Bifurcation Mechanism Research Articles

Stochastic Canards in a Modified Leslie–Gower Population Model

The problem of analyzing the bifurcation mechanisms of complex stochastic oscillations in population dynamics is considered. We study this problem on the basis of the modified Leslie–Gower prey–predator model with randomly forced Holling-II functional response. The paper focuses on the effects of noise in the Canard explosion zone. The phenomenon of noise-induced splitting of Canard cycles is discovered and studied in terms of stochastic [Formula: see text]-bifurcations. In parametric analysis, we use the stochastic sensitivity technique with the apparatus of confidence domains to find the most noise-sensitive Canard. For the phenomenon of stochastic splitting, an underlying deterministic mechanism using critical curves near the cycle orbit and sub-/super-critical zones is revealed.

Dynamical effects of low-frequency and high-frequency current stimuli in a memristive Morris–Lecar neuron model

This paper deduces that the potassium and calcium ion currents in the two-dimensional (2D) Morris–Lecar neuron model can be respectively characterized by a passive memristor and a locally active memristor. Then a memristive Morris–Lecar neuron model with an equivalent circuit scheme is first constructed. The equilibrium trajectory and its stability are analyzed, which displays fold and Hopf bifurcations with proper model parameters. Bursting behavior and mixed-mode oscillations with after-depolarizing potential for low-frequency current stimulus are numerically disclosed. Besides, the bifurcation mechanisms for bursting behavior and mixed-mode oscillations are theoretically deduced. Furthermore, the firing patterns and antimonotonicity phenomenon for high-frequency current stimulus are numerically discovered. This study provides a new concept for building memristive bionic circuits from neuron models with well-defined ion channel currents.

Transition and bifurcation mechanism of firing activities in memristor synapse-coupled Hindmarsh–Rose bi-neuron model

Memristor, especially the locally active one, is widely used in emulating the synapse-inspired activities in neural networks, in which the connection between memristor properties and neuronal electrical activities requires further investigation. In this paper, a bi-neuron model is constructed by bi-directionally connecting a locally active memristor between the membrane potentials of two deterministic 3D HR neuron models. Complex spiking/bursting firing activities and their transitions are disclosed under different memristor control parameters. The bifurcation mechanisms of the revealed synchronous and asynchronous firing activities are explored inside each individual neuron by taking the externally input state variables as modulation parameters. These results demonstrate the crucial effect of the locally active memristor synapse on firing activities of its coupled network. FPGA-based digital circuit experiment is finally performed to verify the numerical results. This research is beneficial to understanding, control and application of firing activities in neural networks.

Semi-analytical computation of bifurcation of orbits near collinear libration point in the restricted three-body problem

A unified semi-analytical solution is presented for constructing the phase space near collinear libration points in the Circular Restricted Three-body Problem (CRTBP), encompassing Lissajous orbits, quasihalo orbits, Axial orbits, and their invariant manifolds, as well as transit and non-transit orbits. Based on classical in-plane and out-of-plane frequency resonance mechanisms, the Lindstedt–Poincaré method could only derive separate analytical solutions for the invariant manifolds of Lissajous orbits and halo orbits, falling short for the invariant manifolds of quasihalo orbits. In this paper, by introducing a coupling coefficient η and a bifurcation equation, a unified series solution for these orbits is systematically developed using a coupling-induced bifurcation mechanism and Lindstedt–Poincaré method. Analyzing the bifurcation equation obtained from different coupling forms reveals bifurcation conditions for all kinds of orbits near collinear libration points. When η = 0, the series solution describes non-bifurcated orbits, while when η ≠ 0, the solution describes bifurcated orbits, including quasihalo orbits, Axial orbits, and their invariant manifolds, as well as newly bifurcated transit and non-transit orbits. This unified semi-analytical framework provides a more comprehensive understanding of the complex phase space structures near collinear libration points in the CRTBP.

