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The stability of predator–prey interactions in ecosystems is influenced by both inherent species interactions and external factors. For instance, the presence of additional food, as an external factor, may affect the system. To further explore this, a stage-structured predator–prey model is constructed, incorporating the influences of fear and delay on prey-population growth, which provides additional food for immature predators and facilitates cooperative behavior between mature and immature predators. The analysis evaluates the positivity, boundedness, equilibrium points, local stability around each equilibrium point, and certain bifurcations of the system. Additionally, numerical simulations are provided to correspond with the results of the theoretical analysis. It is observed that an appropriate level of fear contributes positively to system stability. While cooperation among predators can benefit immature predators, it also has the potential to harm the overall system. The introduction of additional food complicates the system dynamics, although it benefits predators, it places prey at a disadvantage. Furthermore, we observe a correlation between the level of fear and the effects of additional food, as well as the capacity of additional food to mitigate the influence of delay. [ABSTRACT FROM AUTHOR]

In regular dynamics, discrete maps are model presentations of discrete dynamical systems, and they may approximate continuous dynamical systems. Maps are used to investigate general properties of dynamical systems and to model various natural and socioeconomic systems. They are also used in engineering. Many natural and almost all socioeconomic systems possess memory which, in many cases, is power-law-like memory. Generalized fractional maps, in which memory is not exactly the power-law memory but the asymptotically power-law-like memory, are used to model and investigate general properties of these systems. In this paper we extend the definition of the notion of generalized fractional maps of arbitrary positive orders that previously was defined only for maps which, in the case of integer orders, converge to area/volume-preserving maps. Fractional generalizations of Hénon and Lozi maps belong to the newly defined class of generalized fractional maps. We derive the equations which define periodic points in generalized fractional maps. We consider applications of our results to the fractional and fractional difference Hénon and Lozi maps. [ABSTRACT FROM AUTHOR]

In Bénard-Rayleigh convection we consider the pattern defect in orthogonal domain walls connecting a set of convective rolls with another set of rolls orthogonal to the first set. This is understood as an heteroclinic orbit of a reversible system where the x - coordinate plays the role of time. This appears as a perturbation of the heteroclinic orbit proved to exist in a reduced 6-dimensional system studied by a variational method in Buffoni et al. (J Diff Equ, 2023, https://doi.org/10.1016/j.jde.2023.01.026), and analytically in Iooss (Heteroclinic for a 6-dimensional reversible system occuring in orthogonal domain walls in convection. Preprint, 2023). We then prove for a given amplitude ε 2 , and an imposed symmetry in coordinate y, the existence of a one-parameter family of heteroclinic connections between orthogonal sets of rolls, with wave numbers (different in general) which are linked with an adapted shift of rolls parallel to the wall. [ABSTRACT FROM AUTHOR]

The paper devotes to the synthesis of local and global analysis of a discrete model describing the second-order digital filter with nonlinear signal processors. The discrete model gives rise to a two-dimensional non-invertible map, whose basins of attraction have complicated topological structures due to the intrinsic multi-stability. The influences of joint parameters on the local dynamics are presented in great details. Both theoretical and numerical results are plotted on the two-dimensional parametric planes. To show more detailed bifurcation structure, the isoclines are extended to higher periodic orbits for detecting the cusps of resonant entrainments. Invariant manifolds and critical curves are employed to illustrate the global dynamics of the model vividly. The tangency and intersections of invariant manifolds expound the process of erosions of basins of attraction. The global bifurcations of basins of attraction are deduced dynamically by critical curves. [ABSTRACT FROM AUTHOR]

Besides consultants and practitioners, some contributions in the organizational economics literature have advocated substituting internal firms’ bureaucracies with markets to regulate internal transactions. However, usually the effects of competition on performance are considered in terms of competition across firms or industries. By contrast, other contributions point out that competition is pervasive inside firms as well. In this paper, we assume that conflict is directly related to the level of competition and propose a model which analyze the dynamics of performance when the manager decides the level of competition observing the group performance. We study the stability of the equilibria and analyze the bifurcations. We show that the fixed point with null performance is a Milnor attractor, and this may suggests why any attempt to move from this unsatisfactory outcome is unsuccessful. [ABSTRACT FROM AUTHOR]

