Bifurcation Set Research Articles

A Novel Method for Analyzing Global Bifurcation and Global Hysteresis of FGM Truncated Conical Shell Structure

In the dynamic bifurcation and dynamic hysteresis of a system, our first consideration is the tiny changes in time that can lead to variations in the vector field, potentially causing the occurrence of system dynamic bifurcation and dynamic hysteresis. This paper presents a novel method for analyzing the global bifurcation and hysteresis of Functionally Graded Material (FGM) truncated conical shell structure under aerodynamics and in-plane force along meridian near internal resonances. This method involves introducing the ratio of vector fields as a periodic perturbation parameter in the system’s nonlinear second-order ordinary differential governing equations. This allows for the analysis of the nonlinear ordinary differential equations as algebraic equations, yielding algebraic expressions that only involve parameters and do not contain state variables. Subsequently, the global bifurcation set and global hysteresis set of FGM truncated conical shell structure are obtained, providing the parameters interval ranges for the stability and instability of FGM truncated conical shell structure. By utilizing MAPLE software for numerical simulation, the images of the global bifurcation set and global hysteresis set about the amplitude of in-plane load are first simulated numerically. Subsequently, the amplitude graphs of the equilibrium points are numerically simulated for validation. The results demonstrate a perfect alignment between the images of the global bifurcation set and global hysteresis set and the data from the amplitude graphs of the equilibrium point. Due to the periodicity of the periodic perturbation parameter, it shuttles between different persistent regions as other parameters change, leading to the generation of chaotic solutions in the governing equations. The validity of the obtained results is confirmed through comparisons with existing literature.

Firing patterns and fast–slow dynamics in an N-type LAM-based FitzHugh–Nagumo circuit

The diversity of firing patterns and their bifurcation mechanisms of a neuron circuit are crucial for exploiting bionic applications. This paper constructs an N-type locally active memristor (LAM)-based FitzHugh–Nagumo (FHN) circuit by replacing the tunnel-diode in the original FHN circuit. Numerical simulations and hardware measurements reveal that the N-type LAM-based FHN circuit can reproduce rich neuromorphic firing patterns of quasi-periodic/periodic bursting behaviors and chaotic/periodic spiking behaviors. The bursting and spiking behaviors are triggered by a low-frequency stimulus and a high-frequency one, respectively. Besides, the fold and Hopf bifurcation sets are depicted. Then, the bifurcation mechanisms for the quasi-periodic and periodic bursting behaviors via Hopf/Hopf and Hopf/fold bifurcations are theoretically deduced by time-domain waveform and equilibrium trajectory. The numerical results and hardware experiments exhibit the effectiveness of the proposed memristive FHN circuit in reproducing abundant firing patterns of bursting and spiking behaviors.

An entropy based spatial–temporal cube with its application to assess stress of overburden due to mining

Underground coal seam mining significantly alters the stress and energy distribution within the overlying rock, leading to eventual structural degradation. Therefore, it is imperative to quantitatively identify the temporal and spatial characteristics of stress evolution of overlying rock caused by mining. This paper introduces a novel rock stress model integrating entropy and a spatial–temporal cube. Similar material model tests are used to identify the abrupt entropy changes within the mining rock, and the trend analysis is carried out to describe the spatial–temporal evolution law of stress during mining. Experimental findings indicate elevated stress levels in the unmined rock preceding and following the panel, as well as within specific rock strata above it. Definitively, dynamic stress arches within the surrounding rock of the stope predominantly bear and distribute the load and pressure from the overlying rock, and each stress mutation is accompanied by a sudden stress entropy change. Over time, z-score shows that the noticeable reduction in mining-induced overburden stress becomes increasingly pronounced, especially in the water-conducting fracture zone. The model's bifurcation set serves as the comprehensive criterion for the entropy-induced sudden changes in the rock system, signifying overall failure.