Numerical study on bifurcation characteristics of wind-induced vibration for an H-shaped section

In order to reveal the influence of initial excitation on the bifurcation phenomenon of bridge decks, a new perspective of flow characteristics is developed based on the computational fluid dynamics numerical simulation method. Then, the bifurcation mechanism of vortex-induced vibration (VIV) response and nonlinear flutter response of the H-shaped section is investigated. The results show that when the wind speed is 2 m/s, under a small torsional excitation of 0.5°, the flow field of the H-shaped section will develop into the vortex shedding mode of the vertical vibration, resulting in vertical VIV. However, while under a large excitation of 6°, the flow field will directly transform into the vortex shedding mode of the torsional vibration, resulting in torsional VIV. Therefore, the bifurcation phenomenon of the VIV response is observed. When the wind speed is 4 m/s, the H-shaped section exhibits a nonlinear flutter limit cycle oscillation under a large excitation of 8°, but its response can be ignored under a small excitation of 0.5°. This phenomenon is attributed to the significant change in the transition of the vortex shedding mode from a small amplitude to a stable large amplitude, and the flow field lacks enough energy to complete the transition of the vortex shedding mode, resulting in the bifurcation phenomenon of the nonlinear flutter response. When the wind speed is 3.0 m/s, the large excitation will change the vortex shedding frequency of the new H-shaped section, resulting in the torsional VIV.

Computational study on the bifurcation mechanism in the H2CO− + CH3Cl →CH3CH2O + Cl−/H2CO + CH3 + Cl− reaction: The importance of intramolecular vibrational redistributions

We have investigated the bifurcation dynamics of the H2CO− + CH3Cl →CH3CH2O + Cl− (C-substitution channel)/H2CO + CH3 + Cl− (electron transfer channel) reaction using three different molecular dynamics (MD) methods: quasi-classical trajectory (QCT), classical MD, and ring-polymer MD simulations. All MD calculations were performed using the on-the-fly approach at the ab initio molecular orbital theory level. We found that the C-substitution channel is dominant in the classical and ring-polymer MD calculations, while the electron transfer channel is dominant in the QCT calculations. The different bifurcation mechanisms in the types of MD simulations were investigated using supervised machine-learning classification. This difference is attributed to the varying efficiency of the intramolecular vibrational redistribution process among the different MD methods. Thus, in the bifurcation mechanism of the H2CO− + CH3Cl reaction, the bifurcation fractions are primarily determined by the vibrational dynamics occurring in the later stages of the reaction.

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.

Early warning signals for bifurcations embedded in high dimensions

Recent work has highlighted the utility of methods for early warning signal detection in dynamic systems approaching critical tipping thresholds. Often these tipping points resemble local bifurcations, whose low dimensional dynamics can play out on a manifold embedded in a much higher dimensional state space. In many cases of practical relevance, the form of this embedding is poorly understood or entirely unknown. This paper explores how measurement of the critical phenomena that generically precede such bifurcations can be used to make inferences about some properties of their embeddings, and, conversely, how prior knowledge about the mechanism of bifurcation can robustify predictions of an oncoming tipping event. These modes of analysis are first demonstrated on a simple fluid flow system undergoing a Hopf bifurcation. The same approach is then applied to data associated with the West African monsoon shift, with results corroborated by existing models of the same system. This example highlights the effectiveness of the methodology even when applied to complex climate data, and demonstrates how a well-resolved spatial structure associated with the onset of atmospheric instability can be inferred purely from time series measurements.

