect can be adjusted to control pest populations effectively, which provides a new theoretical perspective and is a feasible tool for protecting crops.

In this study, we formulate an eight-compartment mathematical model with vaccination as one of the compartments to analyze the dynamics of COVID-19 transmission. We examine the model’s qualitative properties, such as positivity and boundedness of solutions, and stability analysis of the illness-free equilibrium with respect to the basic reproduction number. We estimate ten significant parameters and also compute the magnitude of the basic reproduction number for India by fitting the proposed model to daily confirmed and cumulative confirmed COVID-19 cases in India. Sensitivity analysis with respect to basic reproduction number is conducted, and the main parameters that impact the widespread of disease are determined. We further extend this model to an optimal control problem by including four non-pharmaceutical and pharmaceutical intervention measures as control functions. Our numerical results show that the four control strategy has greater impact than the three control strategies, two control strategies, and single control strategies on reducing the dynamics of COVID-19 transmission.

When the dual rotor system of the aircraft engine is operating, the mass eccentricity of the power turbine rotor and the misalignment of the shaft coupling or the bearing will cause too large vibration of the rotor; this vibration leads to the rub-impact between the rotor and the casing. The power turbine rotor from the dual rotor system is taken as the research object in this paper. Considering the misalignment, the resulting rub-impact faults, the imbalance of rotor and the disk offset, the equation of motion for the system is developed according to the Lagrangian Equation, and then the Range-Kutta Method is adopted to solve the equation. The influence of the key parameters such as the rotating speed, the misalignment angle and the rub-impact clearance on the dynamics of the system is studied; the finite element analysis was carried out to validate the correctness of the theoretical modeling method. The results show that the rub-impact increases the stiffness of the system; the Hopf bifurcation occurs in the misalignment and rub-impact coupling system; the vibrational stability near the half of the switching speed slumps with the increase of the misalignment angle; with increasing of the stiffness, the number of the chaotic zone increases, and the range of the chaos is widening; enlarging the rub-impact clearance is beneficial to reduce the degree of the rub-impact system and enhance the stability of the system.

The application of fractional partial differential equations in the area of nonlinear optics has greatly facilitated the development of optical fiber communication. The primary objective of this article is to analyze the dynamic behavior of generalized nonlinear fractional Tzitzéica-type equation and obtain its optical soliton solutions. Using the planar dynamic system, phase portraits of the system are studied from a qualitative perspective, and then the polynomial complete discrimination method is utilized to obtain various optical soliton solutions of the equation. Furthermore, the relevant characteristics are examined through digital simulation and image analysis.

The main objective of this paper is to study the chaotic behavior and embedded soliton of cubic-quartic χ(2) and χ(3) nonlinear susceptibilities with multiplicative white noise. Firstly, traveling wave transformation and the trial equation method are used to obtain the nonlinear ordinary differential equations. Secondly, chaotic behavior, bifurcation and sensitivity of the perturbed perturbation system are studied. Thirdly, the embedded soliton solutions of cubic-quartic χ(2) and cubic χ(3) nonlinear model can be obtained.

This article studies the generalized nonlinear Schrödinger equation, which is used to simulate the propagation model of optical pulses in Non-Kerr medium. Building upon the traveling wave transformation, the generalized nonlinear Schrödinger equation is simplified to an ordinary differential equation. By employing the two-dimensional planar dynamic system to analyze, the bifurcation, phase portraits and chaotic behaviors of the system is presented. Furthermore, 2D and 3D phase portraits of the dynamic system with perturbation term are plotted with the Maple software. The optical soliton solutions of the generalized nonlinear Schrödinger equation are constructed by using the polynomial complete discriminant method.

In this paper, the stochastic Schrödinger–Hirota equation in birefringent fibers with spatiotemporal dispersion and parabolic law nonlinearity is studied, which is usually used to describe the mathematical model of optical soliton propagation in dispersive optical fibers from the field of nonlinear optics. Firstly, the stochastic Schrödinger–Hirota equation is converted into two-dimensional planar dynamic system by using the traveling wave transformation. Secondly, the qualitative analysis of the stochastic Schrödinger–Hirota equation is discussed by using the theory of planar dynamical systems. Finally, the chaos pattern of the stochastic Schrödinger–Hirota equation with perturbed system is considered. What is more, phase portraits and sensitivity analysis of the perturbed system are plotted by using the Maple software.