Nonlinear energy harvesting system with multiple stability

The nonlinear energy harvesting systems of the forced vibration with an electron-mechanical coupling are widely used to capture ambient vibration energy and convert mechanical energy into electrical energy. However, the nonlinear response mechanism of the friction induced vibration (FIV) energy harvesting system with multiple stability and stick-slip motion is still unclear. In the current paper, a novel nonlinear energy harvesting model with multiple stability of single-, double- and triple-well potential is proposed based on V-shaped structure spring and the belt conveying system. The dynamic equations for the energy harvesting system with multiple stability and self-excited friction are established by using Euler-Lagrangian equations. Secondly, the static characteristics of the nonlinear restoring force, the friction force, and the potential energy surfaces are obtained to show the nonlinear stiffness, multiple equilibrium points, discontinuous behaviors and multiple well responses. Then, the equilibrium surface of bifurcation sets for the autonomous system is given to show the third-order quasi zero stiffness (QZS3), fifth-order quasi zero stiffness (QZS5), double well (DW) and triple well (TW). The co-dimension bifurcation sets of the self-excited vibration system are analyzed and the corresponding phase portraits for the coexistent of multiple limit cycles are obtained. Furthermore, the analytical formula of amplitude frequency response of the approximated system are obtained by the complex harmonic method. The response amplitudes of charge, current, voltage and power of the forced electron-mechanical coupled vibration system for QZS3, QZS5, DW and TW are analyzed by using the numerically solution. Finally, a prototype of FIV energy harvesting system is manufactured and the experimental system is setup. The experimental works of static restoring forces, damping forcse and the electrical outputs are well agreeable with the numerical results, which testified the proposed FIV energy harvesting model.

Chaos and bifurcations of a two-dimensional hepatitis C virus model with hepatocyte homeostasis.

In this paper, we delve into the intricate local dynamics at equilibria within a two-dimensional model of hepatitis C virus (HCV) alongside hepatocyte homeostasis. The study investigates the existence of bifurcation sets and conducts a comprehensive bifurcation analysis to elucidate the system's behavior under varying conditions. A significant focus lies on understanding how changes in parameters can lead to bifurcations, which are pivotal points where the qualitative behavior of the system undergoes fundamental transformations. Moreover, the paper introduces and employs hybrid control feedback and Ott-Grebogi-Yorke strategies as tools to manage and mitigate chaos inherent within the HCV model. This chaos arises due to the presence of flip and Neimark-Sacker bifurcations, which can induce erratic behavior in the system. Through the implementation of these control strategies, the study aims to stabilize the system and restore it to a more manageable and predictable state. Furthermore, to validate the theoretical findings and the efficacy of the proposed control strategies, extensive numerical simulations are conducted. These simulations serve as a means of confirming the theoretical predictions and provide insight into the practical implications of the proposed control methodologies. By combining theoretical analysis with computational simulations, the paper offers a comprehensive understanding of the dynamics of the HCV model and provides valuable insights into potential strategies for controlling and managing chaos in such complex biological systems.

Mixed-mode oscillations of an atomic force microscope in tapping mode.

In the phenomenon of mixed-mode oscillations, transitions between large-amplitude and small-amplitude oscillations may lead to anomalous jitter in the probe of a tapping mode atomic force microscope (TM-AFM) during the scanning process, thereby affecting the accuracy and clarity of the topographical images of the tested sample's surface. This work delves deeply into various mixed-mode oscillations and the corresponding formation mechanisms in TM-AFM under low-frequency resonant excitation. Through a detailed analysis of bifurcation sets of the fast subsystem, we found that the system's mixed-mode oscillations encompass the typical two coexisting branches and the novel three coexisting branches of equilibrium point attractors. In the stable case, a certain transition pattern in phase trajectory can be observed involving two jumps and four jumps, switching between quiescent and spiking states. In the bi-stable case, the trajectory undergoes distinct transitions decided by whether to pass through or crossover the middle branch of attractors when bifurcation occurs. By applying basin of attraction and fast-slow analysis methods, we unfold the dynamic mechanism of mixed-mode oscillations with distinct switching patterns. Our research contributes to a better understanding of complex oscillations of TM-AFM and provides valuable insights for improving image quality and measurement precision while mitigating detrimental oscillations.

