Numerical Bifurcation Analysis Research Articles

Discontinuity-Induced Dynamics in the Conductance-Based Adaptive Exponential Integrate-and-Fire Model.

In this article, we present a computational study of the Conductance-Based Adaptive Exponential (CAdEx) integrate-and-fire neuronal model, focusing on its multiple timescale nature, and on how it shapes its main dynamical regimes. In particular, we show that the spiking and so-called delayed bursting regimes of the model are triggered by discontinuity-induced bifurcations that are directly related to the multiple-timescale aspect of the model, and are mediated by canard solutions. By means of a numerical bifurcation analysis of the model, using the software package COCO, we can precisely describe the mechanisms behind these dynamical scenarios. Spike-increment transitions are revealed. These transitions are accompanied by a fold and a period-doubling bifurcation, and are organised in parameter space along an isola periodic solutions with resets. Finally, we also unveil the presence of a homoclinic bifurcation terminating a canard explosion which, together with the presence of resets, organises the delayed bursting regime of the model.

Mean-field analysis of synaptic alterations underlying deficient cortical gamma oscillations in schizophrenia.

Deficient gamma oscillations in the prefrontal cortex (PFC) of individuals with schizophrenia (SZ) are proposed to arise from alterations in the excitatory drive to fast-spiking interneurons (E I) and in the inhibitory drive from these interneurons to excitatory neurons (I E). Consistent with this idea, prior postmortem studies showed lower levels of molecular and structural markers for the strength of E I and I E synapses and also greater variability in E I synaptic strength in PFC of SZ. Moreover, simulating these alterations in a network of quadratic integrate-and-fire (QIF) neurons revealed a synergistic effect of their interactions on reducing gamma power. In this study, we aimed to investigate the dynamical nature of this synergistic interaction at macroscopic level by deriving a mean-field description of the QIF model network that consists of all-to-all connected excitatory neurons and fast-spiking interneurons. Through a series of numerical simulations and bifurcation analyses, findings from our mean-field model showed that the macroscopic dynamics of gamma oscillations are synergistically disrupted by the interactions among lower strength of E I and I E synapses and greater variability in E I synaptic strength. Furthermore, the two-dimensional bifurcation analyses showed that this synergistic interaction is primarily driven by the shift in Hopf bifurcation due to lower E I synaptic strength. Together, these simulations predict the nature of dynamical mechanisms by which multiple synaptic alterations interact to robustly reduce PFC gamma power in SZ, and highlight the utility of mean-field model to study macroscopic neural dynamics and their alterations in the illness.

Research on nonlinear instability of blended-wing-body aircraft based on experimental bifurcation analysis

Focusing on the nonlinear instability problem of the blended-wing-body aircraft, first, numerical bifurcation analysis is adopted to obtain catastrophe points and branches where the elevator is taken as a continuous parameter. Second, based on virtual flight tests and bifurcation theory, an experimental bifurcation analysis method is developed. Open-loop and closed-loop experiments are carried out, and the corresponding experimental bifurcation diagrams are obtained. The comparative analysis shows that the experimental bifurcation analysis enables the aircraft to make a response that is close to reality under the influence of nonlinear and unsteady aerodynamics and has a more intuitive embodiment of instability characteristics, providing an accurate prediction of departure. The open-loop experiment predicts a sudden longitudinal nose-up at an angle of attack of 7.4°. The closed-loop experiment can transform the open-loop unstable branch into a stable branch. However, roll instability still occurs at a critical angle of attack of 16.0°. These research results provide important dynamic information for the stable flight and the design of the controller.

Stick–slip oscillations in the low feed linear motion of a grinding machine due to dry friction and backlash

This paper deals with mechanical modelling and numerical bifurcation analysis of stick–slip oscillations that plague extremely low feed grinding operations. Based on experimental observations, a novel two degree of freedom mechanical model of a grinding machine feed drive system is formulated, which incorporates Stribeck-type dry friction, position and velocity controller dynamics, and actuator backlash. Loss and re-establishment of contact between the feed drive elements is modelled through both rigid-body impacts and a contact-stiffness model. The resulting piecewise-smooth equations of motion are subjected to detailed stability and bifurcation analysis with the help of shooting and collocation based numerical continuation tools. Major focus is attributed to the influence of the feed velocity and the control-loop parameters as well as the identification of stable, stick–slip free parameter regimes. Finally, a controller enhancement strategy is proposed, based on event-driven integrator reset rules, to help limit the amplitude of arising limit-cycle oscillations.

Activity patterns in ring networks of quadratic integrate-and-fire neurons with synaptic and gap junction coupling.