In this paper, the cubic–quartic complex Ginzburg–Landau (CGL) equation is investigated by using the trial function method. The traveling wave hypothesis is applied to convert the CGL equation to an ordinary differential equation (ODE), which is equivalent to a dynamic system. The qualitative behavior, bifurcation of the phase portraits for this system is studied. Furthermore, the chaotic motions in system involving external periodic perturbation are considered. Some new optical solitons, such as Jacobian elliptic function solutions, for CGL equation with quadratic–cubic law nonlinearity are constructed using the complete discrimination system for polynomial method. Moreover, two-dimensional graphs, three-dimensional, corresponding contour and density plots for some acquired solutions are depicted to carry out the physical properties of optical pulse propagation.

The Chen-Lee -Liu model has many applications in assorted fields, particularly in the study of nonlinear dynamics, chaos theory, circuit design, signal processing, secure communications, encryption and decryption of chaotic signals, as well as cryptography. The modified extended auxiliary equation mapping method has been applied to the Chen-Lee-Liu model in this article and explores new wave profiles, such as singular periodic solutions, periodic solutions, and kink-type soliton solutions. The complex wave conversion is considered to make a simple differential equation. Three- and two-dimensional images are plotted using Mathematica and MATLAB, and their dispersion and nonlinearity effects are discussed. We also discuss the bifurcation analysis of the studied model. The stability of the equilibrium points is studied, and the phase portrait of the system is presented graphically. The obtained wave profiles might play an important role in telecommunication systems, fiber optics, and nonlinear optics.

This paper deals with an unstirred competitive chemostat model with the Beddington–DeAngelis functional response. With the help of the linear eigenvalue theory and the monotone dynamical system theory, we establish a relatively clear dynamic classification of this system in terms of the growth rates of two species. The results indicate that there exist several critical curves, which may classify the dynamics of this system into three scenarios: 1) extinction; 2) competitive exclusion; and 3) coexistence. Comparing with the classical chemostat model [26], our theoretical results reveal that under the weak–strong competition cases, the role of intraspecific competition can lead to species coexistence. Moreover, the simulations suggest that under different competitive cases, coexistence can occur for suitably small diffusion rates and some intermediate diffusion rates. These new phenomena indicate that the intraspecific competition and diffusion have a great influence on the dynamics of the unstirred chemostat model of two species competing with the Beddington–DeAngelis functional response.

The present model is dealt with prey-predator interactions in two different patches where only prey species are allowed to disperse among the patches. Each of these two patches has different predator population but the predator in Patch-2 only is affected with a disease. The proposed model is biologically welldefined. Also, the feasibility of the equilibrium points and corresponding stability conditions are analysed. It is found that the disease among predator, even in one patch, makes an important role to control the whole system dynamics as it starts to oscillates by regulating the disease transmission rate. Moreover, the disease transmission rate has a stabilizing as well as destabilizing effect on the system dynamics. From the results, it is observed that a high dispersal rate decreases the count of infected predator in a patch in presence of prey dispersal. There is another interesting result: it is observed that the prey dispersal cannot destabilize the coexistence state, i.e., the system which is stable in absence of dispersal remains stable when the prey species disperse between two patches.

Vibration induced by dry friction is ubiquitous in various engineering fields. To explore the vibration characteristics for further studies and/or controls, it is of great theoretical and practical significances to investigate the non-linear dynamic behaviors of the friction systems. This study considers the slight vibration of a two-degree-of-freedom non-linear dry friction excitation system. The differential equations of system motion are established according to Newton’s law of motion. Moreover, the system’s non-linear dynamic is studied when the block velocity is always less than the friction surface velocity. The results indicate that the linearized matrix of the vibration system has a pair of purely imaginary eigenvalues for some critical values of the relevant parameters. The Poincaré-Birkhoff normal forms are utilized to simplify the motion equation under the non-resonant assumption to obtain a simplified equation with only the resonant terms. Furthermore, the truncated part of the simplified equation is analyzed in the case of only linear terms degeneration. Finally, numerical simulations reflect some qualitative conclusions about the system’s local dynamic properties, including equilibrium point, periodic motion, torus motion, and their stability.

