Two-parameter Bifurcation Analysis Research Articles

Ion channels and the diversity of spontaneous firing in anterior pituitary corticotrophs: A dynamical analysis

Pituitary cells release hormones based on firing patterns, mediated by multiple ionic currents. Analyzing a single channel’s effect on firing is insufficient. We used a four-dimensional Hodgkin–Huxley-type corticotroph model to analyze its dynamics from a geometric perspective. We conducted a two-parameter bifurcation analysis of the system and found that activation levels of BK, NS, Kir, Ca, and LCa channels are crucial in generating and transitioning between different firing patterns. Chaotic firing can arise during the transition between firing patterns. We have employed the Poincaré map method and sequence reconstructions to construct return maps to characterize chaos-firing. The membrane potential time series is affected by fluctuations in intracellular calcium ([Ca2+]i). We studied the variable c using fast–slow analysis techniques and phase portraits to explore its relationship with phase portraits. The study revealed that the system exhibits bistability, which is dependent on the initial values of c or V. The findings indicate that firing activities can arise from either a supercritical or subcritical Hopf bifurcation. Analyses show the interaction between the membrane potential and [Ca2+]i at various BK channel activation levels.

Instability in optical injection locking semiconductors lasers using multiparametric bifurcation analysis.

In this paper, we investigate bifurcations of equilibria and transients by using modified rate equations of semiconductor lasers (SCLs) subjected to optical injection. An analytical study is performed to demonstrate some two-parameter bifurcations, inter alia, Bogdanov-Takens and Gavrilov-Guckenheimer bifurcations. A detailed numerical study based on the multiparametric bifurcation method and using 3D-plots and projections reveal a rich locking dynamics of SCLs. In this way, a so-called zero frequency detuning well is highlighted in the vicinity of a Hopf bifurcation confining minimal states of the larger Lyapunov exponent in injection locking curves. Three-parameter bifurcation curves mainly underscore cusp bifurcation and resizing of its multi-equilibrium region by the specific control parameter defined in this model. The bursting phenomenon observed in the transient regime is discussed by using various numerical approaches wherefrom another quantifying method tapping into two-parameter bifurcation analysis is proposed. Thereafter, metastable chaos dynamics supported by spiraling relaxation oscillations is also investigated as well as planar saddle-node bifurcations with three homoclinic orbits for high positive and negative detunings. At last, zero α-factor effects contribute to drastically shrink the unlocking region of SCLs, twofold increase in Hopf bifurcation along with evidencing of complex chaotic sine-shaped and folded torus-shaped attractors.

Fast-slow dynamics analysis in an externally excited smooth and discontinuous oscillator with a pair of irrational nonlinearities

The study of fast-slow oscillations in systems with irrational nonlinearity that may yield abundant dynamical mechanisms is not well developed. This paper aims to investigate the fast-slow dynamics in an excited mass-spring oscillator with a pair of irrational nonlinearities, which can undergo the dynamical transition from smooth to discontinuous characteristics depending on the values of a smoothness parameter. Three different types of fast-slow oscillations are reported in this interesting smooth and discontinuous (SD) oscillator with a pair of irrational nonlinearities. Due to the smooth and discontinuous characteristics of this SD oscillator, we consider its dynamical behaviors under the smooth and discontinuous cases, respectively. Based on the fast-slow analysis and the two-parameter bifurcation analysis, the smooth fast-slow dynamics associated with fold hysteresis and its turnover are revealed. In the discontinuous case, the system can be viewed as a piecewise-smooth dynamical system governed by three different subsystems in different regions divided by two nonsmooth boundaries. In particular, the nonsmooth boundaries can be divided into parts with different dynamical behaviors, including escaping and crossing lines. Unlike the smooth case, there is no change in the stability of the equilibrium in these three subsystems. However, transitions of system trajectory induced by crossing lines can account for the generation of fast-slow oscillations in the piecewise-smooth system. As a result, the smooth and piecewise-smooth fast-slow dynamics in the excited SD oscillator with a pair of irrational nonlinearities are revealed, which deepens the understanding of fast-slow dynamics of the dynamical systems with irrational nonlinearity.