The large amplitude steady state periodic response of truncated conical shell panels under the influence of transverse harmonic excitation is analyzed by considering the nonlinear strain displacement relations. The finite element analysis is based on the kinematics of first-order shear deformation theory and the constrained strain terms have been interpolated using field-consistent modified shape functions to avoid shear locking. The governing equation of motion has been obtained using Hamilton's principle, which has been solved using the Modified shooting method and continuation schemes to yield the complete frequency response. The influence of boundary conditions, the amplitude of the forcing function and curvature on the periodic response was investigated. The hardening or softening nonlinear behavior has been obtained depending upon boundary conditions, geometry and forcing function amplitude. The combined influence of geometric nonlinearity and varying curvature of the truncated conical shell leads to significantly greater negative half-cycle amplitude. The peculiar nature of the restoring force dynamics with increased inward deflection causes the restoring forces to act in a destabilizing sense leading to increase in negative half-cycle amplitude. The periodic stress variation reveals multiple stress reversals during a loading cycle and is very critical for the fatigue design of such components. The multiple slope changes/bifurcations in the frequency response have been examined using response history, frequency spectra and the phase plane plots where it is revealed that for some cases the higher harmonic contributions are even greater than the fundamental harmonic. The deformed configuration at various instants during the periodic cycle reveals modal interaction between first and higher modes. Moreover, the nonlinear frequency response curves depict softening nonlinear behavior for deeper shells which gradually transforms into hardening behavior for shallow shell panels. [ABSTRACT FROM AUTHOR]

This paper proposes a base excited, spring coupled double cantilever beam with attached tip mass based piezoelectric energy harvester. The dimensions of the beams are so chosen that the system experiences 1:1 internal resonance. The governing coupled electro-mechanical equations of motion of the system are obtained by using the Lagrange principle which is reduced to the temporal form by using generalized Galerkin's method. The governing temporal equations of motion are in the form of that of a parametrically excited coupled system with geometric and inertial nonlinearities. These nonlinear equations are solved using the method of multiple scales and the results are compared with those solved by numerical method (4th order Runge Kutta method) for principal parametric resonance conditions. Parametric studies are conducted by varying the tip masses of the beams, coupled spring stiffness, load resistance, external excitation frequency and excitation amplitude. Results show that by introducing spring coupling, the system operational frequency range increases significantly and also energy transfer process takes place due to the internal resonance. Significant enhancement of voltage and power are observed in the present work in comparison to the earlier studied equivalent single beam-based energy harvester. [ABSTRACT FROM AUTHOR]

Waveform transitions have high correlation to spike wave discharges and polyspike wave discharges in seizure dynamics. This research adopts nonlinear dynamics to study the waveform transitions in a cerebral thalamo-coritcal neural network subjected to a square sensory control via discretization and mappings. The continuous non-smooth network outputs are discretized to establish implicit mapping chains or loops for stable and unstable waveform solutions. Bifurcation trees of period-1 to period-2 waveforms as well as independent bifurcation tree of period-3 to period-6 waveforms are obtained theoretically. The independent bifurcation tree should be taken much care during the control since it coexists with global stable waveforms but contains more spikes. Stability and bifurcations of the nonlinear waveform transitions are predicted by eigenvalue analysis of the discretized model. The transient process from unstable waveform to stable waveform is illustrated. The spike adding and period-doubling phenomenon are presented for illustration of the network response after control. The dominant frequency components and the detailed quantity levels of the corresponding amplitudes are exhibited in the harmonic spectrums which can be implemented to controller design for reduction and elimination of the absence seizures. This research presents new perspectives for the waveform transitions and provides theories and data for seizure prediction and regulation. [ABSTRACT FROM AUTHOR]