In this paper, the coupled Kundu–Mukherjee–Naskar equation is considered, which is usually used to describe the oceanic rogue waves, hole waves and Bragg grating fibers. Firstly, the traveling wave transformation is applied to convert the coupled Kundu–Mukherjee–Naskar equation into two-dimensional planar dynamic system. Secondly, bifurcation and two-dimensional phase portrait of the coupled Kundu–Mukherjee–Naskar equation are discussed by using the plane dynamics system. What is more, after adding periodic perturbation term, two-dimensional phase portrait, three-dimensional phase portrait, Poincaré section and sensitivity of its perturbed system are plotted by the Maple software. Finally, the optical soliton solutions of the coupled Kundu–Mukherjee–Naskar equation are obtained by using planar dynamical system method.

New challenges like the outbreak of new diseases, government policies, war and insurgency etc. present distortion, delay and denial of persons’ access to ART, thereby fuelling the spread and increasing the burden of HIV/AIDS. A mathematical model is presented to study the transmission dynamics and control of HIV infection. The qualitative and quantitative analyses of the model are carried out. It is shown that the disease-free equilibrium of the model is globally asymptotically stable whenever the basic reproduction number is less than unity. It is also shown that a unique endemic equilibrium exists whenever the basic reproduction number exceeds unity and that the model exhibits a forward bifurcation. Furthermore, the Lyapunov function is used to show that the endemic equilibrium is globally asymptotically stable for a special case of the model whenever the associated basic reproduction number is greater than unity. The model is calibrated to the data on HIV/AIDS prevalence in Nigeria from 1990 to 2019 and it represents reality. The numerical simulations on the global stability of disease-free equilibrium and endemic equilibrium justify the analytic results. The fraction of the detected individuals who stays in the treatment class plays a significant role as it influences the population of the latently-infected individuals and AIDS class by preventing individuals from progressing into the AIDS class.

The stochastic fraction Hirota–Maccari system is used as a governing model to check the wave propagation in nonlinear optics, plasma physics and hydrodynamics. Firstly, the random fractional order Hirota–Maccari system is transformed into a two-dimensional plane dynamic system by using traveling wave transformation. Secondly, the bifurcation is discussed by using the theory of plane dynamic system, and the sensitivity to the strength and frequency of perturbation is analyzed. In addition, the exact traveling wave solution of the stochastic fractional Hirota–Maccari system is constructed by using the analytical method of the plane dynamic system. Finally, the results of comparison analysis indicated that the fractional derivatives and noises will greatly influence the solution behavior.

We investigate the stochastic dynamics of the prey–predator model of the Low-to-High confinement mode (L-H) transition in magnetically confined fusion plasmas. By considering stochastic noise in the turbulence and zonal flows as well as constant and time-varying input power Q, we perform multiple stochastic simulations of over a million trajectories using GPU computing. Due to stochastic noise, some trajectories undergo the L-H transition while others do not, leading to a mixture of H-mode and dithering at a given time and/or input power. One of the consequences of this is that H-mode characteristics appear at a smaller input power QQc as a second peak. The coexisting H-mode and dithering near Q=Qc leads to a prominent bimodal PDF with a gradual L-H transition rather than a sudden transition at Q=Qc and uncertainty in the input power. Also, a time-dependent input power leads to increased variability (dispersion) in stochastic trajectories and a more prominent bimodal PDF. We provide an interpretation of the results using information geometry to elucidate self-regulation between zonal flows, turbulence, and information causality rate to unravel causal relations involved in the L-H transition.

This paper deals with the analysis of nonlinear vibration phenomena in the arm system of an industrial robot. The presented mathematical model of the robot takes into account the flexibility of the joints, which results in low-frequency vibrations in the arm. The study of vibration phenomena has been carried out using analytical, numerical and experimental methods. Special attention has been paid to bifurcation phenomena. The influence of two bifurcation parameters on the behaviour of the robot arm was studied. It was shown that the amplitude and frequency of the harmonic excitation create a nonlinear effect on the vibration amplitudes of the links. In bifurcation regions, the vibration amplitudes can even differ several times depending on the direction of changes in the bifurcation parameter. In general, the results indicate that, contrary to common practice, it is worth paying attention to nonlinear phenomena when analysing robot vibrations.

