Dynamic Bifurcation Research Articles

Complex rhythm and synchronization of half-center oscillators under electromagnetic induction

<p>Half-center oscillators are typical small circuits that are crucial for understanding CPG. The complex rhythms of CPG are closely related to certain diseases, such as epilepsy. This paper considered the influence of electromagnetic induction on the discharge mode of the half-center oscillators. First, we analyzed the response of individual firing neuron rhythms to electromagnetic induction when the slow-variable parameters vary. We also discussed the changes in the dynamic bifurcation structure when the intensity of electromagnetic induction varies. Furthermore, we determined the effects of mutually inhibitory and self-inhibitory synaptic parameters on the firing rhythm of the half-center oscillators. The different responses induced by electromagnetic induction interventions, showed that mutually inhibitory synapses modulate the firing rhythm weakly and self-inhibition synapses have a significant impact on firing rhythm. Finally, with the change of synaptic parameter values, the combined effects of autapse and mutually inhibitory synapses on the discharge rhythm of half-center oscillators were analyzed in symmetric and asymmetric autapse modes. It was found that the synchronous state of the half-center oscillators had a more robust electromagnetic induction response than the asynchronous state.</p>

Comparative Analysis of Nonlinear Dynamics of an Angular Velocity System of 2-DOF Aerial Manipulator with Different Physical Parameters

The two-degree-of-freedom (2-DOF) aerial manipulator is composed of a quadrotor aircraft and a 2-DOF manipulator, which significantly expands the scope of grabbing and transporting objects. After the manipulator is installed on the quadrotor, the manipulator and the load will cause serious interference to the quadrotor, resulting in difficulty of system control and even instability. This paper presents a mathematical model of the angular velocity system of the 2-DOF aerial manipulator. The model considers the influence of the manipulator and the load on the quadrotor. Based on this model, the nonlinear dynamics of the angular velocity system of the 2-DOF aerial manipulator are analyzed by solving the equilibrium points, calculating the Lyapunov exponents, analyzing the dynamic bifurcation diagram, and drawing the dynamic region distribution map. It is found that angular velocity can produce the dynamic behaviors of sink, period-doubling, and chaos under certain circumstances. By analyzing the nonlinear dynamic behaviors of the angular velocity system under different manipulator postures, different manipulator configurations, different load masses, and different load resistances, the stability of the angular velocity system is analyzed to guide the use of the aerial manipulator more safely and efficiently.

Modeling the Bifurcation Dynamics of Rumor Propagation in the Spatial Environment

The harm caused by rumors is immeasurable. Studying the dynamic characteristics of rumors can help control their spread. In this paper, we propose a nonsmooth rumor model with a nonlinear propagation rate. First, we utilize the positive invariant regions to prove the boundedness of solutions. Second, we analyze the conditions for the existence of equilibrium points in both the left and right systems. Additionally, we confirm the occurrence of saddle-node bifurcation in the left system. Next, by considering the influence of spatial diffusion, we establish the conditions for Turing instability. Then we discuss the conditions for spatial homogeneous and inhomogeneous Hopf bifurcations in the left and right systems, respectively. We differentiate between supercritical and subcritical bifurcations using the Lyapunov coefficient. Furthermore, we examine the conditions for the existence of discontinuous Hopf bifurcation at the demarcation point. Finally, in the numerical simulation section, we validate our theorems on Turing patterns. We also investigate the impact of parameter changes on rumor propagation and conclude that an increase in the psychological inhibitory factor significantly reduces the rate of rumor propagation, providing an effective strategy for curbing rumors. To that end, we fit actual data to our system and the results are excellent, confirming the validity of the system.

Symmetry, Multistability and Antimonotonicity of a Shinriki Oscillator with Dual Memristors

In this paper, a type of modified dual memristive Shinriki oscillator is constructed with a flux-controlled absolute-type memristor and a voltage-controlled generic memristor, and the proposed oscillator with abundant dynamical behaviors, including the multistability and antimonotonicity, is comprehensively studied through dynamical distribution graphs, bifurcation diagrams, Lyapunov exponents and phase portraits. It is found that the passive/active state of memristor, which means different characteristics in the [Formula: see text]–[Formula: see text] domain with positive and negative parameters of the elements, can affect the state of the oscillator. For example, if the memristor is active, the oscillator will change more frequently in the multistable region. Also, it is noted that, for inherent initial-related symmetry and circuit structures with duality, both phenomena have strong symmetric characteristics and opposite evolution trends modulated by values of corresponding components. Especially, the bubbles, which are symmetric about parameters with duality and own complex evolution laws, have rarely been explored in previous works. In addition, the memristive oscillator is modularized based on field programmable gate array (FPGA) technology, and the multiple coexisting attractors are captured, which verifies the accuracy of the numerical results.