Identifying critical regulatory interactions in cell fate decision and transition by systematic perturbation analysis

One of the most significant challenges in biology is to elucidate the roles of various regulatory interactions in cell fate decision and transition. However, it remains to be fully clarified how they cooperate and determine fate transition. Here, a general framework based on statistical analysis and bifurcation theory is proposed to identify crucial regulatory interactions and how they play decisive roles in fate transition. More exactly, specific feedback loops determine occurrence of bifurcations by which cell fate transition can be realized. While regulatory interactions in the feedback loops determine the direction of transition. In addition, two-parameter bifurcation analysis further provides detailed understanding of how the fate transition based on statistical analysis occurs. Statistical analysis can also be used to reveal synergistic combinatorial perturbations by which fate transition can be more efficiently realized. The integrative analysis approach can be used to identify critical regulatory interactions in cell fate transition and reveal how specific cell fate transition occurs. To verify feasibility of the approach, the epithelial to mesenchymal transition (EMT) network is chosen as an illustrative example. In agreement with experimental observations, the approach reveals some critical regulatory interactions and underlying mechanisms in cell fate determination and transitions between three states. The approach can also be applied to analyze other regulatory networks related to cell fate decision and transition.

Coupled systems with quasi-periodic and chaotic dynamics

The interaction of a system with quasi-periodic autonomous dynamics and a chaotic Rössler system is studied. We have shown that with the growth of the coupling, regimes of two-frequency and three-frequency quasiperiodicity, a periodic regime and a regime of oscillation death sequentially arise. With a small coupling strength, doubling bifurcations of three-frequency tori are observed in the system. A chaotic regime, characterized by two additional zero Lyapunov exponents in spectrum, is revealed. Two-parameter Lyapunov exponent analysis and bifurcation analysis are presented. A new bifurcation scenario of transition from the regime of oscillation death to quasi-periodicity in coupled systems is described.

Dynamics analysis and Hamilton energy control of a class of Filippov neuron model

Electromagnetic induction plays a key role in regulating the electrical activity, excitability, and bistable structure of neurons. In this paper, a discontinuous control strategy with membrane potential as the threshold is introduced to the HR neuron model under the influence of electromagnetic field (EMFN model), and then we establish a Filipov EMFN neuron model to realize the regulation effect of electromagnetic field on the neuron system. Specific work of this paper reads as follows: first, the existence and stability of equilibrium points of two subsystems are analyzed by using MatCont software; second, the bistable region and its internal mechanism are discussed in detail by two-parameter bifurcation analysis; third, the mechanism of bistability and a series of complex sliding mode dynamics including sliding segment and sliding bifurcations are further revealed with the help of the fast–slow variable dissection method; finally, based on Hamilton energy, the regulation of threshold on chaotic discharge in the Filippov EMFN neuron model is explored. The numerical simulation results show that the EMFN neuron model can produce the corresponding sliding limit cycle and sliding bursting behaviors under the influence of the threshold control strategy, meanwhile, the chaotic behavior of the new system can be controlled effectively within a certain range. The results provide ideas for controlling the effects of electromagnetic fields on the biological neuronal system and exploring the regulation mechanism of neurological diseases.

Two-parameter bifurcation and energy consumption analysis of the macro traffic flow model

Two-parameter bifurcation and energy consumption analysis of the macro traffic flow model

A new route to pulse-shaped explosion of limit cycles and its induced amplitude-modulated bursting

This paper aims to report a new route to pulse-shaped explosion (PSE) related to the limit cycle attractor in a mechanical system with multiple-frequency excitations. We show that the compound bursting connected by PSE of equilibria can be observed in this system. Then, with the decrease of the amplitude of parametric excitation, the compound bursting pattern begins to change to other types, accompanied by the generation of PSE of limit cycles. Typically, as a novel sharp transition behavior underlying the occurrence of bursting oscillations, the PSE can be created by the disappearance of critical escape transitions. In present study, based on the two-parameter bifurcation analysis, we find that the PSE can also be created in the branch of limit cycles with the disappearance of fold points of homoclinic bifurcation curves, which is quite different from the ones reported in previous works and can be regarded as a new route to the PSE of limit cycles. Based on this, the compound bursting induced by the PSE of limit cycles is investigated. Besides, it should be pointed out that the limit cycle exhibiting PSE can be regarded as an amplitude-modulated limit cycle attractor, and thus we obtain a newly released bursting rhythm, namely the amplitude-modulated bursting, based on the PSE of limit cycles. These findings have significant implications for the understanding of PSE and the diversity of bursting dynamics induced by it.