The long–short wave interaction system (L-SWIS) is an important model describing the interaction of two waves propagating in a generalized elastic medium. In this study, the bifurcation and the influence of random interaction on the exact solution of the stochastic fractional long–short wave interaction system (SFL-SWIS) with multiplicative Brownian motion are studied, where the derivative refers to the modified Riemann–Liouville definition. After the Hamiltonian system is established by traveling wave transformation and first-order integration, we obtain abundant exact parametric solutions of SFL-SWIS. Also, the three-dimensional graph of some obtained results are drawn by using the symbolic calculation software Maple, which is used to show the influence of fractional derivative and random interaction on the solution. The obtained solitary wave solutions and exact solutions include the elliptic function solution, trigonometric function solution, exponential function solution and singular periodic solution. This study is the first to investigate the bifurcation and exact solution of SFL-SWIS, and the obtained results are theoretically and practically important to the propagation of optical signals, plasma waves and ocean waves.

The inherent non-smoothness of the vibroimpact system leads to complex behaviors and a strong sensitivity to parameter changes. Unfortunately, uncertainties and errors in system parameters are inevitable in mechanical engineering. Therefore, investigations of dynamical behaviors for vibroimpact systems with stochastic parameters are highly essential. The present study aims to analyze the dynamical characteristics of the three-degree-of-freedom vibroimpact system with an uncertain parameter by means of the Chebyshev polynomial approximation method. Specifically, the vibroimpact system model considered is one with unilateral constraint. Firstly, the three-degree-of-freedom vibroimpact system with an uncertain parameter is transformed into an equivalent deterministic form using the Chebyshev orthogonal approximation. Then, the ensemble means responses of the stochastic vibroimpact system are derived. Numerical simulations are performed to verify the effectiveness of the approximation method. Furthermore, the periodic and chaos motions under different system parameters are investigated, and the bifurcations of the vibroimpact system are analyzed with the Poincaré map. The results demonstrate that both the restitution coefficient and the random factor can induce the appearance of the periodic bifurcation. It is worth noting that the bifurcations fundamentally differ between the stochastic and deterministic systems. The former has a bifurcation interval, while the latter occurs at a critical point.

We propose a new model for landslide dynamics under the assumption of a delay failure mechanism. Delay failure is simulated as a delayed interaction between adjacent blocks, which mimics the relationship between the accumulation and feeder part of the accumulation slope. The conducted research consisted of three phases. Firstly, the real observed movements of the landslide were examined to exclude the existence or the statistically significant presence of background noise. Secondly, we propose a new mechanical model of an accumulation landslide dynamics, with introduced delay failure, and with variable friction law. Results obtained indicate the onset of a transition from an equilibrium state to an oscillatory regime if delayed failure is assumed for different cases of slope stiffness and state of homogeneity/heterogeneity of the slope. At the end, we examine the influence of different frictional properties (along the sliding surface) on the conditions for the onset of instability. Results obtained indicate that the increase of friction parameters leads to stabilization of sliding for homogeneous geological environment. Moreover, increase of certain friction parameters leads to the occurrence of irregular aperiodic behavior, which could be ascribed to the regime of fast irregular sliding along the slope.

Pynamical is an educational Python package for introducing the modeling, simulation, and visualization of discrete nonlinear dynamical systems and chaos, focusing on one-dimensional maps (such as the logistic map and the cubic map). Pynamical facilitates defining discrete one-dimensional nonlinear models as Python functions with just-in-time compilation for fast simulation. It comes packaged with the logistic map, the Singer map, and the cubic map predefined. The models may be run with a range of parameter values over a set of time steps, and the resulting numerical output is returned as a pandas DataFrame. Pynamical can then visualize this output in various ways, including with bifurcation diagrams, two-dimensional phase diagrams, three-dimensional phase diagrams, and cobweb plots. These visualizations enable simple qualitative assessments of system behavior including phase transitions, bifurcation points, attractors and limit cycles, basins of attraction, and fractals.