We consider a ring network of quadratic integrate-and-fire neurons with nonlocal synaptic and gap junction coupling. The corresponding neural field model supports solutions such as standing and traveling waves, and also lurching waves. We show that many of these solutions satisfy self-consistency equationswhich can be used to follow them as parameters are varied. We perform numerical bifurcation analysis of the neural field model, concentrating on the effects of varying gap junction coupling strength. Our methods are generally applicable to a wide variety of networks of quadratic integrate-and-fire neurons.

Strong coupling yields abrupt synchronization transitions in coupled oscillators

Coupled oscillator networks often display transitions between qualitatively different phase-locked solutions—such as synchrony and rotating wave solutions—following perturbation or parameter variation. In the limit of weak coupling, these transitions can be understood in terms of commonly studied phase approximations. As the coupling strength increases, however, predicting the location and criticality of transition, whether continuous or discontinuous, from the phase dynamics may depend on the order of the phase approximation—or a phase description of the network dynamics that neglects amplitudes may become impossible altogether. Here we analyze synchronization transitions and their criticality systematically for varying coupling strength in theory and experiments with coupled electrochemical oscillators. First, we analyze bifurcations analysis of synchrony and splay states in an abstract phase model and discuss conditions under which synchronization transitions with different criticalities are possible. In particular, we show that such conditions can be understood by considering the relative contributions of higher harmonics to the phase dynamics. Second, we illustrate that transitions with different criticality indeed occur in experimental systems. Third, we highlight that the amplitude dynamics observed in the experiments can be captured in a numerical bifurcation analysis of delay-coupled oscillators. Our results showcase that reduced order phase models may miss important features that one would expect in the dynamics of the full system. Published by the American Physical Society 2024

Theoretical and Numerical Bifurcation Analysis of a Discrete Predator–Prey System of Ricker Type with Weak Allee Effect

Theoretical and Numerical Bifurcation Analysis of a Discrete Predator–Prey System of Ricker Type with Weak Allee Effect

Data-driven and equation-free methods for neurological disorders: analysis and control of the striatum network.

The striatum as part of the basal ganglia is central to both motor, and cognitive functions. Here, we propose a large-scale biophysical network for this part of the brain, using modified Hodgkin-Huxley dynamics to model neurons, and a connectivity informed by a detailed human atlas. The model shows different spatio-temporal activity patterns corresponding to lower (presumably normal) and increased cortico-striatal activation (as found in, e.g., obsessive-compulsive disorder), depending on the intensity of the cortical inputs. By applying equation-free methods, we are able to perform a macroscopic network analysis directly from microscale simulations. We identify the mean synaptic activity as the macroscopic variable of the system, which shows similarity with local field potentials. The equation-free approach results in a numerical bifurcation and stability analysis of the macroscopic dynamics of the striatal network. The different macroscopic states can be assigned to normal/healthy and pathological conditions, as known from neurological disorders. Finally, guided by the equation-free bifurcation analysis, we propose a therapeutic close loop control scheme for the striatal network.

Hand Vibration Reduction Using Nonlinear Vibration Absorber for the Vibro-Impact Hammer Model

Abstract This work investigates the effectiveness of a passive vibration control device to minimize the amplitude of vibrations from a hand-held impact machine (HHIM) transmitted to a hand-arm system (HAS). For this purpose, we employ a simplified lumped parameter model of an HHIM and the hand-arm system consisting of lumped masses, springs, and viscous dampers. For a better understanding of the system dynamics, we use a vibro-impact model for a more accurate representation of HHIM–ground interactions. We then examine the effectiveness of a cubic nonlinear vibration absorber in minimizing these undesirable vibrations via a nonlinear bifurcation analysis of the system. A parametric study on the numerical bifurcation analysis of the system reveals the criticality of different design parameters of the proposed nonlinear vibration absorber on the system dynamics. Our findings demonstrate that an appropriate selection of absorber parameters could significantly change the amplitude and the periodicity of vibrations transmitted to the hand. Furthermore, we observe the existence of complex motions such as period-2, period-4, and chaotic attractors in the system for the given values of the absorber and operating parameters.

Electrothermal Instabilities in Barium-Titanate-Based Ceramics

An electrothermal analysis for barium-titanate-based ceramics is presented, combining the Heywang–Jonker model for the electric resistivity with a heat dissipation mechanism based on natural convection and radiation in a one-dimensional model on the device level with voltage as the control parameter. Both positive-temperature-coefficient (PTC) and negative temperature coefficient (NTC) effects are accounted for through the double Schottky barriers at the grain boundaries of the material. The problem formulated in this way admits uniform and non-uniform multiple-steady-state solutions that do not depend on the external circuit. The numerical bifurcation analysis reveals that the PTC effect gives rise to several multiplicites above the Curie point, whereas the NTC effect is responsible for the thermal runaway (temperature blowup). The thermal runaway phenomenon as a potential thermal shock could be among the possible reasons for the observed thermomechanical failures (delamination fracture). The theoretical results for the NTC regime and the thermal runaway are in agreement with the experimental flash sintering results obtained for barium titanate, and 3% and 8% yttria-stabilized zirconia.