Bifurcation analysis of a fractional-order Cohen–Grossberg neural network with three delays

This paper pays more attention to the dynamic bifurcations of a fractional-order Cohen–Grossberg neural network in the sense of the Caputo fractional derivative with three delays. In the beginning, the dynamical behaviors of the system in several different cases are discussed when time delay can be viewed as zero, and the corresponding Hopf bifurcation critical values are derived. Subsequently, the system’s dynamics with three delays are investigated for three cases where none are zero. The reduction method of transcendental terms is applied to solve the third transcendental term in the characteristic equation. The exact bifurcation conditions are obtained by choosing one of the delays as the bifurcation parameter, respectively. It is worth noting that previous studies on multi-delay systems only discuss the case where the multi-delay can be transformed into a single delay or two delays based on certain transformations. This paper explores how to study the problem of a system with three delays without transforming the delays. In addition, the effects of time delays on the critical points of the system are examined. Finally, to verify the correctness of the proposed results, numerical simulations are addressed.

New chirp soliton solutions for the space-time fractional perturbed Gerdjikov–Ivanov equation with conformable derivative

The space–time perturbed fractional Gerdjikov–Ivanov equation is the main topic of this work, together with quintic nonlinearity and self-steepening, as it involves several intricate physical phenomena including nonlinearity, self-steepening and fractional calculus, where the fractional derivative is described by employing a conformable derivative. In addition, the governing equation is transformed into an integer-order ordinary differential equation by using an appropriate fractional complex transformation. Under certain restrictions, a direct algebraic method is employed to investigate the structures of chirp soliton solutions enfolding hyperbolic functional terms. The dynamic behaviour and bifurcation of equilibria of the system are thoroughly examined; chirp soliton solutions under specified constraints are investigated and the evolving profiles of the obtained solutions are visualized. Moreover, this research offers valuable perspectives on the behaviour of chirp solitons under specific conditions, which have practical applications in nonlinear physical systems and optical communication systems. The significant contribution of this work is the investigation and obtaining of novel chirp soliton solutions with hyperbolic functional terms under particular limitations using a novel approach. It further extends the prior approaches by treating difficult fractional differential equations from a fresh angle, offering new tools, and closely examining soliton solutions.

Stability analysis of Filippov prey–predator model with fear effect and prey refuge

Mathematical ecosystems play a crucial role in our comprehension and conservation of ecology. Within these ecosystems, prey exhibits protective instincts that compel refuging behaviors to avoid predation risk. When the ratio of prey to predators falls below a threshold, prey seeks refuge. However, when prey is abundant relative to predators, these protective instincts are overridden as prey ventures out to forage. Therefore, this study develops a Filippov prey–predator model with fear effect on prey and switching of prey refuge behavior based on the ratio of prey to predators. Analytical and numerical approaches are used to address the dynamic behaviors, bifurcation sets, existence, and stability of various equilibria in this model. Additionally, the regions of sliding and crossing segments are analyzed. The bifurcation sets of pseudo-equilibrium and local and global sliding bifurcations are investigated. The numerical simulations are conducted to investigate the interplay between fear factor and other relevant parameters within the Filippov model, such as the threshold ratio and prey refuge. These investigations shed light on the influence of them in the model. The results indicate that increasing the fear factor results in a decrease in both prey and predator densities, thereby changing the behavior of the dynamics from a limit cycle oscillation to a stable state and vice versa. Notably, despite these population changes, neither species faces complete extinction.