Dynamic analysis of a class of fractional‐order dry friction oscillators

This article investigates a class of Duffing nonlinear dynamic systems with fractional‐order dry friction and conducts in‐depth research on the stability, chaotic characteristics, and erosion of the safety basin of this system; the results are verified through numerical simulation. First, the average method is used to approximate the amplitude–frequency relationship of the system, and the accuracy of the analytical results is verified through numerical experiments. Second, the Melnikov method is used to obtain the conditions for the system to enter chaos in the Smale horseshoe sense, and the Melnikov curve is drawn for further verification. Then, bifurcation diagrams are drawn for the changes in various parameters in the system, with a focus on analyzing the influence of friction factors on chaotic bifurcation. By applying the definition and calculation principle of the maximum Lyapunov exponent, and drawing and utilizing the maximum Lyapunov exponent graph, the chaotic state that the system enters under different parameters is more clearly defined. Finally, the evolution law of the safety basin under various parameter changes, especially dry friction changes, is analyzed, and the erosion and bifurcation mechanism of the safety basin is studied. Comparing with the bifurcation diagram, it reveals that chaos primarily contributes to the erosion of the safety basin.

Bifurcation Mechanism at a Sustain Point of a Long Narrow Economy

In this paper, we investigate population agglomeration in a long narrow economy, in which an odd number of places are evenly distributed over a line segment. The bifurcation analysis of this economy elucidates the mechanism of the emergence of twin cities around the central city. The validity and usefulness of this analysis are confirmed using several well-known economic geography models that display various kinds of bifurcation behaviors. By this analysis, we investigate the historical change in the population distribution in a chain of cities on Japan’s Main Island.

Desynchronization of temporal solitons in Kerr cavities with pulsed injection.

A numerical and analytical study was conducted to investigate the bifurcation mechanisms that cause desynchronization between the soliton repetition frequency and the frequency of external pulsed injection in a Kerr cavity described by the Lugiato-Lefever equation (LLE). The results suggest that desynchronization typically occurs through an Andronov-Hopf (AH) bifurcation. Additionally, a simple and intuitive criterion for this bifurcation to occur is proposed.

Torque curve bifurcation in T-shaped rotor continuously drilling into granular media

This study explores how a continuous drill of T-shaped rotor interacts with granular media and illustrates an universal torque bifurcation mechanism. Experiments shew that the drilling torque linear increases with depth initially, followed by bifurcation. The bifurcation types are intricately linked to the motion parameters of the rotor, while the depth of bifurcation and saturated torque are directly influenced by rotor geometry. DEM simulations link torque bifurcation to a low-stress fluidized region around the rotor, while the region is enclosed by a fluid–solid interface and influenced by the motion parameters. Moreover, force chain analysis demonstrates the irreversible state change of granular media disturbed by the rotor. Leveraging these insights, we propose two scaling laws quantifying the bifurcation point depth and the saturated torque, then perform qualitative analysis to explain the experimental phenomena. These findings may hold significant promise for practical engineering applications.

Bispericyclic Ambimodal Dimerization of Pentafulvene: The Origin of Asynchronicity and Kinetic Selectivity of the Endo Transition State.

The propensity of fulvenes to undergo dimerization has long been known, although the in-depth mechanism and electronic behavior during dimerization are still elusive. Herein, we made an attempt to gain insights into the reactivity of pentafulvene for Diels-Alder (DA) and [6 + 4]-cycloadditions via conventional and ambimodal routes. The result emphasizes that pentafulvene dimerization preferentially proceeds through a unique bifurcation mechanism where two DA pathways merge together to produce two degenerate [4 + 2]-cycloadducts from a single TS. Despite the [6 + 4]-cycloadduct being thermodynamically preferred, [4 + 2]-cycloaddition reactions are kinetically driven. Singlet biradicaloid is involved in through-space 6e- delocalization as a secondary orbital interaction that originates asynchronicity and stabilizes the bispericyclic transition state (TS). The transformation of various actively participating intrinsic bonding orbitals (IBOs) unambiguously forecasts the formation of multiple products from a single TS and rationalizes the mechanism of ambimodal reactions that are rather difficult to probe with other analyses. The changes in active IBOs clearly distinguish the conventional reactions from bifurcation reactions and can be employed to characterize and confirm the ambimodal mechanism. This report gains a crucial theoretical insight into the mechanism of bifurcation, the origin of asynchronicity, and electronic behavior in ambimodal TS, which will certainly be of enormous value for future studies.