Mixed affective states in bipolar disorder (BD) is a common psychiatric condition that occurs when symptoms of the two opposite poles coexist during an episode of mania or depression. A four-dimensional model by Goldbeter (Progr Biophys Mol Biol 105:119–127, 2011; Pharmacopsychiatry 46:S44–S52, 2013) rests upon the notion that manic and depressive symptoms are produced by two competing and auto-inhibited neural networks. Some of the rich dynamics that this model can produce, include complex rhythms formed by both small-amplitude (subthreshold) and large-amplitude (suprathreshold) oscillations and could correspond to mixed bipolar states. These rhythms are commonly referred to as mixed mode oscillations (MMOs) and they have already been studied in many different contexts by Bertram (Mathematical analysis of complex cellular activity, Springer, Cham, 2015), (Petrov et al. in J Chem Phys 97:6191–6198, 1992). In order to accurately explain these dynamics one has to apply a mathematical apparatus that makes full use of the timescale separation between variables. Here we apply the framework of multiple-timescale dynamics to the model of BD in order to understand the mathematical mechanisms underpinning the observed dynamics of changing mood. We show that the observed complex oscillations can be understood as MMOs due to a so-called folded-node singularity. Moreover, we explore the bifurcation structure of the system and we provide possible biological interpretations of our findings. Finally, we show the robustness of the MMOs regime to stochastic noise and we propose a minimal three-dimensional model which, with the addition of noise, exhibits similar yet purely noise-driven dynamics. The broader significance of this work is to introduce mathematical tools that could be used to analyse and potentially control future, more biologically grounded models of BD. [ABSTRACT FROM AUTHOR]

In this article, using the symplectic methods developed by Moreno and Frauenfelder (aimed at analyzing periodic orbits, their stability and their bifurcations), we will carry out numerical studies concerning periodic orbits in the Jupiter–Europa and Saturn–Enceladus systems. We will put emphasis on planar-to-spatial bifurcations, from deformation of the families in Hill’s lunar problem studied by Aydin. We will also provide an algorithm for the numerical computation of Conley–Zehnder indices, which are instrumental in practice for determining which families of orbits connect to which. As an application, we use our tools to analyze a well-known family of Halo orbits that approaches Enceladus at an altitude of 29 km, which bears interest for future space missions that visit the water plumes. [ABSTRACT FROM AUTHOR]

This paper deals with a continuous-time predator-prey model of Leslie-Gower type considering the use of a physical refuge by a fraction of the prey population. The fraction of hidden prey is assumed to be dependent on the presence of predators in the environment. The conditions for the existence of equilibrium points and their local stability are established. In particular, it is shown that the point (0; 0) has a great importance in the dynamics of the model, since it determines a separating curve Σ that divides the behavior of the trajectories. Those trajectories that are above this curve have as their w- limit the point (0; 0), so the extinction of both populations may be possible depending on the initial conditions.

The main purpose of this paper is to study the bifurcations, optical solitons perturbation and chaotic behavior for the coupled Fokas-Lenells system in birefringent fibers. Firstly, by means of the homogeneous balance principle, the Gallian transformation and traveling wave transformations, the coupled Fokas-Lenells system in birefringent fibers is transformed into ordinary differential equation. Secondly, the dynamical properties of the two-dimensional system and their chaotic behavior with perturbed systems are analyzed in detail. The obtained results substantially improve or complement the corresponding conditions in the literature. Finally, in order to further understand the propagation of the coupled Fokas-Lenells system in birefringent fibers, two-dimensional portraits, three-dimensional portraits, density plots and contour plots of the obtained optical soliton solutions are drawn. [ABSTRACT FROM AUTHOR]

Abstract Qualitative changes can occur in the dynamics of nonlinear systems even for small parameter variations. Such changes are manifestations of bifurcation in dynamical systems. In the context of differential game theory, bifurcations offer insights into the underlying mechanisms driving strategic interactions and identify transitions between different types of behavior. Such critical transitions are tipping points that can dramatically change the outcomes of the game. This work explores the possibility of predicting such qualitative shifts, including supercritical Hopf bifurcations, before they occur using a data-driven forecasting technique. This concept is demonstrated for an attacker–defender game in a limited resource scenario and for an active cybersecurity defense game. The time histories of the system dynamics as it approaches a bifurcation allow one player to detect the existence of bifurcations. This capability provides that player insights into the dynamics of the game and potential defense mechanisms in resource-constrained scenarios.