Bifurcation analysis of 3D-PWS systems with two transversal switching boundaries: A case study in power electronics

This paper addresses the nonlinear analysis of PWS dynamical systems with two transverse switching boundaries through a real case study: a cascade of two buck converters connected to a common bus in a dc microgrid. Two-parameter bifurcation analysis is proposed to study local and global dynamics on a power electronic circuit feeding a constant power load and controlled by means of a sliding mode control. This electronic circuit is modeled as a 3D piecewise smooth dynamical system with two transversal switching boundaries, having sliding motion in just one of them. Complete characterization of phenomena associated with the bifurcations induced by two transversal switching boundaries is one of the main contributions of this work. Numerical continuation methods are employed to obtain bifurcation sets and bifurcation diagrams to understand the dynamic behavior of the study system and also to verify the analytical results.

Internal resonance in a MEMS levitation force resonator

In this work, we explore internal resonances in a levitation force microelectromechanical system-based actuator assuming flexible cantilever and clamped–clamped microbeam configurations. The levitation force is generated through a special arrangement of two-side stationary charged substrates and a central grounded stationary strip. The design assumes as well an upper flexible microbeam strip. The DC part of the excitation voltage pushes up the moving microbeam away from its lower stationary electrode. A superimposed harmonic AC voltage lets the flexible strip vibrate around its equilibrium position. Energy exchange among the computed lower vibration frequencies and their respective frequency and force response curves are explored for possibilities of principal parametric and internal resonances interactions. The generated responses are computed using the bifurcation toolbox MatCont. The effect of Von Karman nonlinearity to power the energy exchange within vibration modes is explored and dominant factors in the MEMS design for resonances are found from a two-parameter bifurcation analysis. Our results allow optimizing the micro-actuator device performance for future applications.

Global dynamics analysis of a Filippov Hindmarsh–Rose neuron model

The electromagnetic induction induced by neuron membrane potential plays an important role in regulating its electrical activities, bistability and hidden dynamics. In this paper, a nonsmooth threshold control strategy is proposed, in which the membrane potential is utilized as the threshold to determine the switching function corresponding to the electromagnetic induction intensity and external stimulus current. Consequently, a four-dimensional Filippov-type HR neuron model is established. First, the existence, stability and global bifurcation behaviors of equilibrium points of two subsystems are discussed by using stability theory and numerical simulation. Then, the bistable behavior and evolution modes of subsystems are investigated based on the two-parameter bifurcation analysis. Further, the existence of various equilibrium points and sliding dynamics of the system are analyzed by Filippov convex method and Utkin’s equivalent control method. Finally, the sliding firing modes and multistable features under the control of the threshold are revealed by the method of fast-slow variable dissection. These results will provide useful theoretical support for understanding the hidden dynamic mechanism of neurons and constructing functional neural networks.

Global multistability and mechanisms of a memristive autapse-based Filippov Hindmash-Rose neuron model

Electromagnetic induction in the nervous system can be emulated by memristive autapses, which plays a critical role in regulating physiological functions. A discontinuous control strategy has been proposed by taking membrane potential as the threshold. Accordingly, a four-dimensional Filippov Hindmash-Rose (HR) neuron model is established by improving the memristive autapse with the threshold control strategy. The existence, stability, and bifurcation conditions of the two subsystems are discussed, and the bistable regions and their internal mechanism are further revealed with the help of two-parameter bifurcation analysis and global basins of attraction. Subsequently, the complex sliding mode dynamics of the model including sliding segments, various equilibrium points, and sliding bifurcations are analyzed via differential inclusions theory. Then, extensive numerical results exhibit that the proposed threshold strategy leads to the occurrence of sliding bursting, sliding limit cycle and coexisting attractors, and so on. Moreover, the mechanism of sliding electric activities, mode transition, and multistability under the threshold strategy feedback is revealed based on the fast-slow variable dissection method. Finally, the internal mechanism of multistable evolutionary patterns and stochastic bifurcations induced by Gaussian white noise is confirmed. The obtained results will contribute to the further design of functional neural networks and provide potential guidance for the treatment of patients with intellectual disabilities.