Deficiency of the HGF/Met pathway leads to thyroid dysgenesis by impeding late thyroid expansion

The mechanisms of bifurcation, a key step in thyroid development, are largely unknown. Here we find three zebrafish lines from a forward genetic screening with similar thyroid dysgenesis phenotypes and identify a stop-gain mutation in hgfa and two missense mutations in met by positional cloning from these zebrafish lines. The elongation of the thyroid primordium along the pharyngeal midline was dramatically disrupted in these zebrafish lines carrying a mutation in hgfa or met. Further studies show that MAPK inhibitor U0126 could mimic thyroid dysgenesis in zebrafish, and the phenotypes are rescued by overexpression of constitutively active MEK or Snail, downstream molecules of the HGF/Met pathway, in thyrocytes. Moreover, HGF promotes thyrocyte migration, which is probably mediated by downregulation of E-cadherin expression. The delayed bifurcation of the thyroid primordium is also observed in thyroid-specific Met knockout mice. Together, our findings reveal that HGF/Met is indispensable for the bifurcation of the thyroid primordium during thyroid development mediated by downregulation of E-cadherin in thyrocytes via MAPK-snail pathway.

Lévy noise-induced global bifurcation with the compatible cell mapping method

Stochastic bifurcation of dynamical system induced by Lévy noise is investigated in this paper from the global viewpoint, which is demonstrated based on the global attractor obtained with the compatible cell mapping (CCM) method. Compared with the traditional cell mapping method, the efficiency of CCM method is greatly improved by concentrating the computational domain on the covering set of the global attractor rather than the whole state space. The computation time can be further reduced by a simple parallel procedure with the block matrix strategy. With the proposed method, the evolutionary process of global properties has been presented. It shows that the shape and size of stochastic attractors and stochastic saddles vary along the direction of unstable manifold with the parameters of Lévy noise. Stochastic bifurcation occurs when stochastic attractor collides with stochastic saddle. It can be also found that the global bifurcation under Lévy noise is discontinuous because of the jump property. The results are significantly different from those of Gaussian noise, which reveals a distinct global evolutionary mechanism of stochastic bifurcation. Furthermore, it also indicates that the CCM method with block matrix is an effective tool for global analysis.

Scenarios for the appearance of strange attractors in a model of three interacting microbubble contrast agents

We study nonlinear dynamics in a model of three interacting encapsulated gas bubbles in a liquid. The model is a system of three coupled nonlinear oscillators with an external periodic force. Such bubbles have numerous applications, for instance, they are used as contrast agents in ultrasound visualization. Certain types of bubbles oscillations may be beneficial or undesirable depending on a given application and, hence, the dependence of the regimes of bubbles oscillations on the control parameters is worth studying. We demonstrate that there is a wide variety of types of dynamics in the model by constructing a chart of dynamical regimes in the control parameters space. Here we focus on hyperchaotic attractors characterized by three positive Lyapunov exponents and strange attractors with one or two positive Lyapunov exponents possessing an additional zero Lyapunov exponent, which have not been observed previously in the context of bubbles oscillations. We also believe that we provide a first example of a hyperchaotic attractor with additional zero Lyapunov exponent. Furthermore, the mechanisms of the onset of these types of attractors are still not well studied. We identify two-parametric regions in the control parameter space where these hyperchaotic and chaotic attractors appear and study one-parametric routes leading to them. We associate the appearance of hyperchaotic attractors with three positive Lyapunov exponents with the inclusion of a periodic orbit with a three-dimensional unstable manifold, while the onset of chaotic oscillations with an additional zero Lyapunov exponent is connected to the partial synchronization of bubbles oscillations. We propose several underlying bifurcation mechanisms that explain the emergence of these regimes. We believe that these bifurcation scenarios are universal and can be observed in other systems of coupled oscillators.

