Analytical Bifurcation Research Articles

Analytical analysis and bifurcation of pine wilt dynamical transmission with host vector and nonlinear incidence using sustainable fractional approach

To study the dynamical system, it is necessary to formulate a mathematical model to comprehend the dynamics of the diseases that are prevalent around the world by using fractional calculus. A mathematical model is developed with the hypothesis created by adding control and asymptomatic variables to observe the rate of change of pine wilt and the ABC operator is used to turn the model into a fractional ordered model for continuous monitoring. The Boundedness and uniqueness of the developed model are investigated for bounded findings by using Banach space, which are the key properties of such an epidemic model. A newly developed system is examined both qualitatively and quantitatively to determine its stable position, and the verification of flip bifurcation has been made for developed systems. Derived reproductive numbers using the next-generation technique as well as the sensitivity of each involved parameter are verified. The Atangana–Toufik scheme is employed to find the solution for the developed system using different fractional values, which are advanced tools for reliable bounded solutions. Simulations have been made to see the real behavior and effects of pine wilt disease with control and asymptomatic battels in the community. Also, identify the real situation of the spread as well as the control of pine wilt after employing control and asymptomatic battels due to treatment. Such a type of investigation will be useful in investigating the spread of disease as well as helpful in developing control strategies based on our justified outcomes.

The Demographic-Wealth model for cliodynamics.

Cliodynamics is a still a relatively new research area with the purpose of investigating and modelling historical processes. One of its first important mathematical models was proposed by Turchin and called "Demographic-Fiscal Model" (DFM). This DFM was one of the first and is one of a few models that link population with state dynamics. In this work, we propose a possible alternative to the classical Turchin DFM, which contributes to further model development and comparison essential for the field of cliodynamics. Our "Demographic-Wealth Model" (DWM) aims to also model link between population and state dynamics but makes different modelling assumptions, particularly about the type of possible taxation. As an important contribution, we employ tools from nonlinear dynamics, e.g., existence theory for periodic orbits as well as analytical and numerical bifurcation analysis, to analyze the DWM. We believe that these tools can also be helpful for many other current and future models in cliodynamics. One particular focus of our analysis is the occurrence of Hopf bifurcations. Therefore, a detailed analysis is developed regarding equilibria and their possible bifurcations. Especially noticeable is the behavior of the so-called coexistence point. While changing different parameters, a variety of Hopf bifurcations occur. In addition, it is indicated, what role Hopf bifurcations may play in the interplay between population and state dynamics. There are critical values of different parameters that yield periodic behavior and limit cycles when exceeded, similar to the "paradox of enrichment" known in ecology. This means that the DWM provides one possible avenue setup to explain in a simple format the existence of secular cycles, which have been observed in historical data. In summary, our model aims to balance simplicity, linking to the underlying processes and the goal to represent secular cycles.

Dynamic patterns of electroosmosis peristaltic flow of a Bingham fluid model in a complex wavy microchannel

The purpose of this paper is to present a rigorous analysis of streamline patterns and their bifurcation to a viscoplastic Bingham fluid model that involves heat and mass transfer in an electroosmotic flow through a complex wavy microchannel. The Bingham fluid act as a solid medium in the core layer, which divides the channel into three distinct sections utilized to model the problem as a switched dynamical system between these zones. To track multiple steady states (stagnation points) and related trapping phenomena, we perform both analytical and numerical bifurcation analysis of each subsystem with respect to different physical effects such as electrical double layer thickness and Helmholtz-Smoluchowski velocity. The key feature of the technique presented here is its ability to reveal the peristaltic transport characteristics of the Bingham fluid model in the presence or absence of symmetric flow properties. The primary novelty here is the ability to regulate the location and stability of the equilibrium points in the domain of interest. This leads to the detection of global bifurcations that reflect important dynamic elements of the model. Our results highlighted a new category of complex behavior that controls transitions between qualitatively different transport mechanisms, as well as a class of non-classical trapping phenomena.