Extremely hidden multi-stability in a class of two-dimensional maps with a cosine memristor

We present a class of two-dimensional memristive maps with a cosine memristor. The memristive maps do not have any fixed points, so they belong to the category of nonlinear maps with hidden attractors. The rich dynamical behaviors of these maps are studied and investigated using different numerical tools, including phase portrait, basins of attraction, bifurcation diagram, and Lyapunov exponents. The two-parameter bifurcation analysis of the memristive map is carried out to reveal the bifurcation mechanism of its dynamical behaviors. Based on our extensive simulation studies, the proposed memristive maps can produce hidden periodic, chaotic, and hyper-chaotic attractors, exhibiting extremely hidden multi-stability, namely the coexistence of infinite hidden attractors, which was rarely observed in memristive maps. Potentially, this work can be used for some real applications in secure communication, such as data and image encryptions.

Persistence and Bifurcation Analysis of a Plankton Ecosystem with Cross-Diffusion and Double Delays

To investigate the joint effects of diffusion and multiple delays, a toxin producing phytoplankton–zooplankton system with cross-diffusion is studied and two-parameter bifurcation analysis is carried out. Firstly, for the delay-free system, the stability of equilibria, existence of solutions and priori bound are given. The globally asymptotic stability of equilibrium is investigated for the two situations with or without the diffusion terms. The linear stability analyses show that the cross-diffusion is a key mechanism for Turing instability to occur and Turing instability interval becomes larger with increasing cross diffusion coefficient. Secondly, the focus is on the effects of delays on the system including persistence, stability and Hopf bifurcation. The sufficient conditions for the diffusion system with delays to remain persistent are given. Under certain conditions, the boundary equilibrium is globally asymptotically stable. By analyzing the distribution of roots of the characteristic equation, using the crossing curve methods, the stable changes of positive equilibrium in two-delay plane are obtained. Explicit algorithm is derived for the properties of the bifurcating periodic solutions by normal form theory and the center manifold theorem for partial functional differential equations. Finally, numerical simulations confirm the correctness of theoretical analyses and find that double Hopf bifurcation may occur.

Firing patterns and bifurcation analysis of Purkinje cell dendrite model

This paper mainly studied firing patterns and related bifurcations in the Purkinje cell dendrite model. Based on the methods of equivalent potentials and time scale analysis, the initial six-dimensional (6D) dendrite model is reduced to a 3D form to facilitate the calculation. We numerically show that the dendrite model could exhibit period-adding bifurcation and four bursting patterns for several vital parameters. Then the bifurcation mechanisms and transition of these four bursting patterns are discussed by phase plane analysis, and two-parameter bifurcation analysis of the fast subsystem, respectively. Moreover, we computed the first Lyapunov coefficient to determine the stability of Hopf bifurcation. Ultimately, we analyzed the codimension-two bifurcation of the whole system and gave a detailed theoretical derivation of the Bogdanov–Takens bifurcation.

Viral infection dynamics with mitosis, intracellular delays and immune response.

In this paper, we propose a delayed viral infection model with mitosis of uninfected target cells, two infection modes (virus-to-cell transmission and cell-to-cell transmission), and immune response. The model involves intracellular delays during the processes of viral infection, viral production, and CTLs recruitment. We verify that the threshold dynamics are determined by the basic reproduction number $ R_0 $ for infection and the basic reproduction number $ R_{IM} $ for immune response. The model dynamics become very rich when $ R_{IM} > 1 $. In this case, we use the CTLs recruitment delay $ \tau_3 $ as the bifurcation parameter to obtain stability switches on the positive equilibrium and global Hopf bifurcation diagrams for the model system. This allows us to show that $ \tau_3 $ can lead to multiple stability switches, the coexistence of multiple stable periodic solutions, and even chaos. A brief simulation of two-parameter bifurcation analysis indicates that both the CTLs recruitment delay $ \tau_3 $ and the mitosis rate $ r $ have a strong impact on the viral dynamics, but they do behave differently.