Dynamics of the generalized Rosenzweig–MacArthur model in a changing and patchy environment

In this paper, we propose a generalized Rosenzweig–MacArthur predator–prey model with two patches and Allee effects in predators. In a constant environment, for the single-patch model we show the existence of degenerate Bogdanov–Takens bifurcation with codimension up to 4 and degenerate Hopf bifurcation with codimension up to 3, which indicate that the nilpotent cusp of codimension 4 is the organizing center of the bifurcation set. Our results show that Allee effects in predators can not only induce much more complex bifurcations and dynamics (such as three limit cycles, multitype tristability), but also serve as a destabilizing factor to make predators extinction (deterministically) when the environment is sufficiently enriched, which provides strong support for the so-called “paradox of enrichment”. For the two-patch model, we numerically reveal the existence of eight positive steady states including two new coexisting attractors, which means that population dispersal can help species coexist or increase the population abundance. In a changing environment, for the single-patch model we find that the population can track unstable states, fast positive or slow negative environmental change can be beneficial to predators by delaying or even avoiding extinction; while for the two-patch model we observe some interesting phenomena: fast negative (or slow positive) environmental change can make predators in different patches trend to extinction synchronously, and rate-induced different regime shifts.

Exploring the geometry of the bifurcation sets in parameter space

By studying a nonlinear model by inspecting a p-dimensional parameter space through (p-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p-1)$$\\end{document}-dimensional cuts, one can detect changes that are only determined by the geometry of the manifolds that make up the bifurcation set. We refer to these changes as geometric bifurcations. They can be understood within the framework of the theory of singularities for differentiable mappings and, in particular, of the Morse Theory. Working with a three-dimensional parameter space, geometric bifurcations are illustrated in two models of neuron activity: the Hindmarsh–Rose and the FitzHugh–Nagumo systems. Both are fast-slow systems with a small parameter that controls the time scale of a slow variable. Geometric bifurcations are observed on slices corresponding to fixed values of this distinguished small parameter, but they should be of interest to anyone studying bifurcation diagrams in the context of nonlinear phenomena.

The crease flow on null hypersurfaces

The crease flow, replacing the Hamiltonian system used for the evolution of crease sets on black hole horizons, is introduced and its bifurcation properties for null hypersurfaces are discussed. We state the conditions of nondegeneracy and typicality for the crease submanifolds, and find their normal forms and versal unfoldings (codimension 3). The allowed boundary singularities are thus prescribed by the Arnold–Kazaryan–Shcherbak theorem for 3-parameter versal families, and hence identified as swallowtails and Whitney umbrellas of particular kinds. We further present the bifurcation diagrams describing crease evolution at the crossings of the bifurcation sets and elsewhere, and a typical example is studied. Some remarks on the connection of these results to the crease evolution on black hole horizons are also given.

Global Phase Portraits of Piecewise Quadratic Differential Systems with a Pseudo-Center

This paper deals with the global dynamics of planar piecewise smooth differential systems constituted by two different vector fields separated by one straight line that passes through the origin. From a quasi-canonical family of piecewise quadratic differential systems with a pseudo-focus point at the origin, which has six parameters, we investigate the subfamilies where the origin is indeed a pseudo-center. For such subfamilies, we classify its global phase portraits in the Poincaré disk and the associated bifurcation sets.

Codimension-one bifurcation analysis and chaos of a discrete prey–predator system

In this paper, we explore local dynamics at equilibrium points, the existence of bifurcation sets and codimension-one bifurcation analysis of a discrete prey–predator system with Holling type-II functional response. Further, OGY and Hybrid control strategies are utilized to control chaos in the under study discrete model due to the occurrence of Neimark–Sacker and flip bifurcations. Finally, numerical simulations are given to verify theoretical results.