Bifurcation and bistability in pneumatically actuated periodically porous elastomers

Pneumatic actuation offers a convenient way to tune pattern formation in periodically porous elastomers. In this paper, we study deflation-induced buckling in periodically porous elastomers using both an analytical approach and the finite element method. By applying the Euler–Bernoulli beam model, an analytical bifurcation condition is derived using the energy approach, from which the critical buckling load can be identified analytically. Furthermore, the theoretical results are qualitatively validated by the corresponding finite element ones. It is noted that a unit cell will be assigned in finite element simulations, and usually, such a choice can be varied. In order to ensure the consistency of different unit cells in finite element simulations, we assign two representative cases and identify the proper boundary conditions that should be applied. By applying the same solution procedure, we carry out a stability analysis and analytically reveal the effect of an initial geometrical imperfection on the buckling behavior. Finally, the variation of band gaps in pneumatically activated periodic elastomers is studied by means of the finite-element method. It is hoped that the present analysis could provide useful insight not only into pattern formation in periodically porous elastomers under deflation but also into the choice of the unit cell in numerical simulations.

Dynamical behavior of analytical soliton solutions, bifurcation analysis, and quasi-periodic solution to the (2+1)-dimensional Konopelchenko–Dubrovsky (KD) system

Nonlinear evolution equations (NLEEs) are extensively used to establish the elementary propositions of natural circumstances. In this work, we study the Konopelchenko–Dubrovsky (KD) equation which depicts non-linear waves in mathematical physics with weak dispersion. The considered model is investigated using the combination of generalized exponential rational function (GERF) method and dynamical system method. The GERF method is utilized to generate closed-form invariant solutions to the (2+1)-dimensional KD model in terms of trigonometric, hyperbolic, and exponential forms with the assistance of symbolic computations. Moreover, 3D, 2D combined line graph and their contour graphics are displayed to depict the behavior of obtained solitary wave solutions. The model is observed to have multiple soliton profiles, kink-wave profiles, and periodic oscillating nonlinear waves. These generated solutions have never been published in the literature. All the newly generated soliton solutions are checked by putting them back into the associated system with the soft computation via Wolfram Mathematica. Moreover, the system is converted into a planer dynamical system using a certain transformation and the analysis of bifurcation is examined. Furthermore, the quasi-periodic solution is investigated numerically for the perturbed system by inserting definite periodic forces into the considered model. With regard to the parameter of the perturbed model, two-dimensional and three-dimensional phase portraits are plotted.

Stability and bifurcation analysis of Generation IV reactors via point reactor models with temperature reactivity feedback

While the stability of present-day reactors is well studied, stability analysis of Generation IV reactors is in its very beginning. In this paper, the stability of point nuclear reactor models with temperature reactivity feedbacks is analyzed. The examined models are referred to as G-L reactor models, which are point reactor models with G delayed neutron-emission groups, L temperature lumps (regions) and a linear reactivity feedback for each temperature. G-L reactor models are applicable for modeling four out of the six Generation IV reactor types (Gas Fast Reactor, Lead Fast Reactor, Sodium Fast Reactor and Very High Temperature Reactor).New analytical stability and bifurcation results of general G-L reactor models and of the G1-L2 reactor model are derived. These analytical results add a different perspective to the previous studies of Generation IV reactors, most of which are based only on numerical analysis. In particular, the codimension-1 bifurcations of general G-L reactor models are characterized, and a classification of the linear stability region in the G1-L2 model is derived.Then, stability and bifurcation analysis of a G1-L2 model of a Generation IV HTR-type microreactor is performed, using both analytical methods and the numerical continuation method. The nominal design point of this microreactor is shown to be globally asymptotically stable, while in other regions of the parameter space, a bistability of the operating steady state and a limit cycle, or of two limit cycles, is demonstrated.