Acceleration sensing based on the bifurcation dynamics of parametrically excited mode-localized resonators

A parametrically excited mode-localized accelerometer is designed using the bifurcation phenomenon to improve the robustness of the fluctuation of the driving voltage and damping while maintaining high sensitivity. A dynamic multi-physics model was established while considering both mechanical and electrostatic nonlinearities. The equation was solved by method of multiple scales and verified by harmonic balanced method coupled with the asymptotic numerical method. Two types of bifurcation exist in amplitude frequency response, namely Saddle-Node bifurcation and Supercritical Hopf bifurcation. By introducing Saddle-Node bifurcation, the response amplitude and measurement range can be improved by 100% and 1000%, respectively, while the sensitivity of the amplitude ratio is about 2 orders of magnitude higher than that based on the frequency ratio. At the Supercritical Hopf bifurcation point, a small acceleration will change the topological structure from Supercritical Hopf to Saddle-Node bifurcation. The variation in the amplitude ratio of the Supercritical Hopf point with acceleration is similar to the sign function, which leads to an extremely high sensitivity of 10000%/g in a dynamic range of ±0.001 g. Moreover, the Supercritical Hopf bifurcation point is not affected by the amplitude of the excitation voltage and damping coefficient, which provides a new method for improving the sensing robustness. Ethical Compliance: All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki Declaration and its later amendments or comparable ethical standards. Conflict of Interest declaration: The authors declare that they have NO affiliations with or involvement in any organization or entity with any financial interest in the subject matter or materials discussed in this manuscript.

Reconnection in a pinch

A recently published analysis of current sheets has updated the classic Harris 1D static solution by considering multiple classes of charged particle trajectories in a generalized and dynamic current sheet. It uses a 1D PIC simulation to describe dynamic pinching and bifurcation of the current sheet. These 1D results strongly suggest that properties of the inflowing plasma, including the plasma beta, have an important effect on the equilibrium thickness of the pinched current sheet. Since 1D studies cannot describe magnetic reconnection, the time appears right to carry such 1D studies over to 2D or 3D simulations to explore current sheet thickness effects on reconnection. The Magnetospheric Multiscale Mission (MMS), with its well-resolved multipoint measurements of collisionless plasma and fields, has found that collisionless reconnection is accompanied by non-adiabatic motions of electrons that only occur in magnetic structures with a narrow scale comparable to electron inertial lengths (de). The recent 1D studies suggest that a plasma pinch to such scales may only occur for inflowing magnetized plasmas with relatively low plasma beta. We conclude that a parametric exploration of simulated and observed reconnection inflow conditions, particularly plasma beta, should shed light on the enablement of reconnection in collisionless plasmas.

Setting Safe Operation Conditions for Acetyl Chloride Hydrolysis through Dynamic Modelling and Bifurcation Analysis

Acetyl chloride hydrolysis is a highly sensitive exothermic reaction that has presented several industrial safety issues. In the present study, a multiparameter mathematical model, previously developed and applied to simulate the oscillatory thermal behavior of an experimental continuous stirred tank reactor, was used to determine the static/dynamic bifurcation behavior of this reactive system. The values predicted by the model showed good agreement with the experimental data reported in the literature. Full topological classification of its fixed points and iterative maps was obtained: unique solutions (stable and unstable), multiple solutions, cyclic envelope, and bifurcation objects of codimension 1 (e.g., fold and Hopf’s points) and codimension 2 (e.g., cusp and generalized Hopf and Bogdanov-Takens points) have been uncovered. The emphasis of the analysis is to determine safe operating conditions through understanding these topological features and manipulating the reactor design and operating parameters.

Diverse soil microbial communities may mitigate climate system bifurcation

Permafrost thaw reinforces greenhouse gas production. The resulting growth of greenhouse gases (especially methane) in the atmosphere may trigger a climate system bifurcation. To study how changes in the microbial community may cause greenhouse gas emissions from permafrost and produce bifurcations in climate temperature dynamics we propose a conceptual nonlinear model that couples an atmospheric dynamics model with the population structure of microbial communities. Our model is mathematically well-posed, demonstrating that microbial population diversity can significantly lower planetary surface temperatures, an outcome contingent upon the average population parameters and their standard deviations.