Investigating rock mutation characteristics and damage state warning model based on energy conversion

AbstractRock engineering achieves the secondary stress balance through rock mass structure adjustment, where energy conversion is throughout and associated closely with rock deformation and damage. In this study, a series of triaxial compression tests were conducted on red sandstone to investigate these features. The results showed that the damage state of red sandstone specimens presented five stages under different confining pressure, corresponding to the multistage evolution characteristic of the energy conversion. In the case of the dissipation energy conversion ratio (η), it showed five stages: a gradual increase, decreasing gradually and reaching a minimum value, increasing gradually, increasing with growth rate, and accelerated growth, therein the strong nonlinearity reflected the stability and instability of the internal structure of the rock and had the basic characteristics of the mutation theory, therefore the damage state warning model was established on just that. The relation between the η and time fitted by a four‐rank potential function had a fitting parameter (R2) larger than 0.9, and the bifurcation set of the η calculated by the damage state warning model had twice stages less than 0. The second stage, which occurred near the minimum value of the η and run through the plastic deformation stage, could be used to predict rock damage and fracture, and it was proven feasible by acoustic emission (AE) precursor and better than AE warning. This research can enrich the methods for identifying rock damage state and provide reference for revealing the occurrence and development mechanism of various rock instability disasters.

Bifurcations to bursting oscillations in memristor-based FitzHugh-Nagumo circuit

To characterize neuronal firing activities and elucidate its bifurcation mechanisms, a memristor-based FitzHugh-Nagumo (FHN) circuit is designed based on the FHN circuit architecture combined with a first-order memristive simulator, and its normalized system with periodic and quasi-periodic bursting oscillations is established. With the change of externally applied excitation, the memristor-based FHN system has the time-varying equilibrium point where the number, position and stability evolve slowly over time. In an evolution period of the time-domain waveform, different fold and/or Hopf bifurcations are triggered, resulting in periodic or quasi-periodic bursting oscillations. To explain the intrinsic bifurcation mechanisms, the fold and Hopf bifurcation sets are built and the transitions between the resting and spiking states are demonstrated, thus identifying the Hopf/fold and Hopf/Hopf bursting oscillations. Finally, based on the circuit simulation model, analog circuit simulations and hardware circuit measurements are developed for the memristor-based FHN circuit to confirm MATLAB numerical simulations. In addition, it is worth noting that the proposed circuit is a simple non-autonomous memristive neuron circuit that is particularly easy to physically implement.

Bifurcations and global dynamics of a predator–prey mite model of Leslie type

AbstractIn this paper, we study a predator–prey mite model of Leslie type with generalized Holling IV functional response. The model is shown to have very rich bifurcation dynamics, including subcritical and supercritical Hopf bifurcations, degenerate Hopf bifurcation, focus‐type and cusp‐type degenerate Bogdanov–Takens bifurcations of codimension 3, originating from a nilpotent focus or cusp of codimension 3 that acts as the organizing center for the bifurcation set. Coexistence of multiple steady states, multiple limit cycles, and homoclinic cycles is also found. Interestingly, the coexistence of two limit cycles is guaranteed by investigating generalized Hopf bifurcation and degenerate homoclinic bifurcation, and we also find that two generalized Hopf bifurcation points are connected by a saddle‐node bifurcation curve of limit cycles, which indicates the existence of global regime for two limit cycles. Our work extends some results in the literature.

Bifurcation structure of interval maps with orbits homoclinic to a saddle-focus

UDC 517.9 We study homoclinic bifurcations in an interval map associated with a saddle-focus of (2, 1)-type in Z 2 -symmetric systems. Our study of this map reveals a homoclinic structure of the saddle-focus, with bifurcation unfolding guided by the codimension-two Belyakov bifurcation. We consider three parameters of the map corresponding to the saddle quantity, splitting parameter, and the focal frequency of the smooth saddle-focus in a neighborhood of homoclinic bifurcations. We symbolically encode the dynamics of the map in order to find stability windows and locate homoclinic bifurcation sets in a computationally efficient manner. The organization and possible shapes of homoclinic bifurcation curves in the parameter space are examined, taking into account the symmetry and discontinuity of the map. Sufficient conditions for stability and local symbolic constancy of the map are presented. This study provides insights into the structure of homoclinic bifurcations of the saddle-focus map, furthering comprehension of low-dimensional chaotic systems.