Analytical bifurcation behaviors of a host–parasitoid model with Holling type III functional response

This topic presents a study on a host–parasitoid model with a Holling type III functional response. In population dynamics, when host density rises, the parasitoid response initially accelerates due to the parasitoid’s improved searching efficiency. However, above a certain density threshold, the parasitoid response will reach a saturation level due to the influence of reducing the handling time. Thus, we incorporated a Holling type III functional response into the model to characterize such a phenomenon. The dynamics of the current model are discussed in this paper. We first obtained the existence and local stability conditions of the positive fixed point of the model. Furthermore, we investigated the bifurcation behaviors at the positive fixed point. More specifically, we used bifurcation theory and the center manifold theorem to prove that the model possess both period doubling and Neimark–Sacker bifurcations. Then, the chaotic behavior of the model, in the sense of Marotto, is proven. Furthermore, we apply a state-delayed feedback control strategy to control the complex dynamics of the present model. Finally, numerical examples are provided to support our analytic results.

Generalization of the Kuramoto model to the Winfree model by a symmetry breaking coupling

We construct a nontrivial generalization of the paradigmatic Kuramoto model by using an additional coupling term that explicitly breaks its rotational symmetry resulting in a variant of the Winfree model. Consequently, we observe the characteristic features of the phase diagrams of both the Kuramoto model and the Winfree model depending on the degree of the symmetry breaking coupling strength for unimodal frequency distribution. The phase diagrams of both the Kuramoto and the Winfree models resemble each other for symmetric bimodal frequency distribution for a range of the symmetry breaking coupling strength except for region shift and difference in the degree of spread of the macroscopic dynamical states and bistable regions. The dynamical transitions in the bistable states are characterized by an abrupt (first-order) transition in both the forward and reverse traces. For asymmetric bimodal frequency distribution, the onset of bistable regions depends on the degree of the asymmetry. Large degree of the symmetry breaking coupling strength promotes the synchronized stationary state, while a large degree of heterogeneity, proportional to the separation between the two central frequencies, facilitates the spread of the incoherent and standing wave states in the phase diagram for a low strength of the symmetry breaking coupling. We deduce the low-dimensional equations of motion for the complex order parameters using the Ott-Antonsen ansatz for both unimodal and bimodal frequency distributions. We also deduce the Hopf, pitchfork, and saddle-node bifurcation curves from the evolution equations for the complex order parameters mediating the dynamical transitions. Simulation results of the original discrete set of equations of the generalized Kuramoto model agree well with the analytical bifurcation curves.

Analytical bifurcation and strong resonances of a discrete Bazykin–Berezovskaya predator–prey model with Allee effect

This paper investigates multiple bifurcations analyses and strong resonances of the Bazykin–Berezovskaya predator–prey model in depth using analytical and numerical bifurcation analysis. The stability conditions of fixed points, codim-1 and codim-2 bifurcations to include multiple and generic bifurcations are studied. This model exhibits transcritical, flip, Neimark–Sacker, and [Formula: see text], [Formula: see text], [Formula: see text] strong resonances. The normal form coefficients and their scenarios for each bifurcation are examined by using the normal form theorem and bifurcation theory. For each bifurcation, various types of critical states are calculated, such as potential transformations between the one-parameter bifurcation point and different bifurcation points obtained from the two-parameter bifurcation point. To validate our analytical findings, the bifurcation curves of fixed points are determined by using MatcontM.