In this paper, we explore the bifurcation, chaos, and local stability of a discrete Hepatitis C virus infection model. More precisely, we studied the local stability at fixed points of a discrete Hepatitis C virus model. We proved that at a partial infection fixed point, the discrete HCV model undergoes Neimark-Sacker bifurcation, but no other local bifurcation exists at this fixed point. Moreover, it was also proved that period-doubling bifurcation does not occur at liver-free, disease-free, and total infection fixed points. Furthermore, we also examined chaos control in the understudied discrete HCV model. Finally, obtained theoretical results were confirmed numerically.

Abstract This work investigates the Kraenkel-Manna-Merle (KMM) system, which models the nonlinear propagation of short waves in saturated ferromagnetic materials subjected to an external magnetic field, despite the absence of electrical conductivity. The study aims to explore and derive new solitary wave solutions for this system using two distinct methodological approaches. In the first approach, the KMM system is transformed into a system of nonlinear ordinary differential equations (ODEs) via Lie group transformation. The resulting ODEs are then solved analytically using a similarity invariant approach, leading to the discovery of various types of solitary wave solutions, including bright, dark, and exponential solitons. The second approach involves applying wave and Galilean transformations to reduce the KMM system to a system of two ODEs, both with and without damping effects. This reduced system is further analyzed to investigate its bifurcation behavior, sensitivity to initial conditions, and chaotic dynamics. The analysis reveals the presence of strange multi-scroll chaotic dynamics in the presence of damping and off-boosting dynamics without damping. In addition to these approaches, the study also applies the planar dynamical theory to obtain further new soliton solutions of the KMM system. These solitons include bright, kink, dark, and periodic solutions, each of which has been visualized through 3D and 2D graphs. The results of this research provide new insights into the dynamics of the KMM system, with potential applications in magnetic data storage, magnonic devices, material science, and spintronics.

Qualitative changes can occur in the dynamics of nonlinear systems even for small parameter variations. Such changes are manifestations of bifurcation in dynamical systems. In the context of differential game theory, bifurcations offer insights into the underlying mechanisms driving strategic interactions and identify transitions between different types of behavior. Such critical transitions are tipping points that can dramatically change the outcomes of the game. This work explores the possibility of predicting such qualitative shifts, including supercritical Hopf bifurcations, before they occur using a data-driven forecasting technique. This concept is demonstrated for an attacker–defender game in a limited resource scenario and for an active cybersecurity defense game. The time histories of the system dynamics as it approaches a bifurcation allow one player to detect the existence of bifurcations. This capability provides that player insights into the dynamics of the game and potential defense mechanisms in resource-constrained scenarios. [ABSTRACT FROM AUTHOR]

In this paper, we explore the bifurcation, chaos, and local stability of a discrete Hepatitis C virus infection model. More precisely, we studied the local stability at fixed points of a discrete Hepatitis C virus model. We proved that at a partial infection fixed point, the discrete HCV model undergoes Neimark-Sacker bifurcation, but no other local bifurcation exists at this fixed point. Moreover, it was also proved that period-doubling bifurcation does not occur at liver-free, disease-free, and total infection fixed points. Furthermore, we also examined chaos control in the understudied discrete HCV model. Finally, obtained theoretical results were confirmed numerically. [ABSTRACT FROM AUTHOR]