Infection-induced increases to population size during cycles in a discrete-time epidemic model

One-dimensional discrete-time population models, such as those that involve Logistic or Ricker growth, can exhibit periodic and chaotic dynamics. Expanding the system by one dimension to incorporate epidemiological interactions causes an interesting complexity of new behaviors. Here, we examine a discrete-time two-dimensional susceptible-infectious (SI) model with Ricker growth and show that the introduction of infection can not only produce a distinctly different bifurcation structure than that of the underlying disease-free system but also lead to counter-intuitive increases in population size. We use numerical bifurcation analysis to determine the influence of infection on the location and types of bifurcations. In addition, we examine the appearance and extent of a phenomenon known as the ‘hydra effect,’ i.e., increases in total population size when factors, such as mortality, that act negatively on a population, are increased. Previous work, primarily focused on dynamics at fixed points, showed that the introduction of infection that reduces fecundity to the SI model can lead to a so-called ‘infection-induced hydra effect.’ Our work shows that even in such a simple two-dimensional SI model, the introduction of infection that alters fecundity or mortality can produce dynamics can lead to the appearance of a hydra effect, particularly when the disease-free population is at a cycle.

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.

Theoretical and numerical analysis of period-doubling bifurcation in sandwich systems

Period-doubling bifurcation will appear when a stiff film is embedded into a compliant matrix or constrained to a soft substrate and undergoes in-plane compression. Though period-doubling bifurcation in bilayer systems have received extensive investigation, the mechanism of intricate post-buckling phenomena in sandwich systems remains an open question due to the inherent high nonlinearity. In this paper, a theoretical model for period-doubling bifurcation in sandwich systems is established and verified by finite element simulations. The proposed theoretical model is generalized to neo-Hookean materials and transformed into an equivalent nonlinear oscillator through an internal-to-external force transformation. Theoretical analysis reveals that an increase in the degree of up-down symmetry suppresses period-doubling bifurcation. Especially, when both the substrate and superstrate are identical, the influence of nonlinear deformation partly reflected by even high terms on the pressure difference on the film disappears, as a result, a novel period-doubling-like pattern appears due to the competitive effects from both sides strike a balance.

Global bifurcation for a Beddington–DeAngelis and Tanner predator–prey reaction–diffusion system with prey‐taxis

The aim of this paper is to establish a precise illustration for the structure of the nonconstant steady states for a Beddington–DeAngelis and Tanner predator–prey reaction–diffusion system with prey‐taxis. We treat the nonlinear prey‐taxis as a bifurcation parameter to analyze the bifurcation structure of the system. Furthermore, the exported global bifurcation theorem, under a rather natural condition, offers the existence of nonconstant steady states. In the proof, a priori estimates of steady states will play an important role. The local stability analysis with a numerical simulation and bifurcation analysis are given. Finally, some conclusions including biological meanings are performed to summarize our main analytic results and future investigations.

Bifurcation analysis of a discrete-time prey-predator model

This paper investigates the importance of studying the dynamics of predator-prey systems and the specific significance of Neimark-Sacker and period-doubling bifurcations in discrete-time prey-predator models. By conducting a numerical bifurcation analysis and examining bifurcation diagrams and phase portraits, we present important results that differentiate our study from others in the field. Firstly, our analysis reveals the occurrence of Neimark-Sacker and period-doubling bifurcations in the model under certain parameter values. These bifurcations lead to the emergence of stable limit cycles characterized by complex and unpredictable dynamics. This finding emphasizes the inherent complexity and nonlinearity of predator-prey systems and contributes to a deeper understanding of their dynamics. Additionally, our study highlights the advantages and limitations of employing discrete-time models in population dynamics research. The use of discrete-time models allows for a more tractable analysis while still capturing significant aspects of ecological systems. In conclusion, this study holds importance in shedding light on the dynamics of predator-prey systems and the specific role of Neimark-Sacker and period-doubling bifurcations. Our findings contribute to the understanding of predator-prey systems and offer implications for ecological management strategies.