Periodic necking of misfit hyperelastic filaments embedded in a soft matrix

The necking instability is a precursor to tensile failure and rupture of materials. A quasistatically loaded free-standing uniaxial specimen typically exhibits necking at a single location, thus corresponding to a long wavelength bifurcation mode. If confined to a substrate or embedded in a matrix the same filament can exhibit periodic necking and fragmentation thus creating segments of finite length. While such periodic instabilities have been extensively studied in ductile metal filaments and thin sheets, less is known about necking in hyperelastic materials. Nonetheless, in recent years, there has been a renewed interest in the role of necking in novel materials, for the advancement of fabrication processes and to explain fragmentation phenomena observed in 3D printed active biological matter. In both cases materials are not well described by the existing frameworks that employ J2 deformation theory of plasticity, and existing studies do not account for the significant role of misfit stretches that may emerge in these systems through chemical, or biological contraction. To address these limitations, in this paper we begin by experimentally demonstrating the role of the surrounding matrix on the necking and fragmentation of a compliant filament embedded in a tunable rubber matrix. Using a generalized hyperelastic model with strain softening, our analytical bifurcation analysis explains the experimental observations and is shown to agree with numerical predictions. The analysis reveals three distinct bifurcation modes: the long wavelength necking, thus recovering the Considère criterion; the periodic necking observed in our experiments; and a short wavelength mode that is characterized by localization along the center cord of the filament and is independent of the film-to-matrix stiffness ratio. We find that the softening coefficient and the filament misfit stretch can significantly influence the stability threshold and observed wavelength, respectively. Our results can guide the design and fabrication of novel composite materials and can explain the fragmentation processes observed in active biological materials.

Elasto-capillary necking, bulging and Maxwell states in soft compressible cylinders

Localized pattern formations and “two-phase” deformations are studied theoretically in soft compressible cylinders subject to surface tension and axial loading through several force-controlled loading scenarios. By drawing upon known results for separate yet mathematically similar elastic localization problems, a concise family of analytical bifurcation conditions for localized bulging or necking are derived in terms of a general compressible strain energy function. The effect of material compressibility and strain-stiffening behaviour on the bifurcation point is analysed, and comparisons between our theoretical bifurcation conditions and the corresponding numerical simulation results of Dortdivanlioglu and Javili (Extreme Mech. Lett. 55, 2022) are made. It is then explained how the fully developed “two-phase” (Maxwell) state which evolves from the initial localized bifurcation solution can be comprehensively understood using the simple analytical expressions for the force parameters corresponding to the primary axial tension deformation. The power of this simple analytical approach in validating numerical simulation results for elastic localization and phase-separation-like problems of this nature is highlighted.

Period-3 motions to chaos in a periodically forced nonlinear-spring pendulum.

In this paper, the complete bifurcation dynamics of period-3 motions to chaos are obtained semi-analytically through the implicit mapping method. Such an implicit mapping method employs discrete implicit maps to construct mapping structures of periodic motions to determine complex periodic motions. Analytical bifurcation trees of period-3 motions to chaos are determined through nonlinear algebraic equations generated through the discrete implicit maps, and the corresponding stability and bifurcations of periodic motions are achieved through eigenvalue analysis. To study the periodic motion complexity, harmonic amplitudes varying with excitation amplitudes are presented. Once more, significant harmonic terms are involved in periodic motions, and such periodic motions will be more complex. To illustrate periodic motion complexity, numerical and analytical solutions of periodic motions are presented for comparison, and the corresponding harmonic amplitudes and phases are also presented for such periodic motions in the bifurcation trees of period-3 motions to chaos. Similarly, other higher-order periodic motions and bifurcation dynamics for the nonlinear spring pendulum can be determined. The methods and analysis presented herein can be applied for other nonlinear dynamical systems.

Production of microalgae biomass in a continuous stirred bioreactor: Analysis of microalgae-bacteria competition mediated by nitrogen and organic carbon

A model-based analysis of the competition between microalgae and bacteria mediated by nitrogen and organic carbon in a continuous bioreactor with recycle is presented. The analysis is aimed to evaluate if the ability of microalgae to uncouple nitrogen and carbon uptake can allow to prevent the contamination of bacteria with negligible carbon accumulation or to sustain the coexistence of microalgae and bacteria. To this purpose, a Droop-like model and a multiplicative Monod model are used to describe microalgae and bacteria growth kinetics, respectively. The regions of the bioreactor parameter space ensuring steady state stability of microalgae monocultures, microalgae-bacteria consortia and bacteria monocultures are identified by analytical bifurcation analysis of the mass balance equations governing bioreactor dynamics. Expressions in closed form are derived for the stability boundaries of the steady states as bioreactor parameters vary, providing guidelines for plant design and control.