Dual mechanisms of Bcl-2 regulation in IP3-receptor-mediated Ca2+ release: A computational study* *Project supported by Shanxi Province Science Foundation for Youths (Grant No. 201901D211159) and the National Natural Science Foundation of China (Grant Nos. 11504214, 11874310, and 12090052).

Inositol 1,4,5-trisphosphate receptors (IP3R)-mediated calcium ion (Ca2+) release plays a central role in the regulation of cell survival and death. Bcl-2 limits the Ca2+ release function of the IP3R through a direct or indirect mechanism. However, the two mechanisms are overwhelmingly complex and not completely understood. Here, we convert the mechanisms into a set of ordinary differential equations. We firstly simulate the time evolution of Ca2+ concentration under two different levels of Bcl-2 for the direct and indirect mechanism models and compare them with experimental results available in the literature. Secondly, we employ one- and two-parameter bifurcation analysis to demonstrate that Bcl-2 can suppress Ca2+ signal from a global point of view both in the direct and indirect mechanism models. We then use mathematical analysis to clarify that the indirect mechanism is more efficient than the direct mechanism in repressing Ca2+ signal. Lastly, we predict that the two mechanisms restrict Ca2+ signal synergistically. Together, our study provides theoretical insights into Bcl-2 regulation in IP3R-mediated Ca2+ release, which may be instrumental for the successful development of therapies to target Bcl-2 for cancer treatment.

Dynamics of a Large-Scale Spiking Neural Network with Quadratic Integrate-and-Fire Neurons.

Since the high dimension and complexity of the large-scale spiking neural network, it is difficult to research the network dynamics. In recent decades, the mean-field approximation has been a useful method to reduce the dimension of the network. In this study, we construct a large-scale spiking neural network with quadratic integrate-and-fire neurons and reduce it to a mean-field model to research the network dynamics. We find that the activity of the mean-field model is consistent with the network activity. Based on this agreement, a two-parameter bifurcation analysis is performed on the mean-field model to understand the network dynamics. The bifurcation scenario indicates that the network model has the quiescence state, the steady state with a relatively high firing rate, and the synchronization state which correspond to the stable node, stable focus, and stable limit cycle of the system, respectively. There exist several stable limit cycles with different periods, so we can observe the synchronization states with different periods. Additionally, the model shows bistability in some regions of the bifurcation diagram which suggests that two different activities coexist in the network. The mechanisms that how these states switch are also indicated by the bifurcation curves.

Multi-time scale dynamics of mixed depolarization block bursting

Two-coupled excitatory neurons model containing both calcium-activated non-specific cationic and persistent sodium current in the pre-Botzinger complex (pre-BotC) is studied. A new type of mixed burst similar to a depolarization block bursting (DB-bursting) is observed in the model of pre-BotC neurons. The multi-time scale dynamics, one- and two-parameter bifurcation analysis, are used to study the types of mixed burst and their transition mechanisms. The results show that the periodic fluctuation of calcium concentration has a great influence on the generation of mixed bursting. The change of time scale caused by fluctuation of calcium concentration and the relative position of bifurcation curves have great influence on patterns of bursting.

Two-parameter bifurcation analysis of an aircraft nose landing gear model

The Hopf–Hopf bifurcations of an aircraft nose landing gear model are investigated in this paper based on two pairs of continuation parameters. It is shown that due to the effect of higher-order terms, there exists a deviation between the bifurcation curve of the original system and the bifurcation curve of its truncated amplitude system, which results in a contraction in the bistable region compared with the previous work. Two codimension 3 strong resonant points can be detected on the Neimark–Sacker bifurcation curve. Period-3 and period-4 limit cycles with torsional-lateral shimmy can emerge when the vertical force and the taxiing velocity are chosen near the resonant points. The occurrence of 1:3 and 1:4 resonant Hopf–Hopf bifurcations can induce tire-ground vibrations in torsional and lateral direction, which can greatly degrade the performance and safety and possibly destroy the airstrip. So it is of significance to investigate the resonate behaviors of the aircraft nose landing gear to ensure the safe landing.