Bifurcations and synchrony in a ring of delayed Wilson–Cowan oscillators

Ring structures are crucial in network neuroscience, enabling the integration of neural information through closed loop circuits within feedback systems. Here, we use numerical bifurcation analysis to explore time delay effects on a ring of delay-coupled Wilson–Cowan masses. Investigating a low-dimensional ‘self-coupled’ version of the aforementioned system, we uncover the bifurcation structure of the synchronization manifold, and unveil a diverse array of dynamic synchronization patterns that emerge as a consequence of Hopf branch crossings and subsequent higher-order bifurcations. Analysis of the full system reveals transverse instabilities in the synchronized state for large regions of parameter space, with the ring network architecture promoting various dynamics depending on the balance between coupling strength and delay time. Under weak coupling, emergent oscillations are generally synchronous or anti-phase synchronous, with transitions between them triggered by a torus bifurcation of a periodic orbit. Regions of synchronous and anti-phase synchronous solutions are delineated by weakly chaotic borders due to the breakdown of the torus. As coupling strength increases, the bifurcation diagram displays more overlapped branching structure, resulting in increasingly complicated, multistable dynamics.

Dynamics of pumping effect in 1 pipe-2 tanks flow configuration with periodic excitation

This paper is dedicated to analyzing the pumping effect in rigid 1 pipe-2 tanks flow configuration proposed by G. Propst. Our main purpose is to study the dynamic behaviors of the periodic solutions of the governing differential equation, as the periodic solution will result in the periodically forced pump in the configuration. We establish a sufficient condition for the existence of the harmonic solution by applying the topological degree theory. Additionally, we utilize the regular perturbation method and averaging method to examine the stability of the harmonic solution, as well as the existence and stability of the second-order subharmonic solution of the equation. Through numerical bifurcation analysis, we present the bifurcation diagrams of the periodic solutions, which reveal that higher-order subharmonic solutions with varying stability can be generated through period-doubling bifurcation of the periodic solutions.

Generalized FitzHugh–Nagumo model with tristable dynamics: Deterministic and stochastic bifurcations

We propose an extension of the Fitzhugh-Nagumo model, which possesses a regime of three coexisting stable states: resting equilibrium and two stable oscillatory states. Such a regime is absent in the original Fitzhugh-Nagumo model but it is known to exist in higher-dimensional conductance based neuronal models. Thus, the proposed system provides a simpler two-dimensional model with such a property. Using numerical bifurcation analysis as well as Lindsted’s method, we explore parameter regions and bifurcations leading to the tristability. Considering the effects of channel fluctuations as Gaussian white noise, phenomenological bifurcations of the corresponding stochastic system are analyzed using a Fokker–Planck approach. We investigate how the interplay between the system parameters and the noise intensity induces a switching of neural activities between silence, subthreshold, and spiking.

Bifurcation Analysis Software and Chaotic Dynamics for Some Problems in Fluid Dynamics Laminar–Turbulent Transition

The analysis of bifurcations and chaotic dynamics for nonlinear systems of a large size is a difficult problem. Analytical and numerical approaches must be used to deal with this problem. Numerical methods include solving some of the hardest problems in computational mathematics, which include system spectral and algebraic problems, specific nonlinear numerical methods, and computational implementation on parallel architectures. The software structure that is required to perform numerical bifurcation analysis for large-scale systems was considered in the paper. The software structure, specific features that are used for successful bifurcation analysis, globalization strategies, stabilization, and high-precision implementations are discussed. We considered the bifurcation analysis in the initial boundary value problem for a system of partial differential equations that describes the dynamics of incompressible ABC flow (3D Navier–Stokes equations). The initial stationary solution is characterized by the stability and connectivity to the main solutions branches. Periodic solutions were considered in view of instability transition problems. Finally, some questions of higher dimensional attractors and chaotic regimes are discussed.

Dynamic properties of a reaction–diffusion predator–prey model with nonlinear harvesting: A linear and weakly nonlinear analysis

In this work, we have studied the spatiotemporal dynamics of a predator–prey model with selective nonlinear saturated harvesting of both species. The study of the non-diffusive system includes analytical and numerical analysis of local and global bifurcations. Bifurcation analysis of the non-diffusive system unveils that the system may exhibit transcritical, Hopf, saddle–node and homoclinic bifurcation with increased carrying capacity and harvesting efforts. The simulation results further reveal that the paradox of enrichment may disappear in a harvested system for a higher value of carrying capacity. The spatiotemporal study consists of an analysis of different pattern-forming instabilities. A critical ratio of the diffusivities of predator to prey for the occurrence of Turing instability is established. Further, a weakly nonlinear analysis is provided to derive the amplitude equation and predict the form of the pattern. The key highlight of the work is that prey harvesting facilitates spatiotemporal chaos, and predator diffusion controls it. Furthermore, intensive prey harvesting forces spatial segregation, whereas predator harvesting relaxes it. It is also observed that spatial segregation enriches the overall species biomass. A unique observation of this study is the occurrence and nonoccurrence of spatiotemporal chaos in the Hopf-Turing parametric space.