Reassessment of contact restrictions and testing campaigns against COVID-19 via spatio-temporal modeling

Since the earliest outbreak of COVID-19, the disease continues to obstruct life normalcy in many parts of the world. The present work proposes a mathematical framework to improve non-pharmaceutical interventions during the new normal before vaccination settles herd immunity. The considered approach is built from the viewpoint of decision makers in developing countries where resources to tackle the disease from both a medical and an economic perspective are scarce. Spatial auto-correlation analysis via global Moran’s index and Moran’s scatter is presented to help modulate decisions on hierarchical-based priority for healthcare capacity and interventions (including possible vaccination), finding a route for the corresponding deployment as well as landmarks for appropriate border controls. These clustering tools are applied to sample data from Sri Lanka to classify the 26 Regional Director of Health Services (RDHS) divisions into four clusters by introducing convenient classification criteria. A metapopulation model is then used to evaluate the intra- and inter-cluster contact restrictions as well as testing campaigns under the absence of confounding factors. Furthermore, we investigate the role of the basic reproduction number to determine the long-term trend of the regressing solution around disease-free and endemic equilibria. This includes an analytical bifurcation study around the basic reproduction number using Brouwer Degree Theory and asymptotic expansions as well as related numerical investigations based on path-following techniques. We also introduce the notion of average policy effect to assess the effectivity of contact restrictions and testing campaigns based on the proposed model’s transient behavior within a fixed time window of interest.

Analytical and Numerical Bifurcation Analysis of Circuits Based on Nonlinear Resonators

This work presents new frequency-domain methods for the analysis and simulation of circuits based on a nonlinear resonator, with operation ranges delimited by turning points. Insightful analytical conditions fulfilled at these turning points are derived, which will enable an identification of the effect of each circuit element on their location in the solution curve. The cusp points, or co-dimension two bifurcations at which two turning points merge into one, thus delimiting the multivalued intervals, are directly calculated for the first time to our knowledge. In addition, a new numerical method, compatible with the use of commercial harmonic balance, is presented for the straightforward tracing of the multivalued solution curves, together with a new procedure to determine the locus of turning points in terms of any two analysis parameters. This relies on the use of a new mathematical condition to obtain a surface of turning points in the space defined by the two parameters and the input power. The methods have been applied to a wireless power-transfer system based on a recently proposed configuration, obtaining very good agreement with the experimental results.

Sequential Periodic Motions in a Vibration-Assisted, Regenerative, Nonlinear Turning-Tool System

This paper studies the dynamics and bifurcations of a vibration-assisted, regenerative, nonlinear turning-tool system using an implicit mapping method. Machine vibration has been studied for a century for the improvement of machine accuracy and metal removal rate. In fact, this problem is unsolved yet. This is because such dynamical systems are involved in nonlinearity, discontinuity and time-delay. Thus, a comprehensive understanding of nonlinear machining dynamics with time-delay is indispensable. In this paper, period-[Formula: see text] motions in the turning machine-tool system are studied through specific mapping structures, and the corresponding stability and bifurcations of the period-[Formula: see text] motion are determined through the eigenvalue analysis. The analytical bifurcation scenarios for two sets of sequential period-[Formula: see text] motions in a turning-tool system are presented. Numerical simulations of period-[Formula: see text] motions are carried out to verify the prediction of periodic motions. The complex dynamics of vibration-assisted machining with strong nonlinearity are presented, which can provide a good overview for nonlinear dynamics of machine-tool systems.