Simulation of the dynamic cracking of brittle materials using a nonlocal damage model with an effective strain rate effect

In this study, a nonlocal multiscale damage model that considers the effect of a strain rate is presented. Moreover, this model is combined with the scaled boundary finite element method (SBFEM) and quadtree mesh for dynamic crack propagation in quasi-brittle materials. The scaling centre of each SBFEM subdomain was selected as the material point, and the bond connecting two material points was referred to as the material bond. Microscopic damage was quantified based on the stretch rate of the material bond, and macroscale topological damage was calculated by taking the weighted average of the microscopic damage over the bonds within the influence domain. The bell distribution function is utilised as the weight function in the calculation of the nonlocal weight function. To account for the impact of the strain rate during dynamic loading, an effective rate R is introduced to enhance multiscale damage by modifying the damage threshold and bond softening coefficient throughout the analysis process. By analysing the energetic degradation function of the phase field, which establishes a connection between the macroscale topological damage and energy-based damage, a nonlocal multiscale damage model was incorporated into the framework of the SBFEM. Quadtree mesh discretization was employed to achieve rapid and high-quality multilevel mesh generation by effectively utilising the hanging nodes. Three typical dynamic cracking examples were simulated using the proposed model, and the results were compared with the relevant experimental results and references. The results show that the method is suitable for accurate dynamic cracking simulation in quasi-brittle materials. Moreover, the results indicate that the proposed method can spontaneously simulate dynamic crack bifurcation without introducing additional crack bifurcation criteria. For a compact tensile test of concrete under different loading rates, the method proposed in this study can accurately simulate the corresponding failure modes. The results also show that without considering the effect of the strain rate, the damage develops rapidly in an unconstrained state, which is inconsistent with the actual case.

One-dimensional mechanism of gaseous deflagration-to-detonation transition

A one-dimensional mechanism of deflagration to detonation transition is identified and investigated by an asymptotic analysis in the double limit of large activation energy and small Mach number of the laminar flame velocity. The unsteady analysis concerns the self-accelerating tip of an elongated flame in a smooth walled tube. The flame on the tip, considered as plane and orthogonal to the tube axis, is pushed from behind by the longitudinal flow resulting from the cumulative effect of the radial flows of burned gas issued from the lateral flame of the finger-like front (called backflow in the following). The analysis of the one-dimensional dynamics is performed by coupling the flame structure with the downstream-running compression waves propagating in the external flows. A critical elongation is identified from which the slightest increase in elongation leads to a pressure runaway producing the flame blow-off. The dynamics of the inner structure of the laminar flame on the tip which is accelerated by the self-induced backflow is characterized by a finite-time singularity of the reacting flow in the form of a dynamical saddle-node bifurcation.

Hopf Bifurcation and Global Dynamics Analysis of Generalized Sprott L System

Hopf Bifurcation and Global Dynamics Analysis of Generalized Sprott L System

Dynamic bifurcation of a column when vertically impacting on a rigid plane

The dynamic bifurcation of a column is investigated when it impacts a rigid plane vertically, which is completely different from the classical Eulerian static buckling. The variation law of the dimensionless critical buckling time with the dimensionless impact velocity is investigated, and the difference in dimensionless critical buckling times between columns without gravity and those with gravity is investigated. The concept of the dimensionless critical buckling velocity is proposed, which is the dimensionless form of the minimum velocity at which a column can buckle. Regardless of the different perspectives considered, the findings of the study show that either the dimensionless critical buckling time or the dimensionless critical buckling velocity can be used to determine whether buckling has occurred. The dimensionless displacement response of the column is calculated at the first contact between the column and the rigid plane. Different dimensionless initial defects in the column will result in different dimensionless displacement responses. The nonlinear effect has some influence on the analysis results.

Exploring the Dynamics of Simple Inhibition Systems in Continuous Stirred-Tank Reactor: Mathematical Modelling and Bifurcation Analysis

The simple enzyme inhibition systems consist of competitive inhibition, noncompetitive inhibition and uncompetitive inhibition. In this work, we incorporated these simple inhibition systems in the continuous stirred-tank reactor (CSTR) and analysed the models using some techniques from dynamical systems and bifurcation analysis. Our aim is to investigate the behaviours of such systems and compare their overall dynamics. The phase portrait is constructed to simulate possible behaviours such as stable steady states, stable limit cycle, bistability between the steady state and the stable limit cycle and bistability between two steady states. The systems undergo bifurcational changes in dynamics as enzyme concentration, dilution rate and proportional control constant are varied. Moreover, we conducted a codimension two bifurcation analysis to examine the joint effects of dilution rate and proportional control constant on the systems behaviours. Our results revealed distinct dynamics for each inhibition system. Increasing the dilution rate led to a transition from low to high substrate concentrations, with competitive inhibition showing the highest tipping (or bifurcation) point where dynamical regimes change due to intense substrate-inhibitor competition. Elevating enzyme concentration reduced substrate concentration, particularly in non-competitive inhibition systems due to higher conversion rates. Furthermore, the proportional control constant had varying effects depending on the specific inhibition system. These findings emphasize the on the combined influences of distinct chemical procoesses in controlling reactor heat and optimizing bioprocess efficiency, considering the unique characteristics of each inhibition system. Overall, the dynamical study on these simple inhibition systems enables us to improve our understanding on the chemical processes involving enzymes with multiple types of inhibitors and may give some insights in its controlling process.