Awareness plays a vital role in informing and educating people about infection risk during an outbreak and hence helps to reduce the epidemic's health burden by lowering the peak incidence. Therefore, this paper studies a susceptible-aware-infected-recovered (SAIR) epidemic model with the novel combinations of Michaelis-Menten functional type nonlinear incidence rates for unaware and aware susceptible with the inclusion of time delay as a latent period and a saturated treatment rate for infected people. The model is analyzed mathematically to describe disease transmission dynamics in two obtained equilibria: disease-free and endemic. We derive the basic reproduction number R 0 and investigate the local and global stability behavior of obtained equilibria for the time delay. A bifurcation analysis is performed using center manifold theory when there is no time delay, revealing the forward bifurcation when R 0 varies from unity. Moreover, the presence of Hopf bifurcation around EE is shown depending on the bifurcation parameter time delay. Lastly, the numerical simulations validate the analytical findings. [ABSTRACT FROM AUTHOR]

This work investigates the Kraenkel-Manna-Merle (KMM) system, which models the nonlinear propagation of short waves in saturated ferromagnetic materials subjected to an external magnetic field, despite the absence of electrical conductivity. The study aims to explore and derive new solitary wave solutions for this system using two distinct methodological approaches. In the first approach, the KMM system is transformed into a system of nonlinear ordinary differential equations (ODEs) via Lie group transformation. The resulting ODEs are then solved analytically using a similarity invariant approach, leading to the discovery of various types of solitary wave solutions, including bright, dark, and exponential solitons. The second approach involves applying wave and Galilean transformations to reduce the KMM system to a system of two ODEs, both with and without damping effects. This reduced system is further analyzed to investigate its bifurcation behavior, sensitivity to initial conditions, and chaotic dynamics. The analysis reveals the presence of strange multi-scroll chaotic dynamics in the presence of damping and off-boosting dynamics without damping. In addition to these approaches, the study also applies the planar dynamical theory to obtain further new soliton solutions of the KMM system. These solitons include bright, kink, dark, and periodic solutions, each of which has been visualized through 3D and 2D graphs. The results of this research provide new insights into the dynamics of the KMM system, with potential applications in magnetic data storage, magnonic devices, material science, and spintronics. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel Caputo-type fractional-order cascade tri-neuron Hopfield neural network (HNN) taking no connection between the first and third neuron. We analyse the symmetry and dissipativity of the system using divergence and transformations. The stability of the equilibrium points is checked by fixing the synaptic weights. To further analyse the dynamics of the HNN system, we derive a numerical solution by using the Adams–Bashforth–Moulton method along with its stability analysis. We performed several graphical simulations, considering two synaptic weights as adjustable variables, and explored the fact that the HNN system shows various periodic and chaotic attractors. The reason for proposing a fractional-order HNN is that such a system has limitless memory, which can improve the system’s controllability for a wide range of real-world phenomena with important applications. Also, the proposed fractional-order HNN shows better convergence compared to the integer-order case. [ABSTRACT FROM AUTHOR]

Membrane reactors for the production of hydrogen are interesting devices that allow for the decentralized production of pure hydrogen while reducing the energy cost of the traditional steam reforming process. The performance of these systems is strongly dependent on the permeability and selectivity of the membrane employed for the separation of the produced hydrogen. Several studies have shown that the actual rate of hydrogen permeation is lower than the one that would be expected based on studies of membrane permeability carried out with pure hydrogen. This difference has been attributed to the competitive adsorption of other species, specifically CO, on the membrane surface. Here we show that this phenomenon could also lead to steady state multiplicity and analyze the dynamics of membrane reactors in the presence of competitive adsorption by CO. Emphasis has been placed on the effect of temperature and gas composition on possible multistable behavior. The results have been interpreted in terms of a transition between kinetics- and permeation-controlled regimes, in which the behavior of the reactor is limited, respectively, by the rate of the chemical reactions or the rate of hydrogen permeation. [Display omitted] • Dynamics of membrane reactors in the presence of permeance inhibition were analyzed. • Effect of gas composition and temperature on multistable behavior was investigated. • Relationship between observed multistability and membrane inhibition was assessed. • Multistability due to competition between rate of H 2 production and permeation is observed. [ABSTRACT FROM AUTHOR]