Analytical optical pulses and bifurcation analysis for the traveling optical pulses of the hyperbolic nonlinear Schrödinger equation

Analytical forms of optical pulses for the hyperbolic nonlinear Schrodinger equation are studied by the extended tanh expansion method and another new method successfully. Some graphical presentations of these analytical optical pulses are presented. Bifurcation analysis of the optical pulses of the hyperbolic nonlinear Schrodinger equation is also presented. All possible phase plots are depicted based on physical parameters. The hyperbolic nonlinear Schrodinger equation supports solitary, periodic, kink, anti-kink and most important superperiodic optical pulses. This study is applicable to understand the features of nonlinear pulses in optical fiber.

Localised bifurcation in soft cylindrical tubes under axial stretching and surface tension

We investigate localised bulging or necking in an incompressible, hyperelastic cylindrical tube under axial stretching and surface tension. Three cases are considered in which the tube is subjected to different constraints. In case 1 the inner and outer surfaces are traction-free and under surface tension, whilst in cases 2 and 3 the inner and outer surfaces (respectively) are fixed to prevent radial displacement and surface tension. However, each free surface in these latter two cases is still under surface tension. We first state the analytical bifurcation conditions for localisation and then validate them numerically whilst determining whether localisation is preferred over bifurcation into periodic modes. It is shown that bifurcation into a localised solution is unattainable in case 1 but possible and favourable in cases 2 and 3. In contrast, in case 1 any bifurcation must necessarily take the form of a periodic mode with a non-zero wave number. Our results are validated using Finite Element Method (FEM) simulations.

Analytical and numerical bifurcation analysis of a forest ecosystem model with human interaction

We perform both analytical and numerical bifurcation analysis of an alternating forest and grassland ecosystem model coupled with human interaction. The model consists of two nonlinear ordinary differential equations incorporating the human perception of the value of the forest. The system displays multiple steady states corresponding to different forest densities as well as regimes characterized by both stable and unstable limit cycles. We derive analytically the conditions with respect to the model parameters that give rise to various types of codimension-one criticalities such as transcritical, saddle-node, and Andronov–Hopf bifurcations and codimension-two criticalities such as cusp and Bogdanov–Takens bifurcations at which homoclinic orbits occur. We also perform a numerical continuation of the branches of limit cycles. By doing so, we reveal turning points of limit cycles marking the appearance/disappearance of sustained oscillations. Such critical points that cannot be detected analytically give rise to the abrupt loss of the sustained oscillations, thus leading to another mechanism of catastrophic shifts.

Trade-off between filtering and symmetry breaking mean-field coupling in inducing macroscopic dynamical states

We study the manifestation of the competing interaction between the mean-field intensity and the symmetry breaking coupling on the phenomenon of aging transition in an ensemble of limit-cycle oscillators comprising of active and inactive oscillators. Further, we also introduce filtering in both the intrinsic and extrinsic variables of the mean-field diffusive coupling to investigate the counter-intuitive effect of both filterings. We find that large values of the mean-field intensity near unity favor the oscillatory nature of the ensemble, whereas low values favor the onset of the aging transition and heterogeneous dynamical states such as cluster oscillation death and chimera death states even at low values of the symmetry breaking coupling strength. Heterogeneous dynamical states predominates at large values of the coupling strength in all available parameter spaces. We also uncover that even a weak intrinsic filtering favors the aging transition and heterogeneous dynamical states, while a feeble extrinsic filtering favors the oscillatory state. Chimera death state is observed among the active oscillators for the first time in the aging literature. Our results can lead to engineering the dynamical states as desired by an appropriate choice of the control parameters. Further, the transition from the oscillatory to the aging state occurs via an inverse Hopf bifurcation, while the transition from the aging state to the cluster oscillation death states emerges through a supercritical pitch-fork bifurcation. The deduced analytical bifurcation curves are in good agreement with the numerical boundaries of the observed dynamical states.