Caputo and Conformable Fractional Order Guava Model for Biological Pest Control: Discretization, Stability and Bifurcation

Abstract Two predator-prey model describing the guava borers and natural enemies are studied in this paper. Positivity, existence, and uniqueness of the solution, global and local stability analysis of the fixed points of the first model based on the Caputo fractional operator are studied. By adding piecewise constant functions to the second model including conformable fractional operator allows us to transition discrete dynamical system via discretization process. Applying Schur-Cohn criterion to the discrete system, we hold some regions where the equilibrium points in the discretized model are local asymptotically stable. We prove that discretized model displays supercritical Neimark–Sacker bifurcation at the equilibrium point. Theoretical and numerical results show that the discretized system demonstrates richer dynamic properties such as quasi-periodic solutions, bifurcation, and chaotic dynamics than the fractional order model with Caputo operator. All theoretical results are interpreted biologically and the optimum time interval for the harvesting of the guava fruit is given.

Bifurcation dynamics in a stochastic population model with Beverton–Holt growth

The objective of this paper is to investigate the survivability of the Beverton–Holt growth model under multiplicative noise and further explore both dynamical and phenomenological bifurcations. The theoretical results demonstrate that for the parameter Λ≤0, population extinction occurs, whereas for Λ>0, a unique stationary distribution is observed with an explicit expression provided for its density function. Specifically, the parameter Λ serves as a threshold for determining the persistence and extinction of the population, while simultaneously acting as the dynamical bifurcation point of the model. Moreover, as the parameter Λ crosses α22 and α2, two phenomenological bifurcations occur in the stochastic model.

Period-doubling bifurcation to chaos in a pure-sextic soliton fiber laser

We numerically investigate the dynamics of period-doubling bifurcation to chaos in a pure-sextic soliton (PSS) fiber laser dominated by sixth-order dispersion (SOD). By properly setting the cavity parameters, i.e., pump power and SOD amount, it is found that the PSS experiences multiple bifurcations before entering into chaos. The imperfect balance of SOD and self-phase modulation (SPM) provides a new scenario of period-doubling bifurcation dynamics that PSS exhibits the dynamical energy redistribution between the main lobe and the oscillating tails. In particular, a critical state that the PSS evolves spontaneously between the period-doubling bifurcation and chaos has also been demonstrated. These findings will give new insights into the period-doubling bifurcation dynamics as well as chaos in dissipative optical systems.

Population heterogeneity in vaccine coverage impacts epidemic thresholds and bifurcation dynamics

Population heterogeneity, especially in individuals' contact networks, plays an important role in transmission dynamics of infectious diseases. For vaccine-preventable diseases, outstanding issues like vaccine hesitancy and availability of vaccines further lead to nonuniform coverage among groups, not to mention the efficacy of vaccines and the mixing pattern varying from one group to another. As the ongoing COVID-19 pandemic transitions to endemicity, it is of interest and significance to understand the impact of aforementioned population heterogeneity on the emergence and persistence of epidemics. Here we analyze epidemic thresholds and characterize bifurcation dynamics by accounting for heterogeneity caused by group-dependent characteristics, including vaccination rate and efficacy as well as disease transmissibility. Our analysis shows that increases in the difference in vaccination coverage among groups can render multiple equilibria of disease burden to exist even if the overall basic reproductive ratio is below one (also known as backward bifurcation). The presence of other heterogeneity factors such as differences in vaccine efficacy, transmission, mixing pattern, and group size can each exhibit subtle impacts on bifurcation. We find that heterogeneity in vaccine efficacy can undermine the condition for backward bifurcations whereas homophily tends to aggravate disease endemicity. Our results have practical implications for improving public health efforts by addressing the role of population heterogeneity in the spread and control of diseases.