This study investigates the nonlinear dynamic response behavior of a rotating ring that forms an essential element of MEMS (Micro Electro Mechanical Systems) ring-based vibratory gyroscopes that utilize oscillatory nonlinear electrostatic forces. For this purpose, the dynamic behavior due to nonlinear system characteristics and nonlinear external forces was studied in detail. The partial differential equations that represent the ring dynamics are reduced to coupled nonlinear ordinary differential equations by suitable addition of nonlinear mode functions and application of Galerkin’s procedure. Understanding the effects of nonlinear actuator dynamics is essential for characterizing the dynamic behavior of such devices. For this purpose, a suitable theoretical model to generate a nonlinear electrostatic force acting on the MEMS ring structure is formulated. Nonlinear dynamic responses in the driving and sensing directions are examined via time response, phase diagram, and Poincare’s map when the input angular motion and nonlinear electrostatic force are considered simultaneously. The analysis is envisaged to aid ongoing research associated with the fabrication of this type of device and provide design improvements in MEMS ring-based gyroscopes.

In this paper, we explore local behavior at fixed points, chaos and bifurcations of a discrete COVID-19 epidemic model in the interior of R+5. It is explored that for all involved parametric values, COVID-19 model has boundary fixed point and also it has an interior fixed point under certain parametric condition(s). We have investigated local behavior at boundary and interior fixed points of COVID-19 model by linear stability theory. It is also explored the existence of possible bifurcations at respective fixed points, and proved that at boundary fixed point there exists no flip bifurcation but at interior fixed point it undergoes both flip and hopf bifurcations, and we have explored said bifurcations by explicit criterion. Moreover, chaos in COVID-19 model is also investigated by feedback control strategy. Finally, theoretical results are verified numerically.

As a result of its wide range of applications, FSI has grabbed the attention of researchers and scientists. In this study we consider an incompressible, laminar fluid flowing through the bifurcated channel. The wavy walls of the channel are considered elastic. Moreover, a magnetic field is applied towards the axial direction of the flow. Using a two-way fluid-structure interaction, an Arbitrary Lagrangian-Eulerian (ALE) formulation is used for coupling the problem. The problem is discretized using P2 and P1 finite element methods to approximate the displacement, pressure, and velocity. The linearized system of equations is solved using Newton’s iterative scheme. The analysis is carried out for the Reynolds number and Hartman number. The ranges of the studied parameters are Reynolds number 300≤Re≤1000 and Hartmann number 0≤Ha≤10. The hemodynamic effects on the bifurcated channel and elastic walls are calculated using velocity, pressure, wall shear stresses (WSS), and loads at the walls. The study shows there is an increase in boundary load as the values of the Hartman number increase hence WSS increases. On the other hand, an increase in the Reynolds number increases the resistance forces hence velocity and WSS decrease. Also, numerical values of WSS for rigid and elastic walls are calculated. Studies showed that WSS decreases for the FSI case when compare to CFD (computational fluid dynamic) case.

Analysis of a pulse voltage regulator from the point of view of transition to an unstable position and possible bifurcation states. Calculation of the stability condition when changing the parameters of the system. Obtaining the pulse voltage regulator stability surface. This voltage stabilizer can have the basic intrinsic parameters of the system, such as, for example, the load resistance, the value of which can change over time according to unknown laws. In the course of the work, the stability limits for static and dynamic bifurcations were obtained with respect to many parameters of the system, including with respect to the value of the load conductivity. To obtain numerical results, a system with the following main parameters was used: U = 4,1 V, d′0 = 0,4, C = 600 μF, L = 2 mH, f1 = 0,8 Ohm, G = 0,5 V–1, RL = 2 Ohm.

In this manuscript, the optical solitons of nonlinear Schrödinger equation (NLSE) with cubic–quintic law nonlinearity, in the presence of self-frequency shift and self-steepening, has been studied. The ultrahigh capacity propagation and transit of laser light pulses in optical fibres were described using this form of equation. To extract new results, two strong methodologies has been used. To extract the exact solution unified method has been employed. The solutions obtained by this analytical method, are in form of polynomial and rational function solution. Moreover, the validity of non-singular solutions has guaranteed by a limitation condition that is graphically illustrated in 3D. The 2D graphical representation are also used to demonstrate the influence of parameters on the predicted non-singular solutions. The other technique, used for qualitative analysis, is bifurcation. The system has been transformed into a planer dynamical system, which has been transformed into a hamiltonian system. All the possible phase portrait has been plotted by complete discrimination method. The acquired results are novel and have not been recorded before and they indicate that the proposed methodologies may be used to investigate innovative soliton solutions and phase portraits for any NLSE.