Two-Parameter Bifurcations and Hidden Attractors in a Class of 3D Linear Filippov Systems

We take into consideration two different kinds of two-parameter bifurcations in a class of 3D linear Filippov systems, namely pseudo-Bautin bifurcation and boundary equilibrium bifurcations for two scenarios. The bifurcation conditions for generating rich dynamic behaviors are established. The main objective is to investigate the effects of two parameters interacting simultaneously on a variety of dynamic phenomena. In order to analyze the pseudo-Bautin bifurcation, we build the Poincaré map and analyze the number of fixed points whose types are related to the crossing limit cycles. In order to analyze boundary equilibrium bifurcations for two scenarios, we perform an analysis on the existence and admissibility of equilibria. Besides, a comprehensive investigation on hidden attractors induced by boundary equilibrium bifurcations is conducted. The novelty resides in overcoming the constraints of previous studies that solely take into account the dynamics of individual parameter variations. We innovatively characterize the two-parameter bifurcation mechanism of a new class of Filippov systems, and qualitatively demonstrate the coexistence of hidden attractor and stable pseudo-equilibrium.

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.

Primary bifurcation mechanism, heat transfer and bicriticality in time temperature-modulated Rayleigh–Bénard convection in a Hele–Shaw cell

In this paper, we perform a detailed numerical study of the linear dynamics of Rayleigh–Bénard convection in a fluid layer confined in a Hele–Shaw cell and subjected to a temperature gradient with a stationary and a periodic time-dependent component. Using Floquet theory for time and the spectral method for space, we show that the system exhibits three classes of time-dependent solutions, namely harmonic, sub-harmonic and quasi-periodic instabilities. These compete into the primary bifurcation driven by two different mechanisms depending on the modulation amplitude. Indeed, harmonic convective rolls, driven by the stationary thermal gradient, arise below critical modulation amplitudes beyond which the well-known parametric resonance mechanism is responsible for the development of convective rolls. In addition, it is revealed that the convective flow reversal could identify this exchange of instability mechanisms occurring via either bicritical states with codimension-two bifurcation points between harmonic and subharmonic modes or via displacement of Floquet multipliers in the Floquet-complex plane. This latter scenario sets in between subharmonic and quasiperiodic instability modes in the high frequency limit by varying the phase-lag between the temperatures of the lower and upper boundaries. Moreover, it is shown that the nature of the instability mechanism alters considerably the heat transfer dynamics of the system.

Bifurcation mechanisms underlying the nested mixed-mode oscillations in a canard-generating driven Bonhoeffer–van der Pol oscillator

In previous works (Inaba and Kousaka, 2020; Inaba and Tsubone, 2020; Inaba et al, 2022) the bifurcation structures referred to as nested mixed-mode oscillations (MMOs) were discovered; these structures can be generated by a driven slow–fast Bonhoeffer–van der Pol (BVP) oscillator. It is known that this oscillator can generate canards in the absence of perturbations. In this work, we investigate nested MMOs in a canard-generating driven BVP oscillator near a supercritical Hopf bifurcation point subjected to weak periodic perturbations. The analysis of this system is undertaken using the techniques presented by Kawakami, (1984). These techniques include the use of a shooting algorithm to derive the parameters of the forced oscillators at which bifurcations occur by solving the simultaneous equations that identify the conditions in which the solution of the forced oscillators is periodic and the characteristic multiplier(s) with the largest absolute value of the periodic solution on the Poincaré section crosses the unit circle. We also obtain several two-parameter bifurcation diagrams that describe the system investigated here. One motivation for this study is to demonstrate that the bifurcation structures present in nested MMOs are likely to be a widespread phenomenon. We hypothesize here that nested MMOs represent a stable phenomenon; this is likely to be the case because they appear and disappear consistently and are bound by sequential saddle–node and period-doubling bifurcation curves; this hypothesis is supported by the observation that codimension-two bifurcation points or cusps do not appear in the region close to these bifurcation boundaries.