Bifurcation analysis and chaos in a discrete Hepatitis B virus model

In this paper, we have delved into the intricate dynamics of a discrete-time Hepatitis B virus (HBV) model, shedding light on its local dynamics, topological classifications at equilibrium states, and pivotal epidemiological parameters such as the basic reproduction number. Our analysis extended to exploring convergence rates, control strategies, and bifurcation phenomena crucial for understanding the behavior of the HBV system. Employing linear stability theory, we meticulously examined the local dynamics of the HBV model, discerning various equilibrium states and their topological classifications. Subsequently, we identified bifurcation sets at these equilibrium points, providing insights into the system's stability and potential transitions. Further, through the lens of bifurcation theory, we conducted a comprehensive bifurcation analysis, unraveling the intricate interplay of parameters that govern the HBV model's behavior. Our investigation extended beyond traditional stability analysis to explore chaos and convergence rates, enriching our understanding of the dynamics of the understudied HBV model. Finally, we validated our theoretical findings through numerical simulations, confirming the robustness and applicability of our analysis in real-world scenarios. By integrating biological and epidemiological insights into our mathematical framework, we offered a holistic understanding of the dynamics of HBV transmission dynamics, with implications for public health interventions and disease control strategies.

Structurally Unstable Synchronization and Border-Collision Bifurcations in the Two-Coupled Izhikevich Neuron Model

This study investigates a structurally unstable synchronization phenomenon observed in the two-coupled Izhikevich neuron model. As the result of varying the system parameter in the region of parameter space close to where the unstable synchronization is observed, we find significant changes in the stability of its periodic motion. We derive a discrete-time dynamical system that is equivalent to the original model and reveal that the unstable synchronization in the continuous-time dynamical system is equivalent to border-collision bifurcations in the corresponding discrete-time system. Furthermore, we propose an objective function that can be used to obtain the parameter set at which the border-collision bifurcation occurs. The proposed objective function is numerically differentiable and can be solved using Newton’s method. We numerically generate a bifurcation diagram in the parameter plane, including the border-collision bifurcation sets. In the diagram, the border-collision bifurcation sets show a novel bifurcation structure that resembles the “strike-slip fault” observed in geology. This structure implies that, before and after the border-collision bifurcation occurs, the stability of the periodic point discontinuously changes in some cases but maintains in other cases. In addition, we demonstrate that a border-collision bifurcation set successively branches at distinct points. This behavior results in a tree-like structure being observed in the border-collision bifurcation diagram; we refer to this structure as a border-collision bifurcation tree. We observe that a periodic point disappears at the border-collision bifurcation in the discrete-time dynamical system and is simultaneously replaced by another periodic point; this phenomenon corresponds to a change in the firing order in the continuous-time dynamical system.

Dynamical systems of self-organized segregation

ABSTRACT We re-consider Schelling’s (1971) bounded neighborhood model as put into the form of a dynamical system by Haw and Hogan (2018). The aim is to determine how tolerance can prevent (or lead to) segregation. In the case of a single neighborhood, we explain the occurring bifurcation set, thereby correcting a scaling error. In the case of two neighborhoods, we correct a major error and derive a dynamical system that does satisfy the modeling assumptions made by Haw and Hogan (2020), staying as close as possible to their construction. We find that stable integration is then only possible if the populations in the two neighborhoods have the option to be in neither neighborhood. In the absence of direct movement between the neighborhoods, the problem is furthermore equivalent to independent single neighborhood problems.

A double-zero bifurcation in a Lorenz-like system

The Lorenz system presents a double-zero bifurcation (a double-zero eigenvalue with geometric multiplicity two). However, its study by means of standard techniques is not possible because it occurs for a non-isolated equilibrium. To circumvent this difficulty, we add in the third equation a new term, Dz2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Dz^2$$\\end{document}. In this Lorenz-like system, the analysis of the double-zero bifurcation of the equilibrium at the origin guarantees, for certain values of the parameters, the existence of a heteroclinic cycle between the two equilibria located on the z-axis. The numerical continuation in parameter space of the locus of heteroclinic connections allows to detect various degeneracies of codimension two and three, some of which have not been previously studied in the literature. These bifurcations are organizing centers of the complicated dynamics exhibited by this system. Furthermore, studying how the bifurcation sets evolve when D tends to zero, we are able to explain, in the Lorenz system, the origin of several global connections which are related to T-point heteroclinic loops.

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.