A nonlinear evolutionary partial differential equation is considered that is obtained as a natural (from the physical point of view) generalization of the well-known Cahn–Hilliard equation. The generalized version is supplemented with terms responsible for convection and dissipation. The new version of the equation is considered together with homogeneous Neumann boundary conditions. For such a boundary value problem, local bifurcations of codimensions one and two are studied. In both cases, the existence, stability, and asymptotic representation of spatially inhomogeneous equilibrium states, as well as invariant manifolds formed by such solutions of the boundary value problem, are analyzed. To substantiate the results, methods of the modern theory of infinite-dimensional dynamical systems are used, including the method of integral manifolds and the apparatus of the Poincaré theory of normal forms. Differences between the results of the analysis of bifurcations in the Neumann boundary value problem and the conclusions of the analysis of the periodic boundary value problem studied by the authors of this paper in previous publications are indicated. [ABSTRACT FROM AUTHOR]

We study the dynamics of a recent model of Rothman for long timescale carbon cycle. We reproduce and extend various results of the Rothman model. We present numerical results showing that the model exhibits both stable and unstable limit cycles via Hopf bifurcations as the parameters are varied. We numerically find normal forms of Bautin bifurcations to confirm their criticality. We also extend the analysis of the normal form coefficients to identify where the fold limit cycle bifurcation occurs [ABSTRACT FROM AUTHOR]

The classical lateral–torsional instability of a cantilever beam with continuous elastic lateral restraint and a transverse point-load applied at the free end is discussed here through the lens of asymptotic simplifications. One of our main goals is to provide analytical approximations for the critical buckling load, as well as its dependence on various key non-dimensional groups. The first part of this study is concerned with a scenario in which a doubly symmetric beam is constrained to rotate about a fixed axis situated at an arbitrary height above the shear centroidal axis. The second part examines a beam that features a continuous elastic lateral restraint spanning its entire length. Assuming that the stiffness of the constraint, κ (say), is finite, the buckling equations in the second case are described by a system of two coupled fourth-order differential equations in the lateral displacement and the angle of twist. We show that as κ → ∞ , the problem discussed in the first part provides an outer solution for the aforementioned system; the relevant boundary conditions for the next-to-leading-order outer approximation are also derived using matched asymptotics. Our theoretical findings are confirmed by comparisons with direct numerical simulations of the full buckling problem. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of an ellipsoid of revolution rolling on a horizontal plane under the assumption that there is no slipping at the point of contact and no spinning about the vertical. A reduction of the equations of motion to a fixed level set of first integrals is performed. Permanent rotations corresponding to the rolling of an ellipsoid in a circle or in a straight line are found. A linear stability analysis of permanent rotations is carried out. A complete classification of possible trajectories of the reduced system is performed using a bifurcation analysis. A classification of the trajectories of the center of mass of the ellipsoid depending on parameter values and initial conditions is performed. [ABSTRACT FROM AUTHOR]

The main direction of the fourth part of the work is the study of an autonomous dynamic system in the field of aviation technology, based on the provisions of the qualitative theory of differential equations with the involvement of the phase portrait method. The materials and results of the article are based on the starting points in the decision-making problem (DPR), specified [1], [3] empirically - the formalized direction of research in the comparison of preprego-autoclaved (surfactant) and non-autoclaved VARTM technologies in the manufacture of carbon-plastic aircraft structures according to the prevalence (first of all-quality) (AK) type of highly loaded stringer panels of the wing caisson (VSP) of mainline aircraft. That is, we are talking about a large multi-stage (based on ChTP) technological system. The research as a whole is based on systems analysis (SA) and interpreted structure-functional modeling of ATP. The evaluation toolkit and criterion-evaluation apparatus based on the macro-level ADS and a number of technological structures and other configurations are used [2]. Special points, stability and phase portrait of ADS (quadratic function) were investigated, with the involvement of interpreted elements of the topology of many species, as well as bifurcation points of the system. [ABSTRACT FROM AUTHOR]