Saddle-node Bifurcation Curves Research Articles

Excitation and manipulation of super cavity solitons in multi-stable passive Kerr resonators

We report on the theoretical analysis as well as the numerical simulations about the nonlinear dynamics of cavity solitons in a passive Kerr resonator operating in the multistable regime under the condition of a sufficiently strong pump. In this regime, the adjacent tilted cavity resonances might overlap, thus leading to the co-existence of combinatory states of temporal cavity solitons and the extended modulation instability patterns. The cavity in the regime of multistablity may sustain distinct families of cavity solitons, vividly termed as super cavity solitons with much higher intensity and broader spectra if compared with those in the conventional bi-stable regime. The description of such complex cavity dynamics in the multstable regime requires either the infinite-dimensional Ikeda map, or the derived mean-field coupled Lugiato-Lefever equations by involving multiple contributing cavity resonances. With the latter model, we revealed the existence of different orders of super cavity solitons, whose stationary solutions were obtained by using the Newton-Raphson algorithm. Along this line, with the continuation calculation, we have plotted the Hopf / saddle-node bifurcation curves, thus identifying the existing map of the stable and breathing (super) cavity solitons. With this defined parameter space, we have proposed an efficient method to excite and switch the super cavity solitons by adding an appropriate intensity (or phase) perturbation on the pump. Such deterministic cavity soliton manipulation technique is demonstrated to underpin the multi-level coding, which may enable the large capacity all-optical buffering based on the passive fiber ring cavities.

Effect of higher-order interactions on chimera states in two populations of Kuramoto oscillators.

We investigate the effect of the fraction of pairwise and higher-order interactions on the emergent dynamics of the two populations of globally coupled Kuramoto oscillators with phase-lag parameters. We find that the stable chimera exists between saddle-node and Hopf bifurcations, while the breathing chimera lives between Hopf and homoclinic bifurcations in the two-parameter phase diagrams. The higher-order interaction facilitates the onset of the bifurcation transitions at a much lower disparity between the inter- and intra-population coupling strengths. Furthermore, the higher-order interaction facilitates the spread of breathing chimera in a large region of the parameter space while suppressing the spread of the stable chimera. A low degree of heterogeneity among the phase-lag parameters promotes the spread of both stable chimera and breathing chimera to a large region of the parameter space for a large fraction of the higher-order coupling. In contrast, a large degree of heterogeneity is found to decrease the spread of both chimera states for a large fraction of the higher-order coupling. A global synchronized state is observed above a critical value of heterogeneity among the phase-lag parameters. We have deduced the low-dimensional evolution equations for the macroscopic order parameters using the Ott-Antonsen Ansatz. We have also deduced the analytical saddle-node and Hopf bifurcation curves from the evolution equations for the macroscopic order parameters and found them to match with the bifurcation curves obtained using the software XPPAUT and with the simulation results.

Degenerate codimension-2 cusp of limit cycles in a Holling–Tanner model with harvesting and anti-predator behavior

In this paper, a Holling–Tanner predator–prey model with constant-yield prey harvesting and anti-predator behavior is investigated. It is shown that the codimension of a nilpotent cusp is 3 and a degenerate Bogdanov–Takens bifurcation of codimension 3 acts as an organizing center for rich dynamical behaviors including saddle–node bifurcation, Hopf bifurcation, saddle–node bifurcation of limit cycles, and homoclinic bifurcation. Moreover, a Hopf bifurcation of codimension 2 is found, a codimension-2 cusp of limit cycles is firstly observed, and the corresponding normal form coefficients are also obtained, which indicate that there exists an acute angle region of three coexistent limit cycles. In particular, a reversed S-shaped saddle–node bifurcation curve of limit cycles is found. In addition, we observe that the anti-predator behavior may cause nearly extinction of the predator population while the prey population reaches its maximum which is less than the carrying capacity due to the effect of harvesting; however, the constant-yield prey harvesting may drive both species to extinction suddenly. Finally, numerical simulations including bifurcation diagrams and phase portraits are given to illustrate the theoretical results.

SIR Model with Vaccination: Bifurcation Analysis

There are few adapted SIR models in the literature that combine vaccination and logistic growth. In this article, we study bifurcations of a SIR model where the class of Susceptible individuals grows logistically and has been subject to constant vaccination. We explicitly prove that the endemic equilibrium is a codimension two singularity in the parameter space (mathcal {R}_0, p), where mathcal {R}_0 is the basic reproduction number and p is the proportion of Susceptible individuals successfully vaccinated at birth. We exhibit explicitly the Hopf, transcritical, Belyakov, heteroclinic and saddle-node bifurcation curves unfolding the singularity. The two parameters (mathcal {R}_0, p) are written in a useful way to evaluate the proportion of vaccinated individuals necessary to eliminate the disease and to conclude how the vaccination may affect the outcome of the epidemic. We also exhibit the region in the parameter space where the disease persists and we illustrate our main result with numerical simulations, emphasizing the role of the parameters.

Generalization of the Kuramoto model to the Winfree model by a symmetry breaking coupling

We construct a nontrivial generalization of the paradigmatic Kuramoto model by using an additional coupling term that explicitly breaks its rotational symmetry resulting in a variant of the Winfree model. Consequently, we observe the characteristic features of the phase diagrams of both the Kuramoto model and the Winfree model depending on the degree of the symmetry breaking coupling strength for unimodal frequency distribution. The phase diagrams of both the Kuramoto and the Winfree models resemble each other for symmetric bimodal frequency distribution for a range of the symmetry breaking coupling strength except for region shift and difference in the degree of spread of the macroscopic dynamical states and bistable regions. The dynamical transitions in the bistable states are characterized by an abrupt (first-order) transition in both the forward and reverse traces. For asymmetric bimodal frequency distribution, the onset of bistable regions depends on the degree of the asymmetry. Large degree of the symmetry breaking coupling strength promotes the synchronized stationary state, while a large degree of heterogeneity, proportional to the separation between the two central frequencies, facilitates the spread of the incoherent and standing wave states in the phase diagram for a low strength of the symmetry breaking coupling. We deduce the low-dimensional equations of motion for the complex order parameters using the Ott-Antonsen ansatz for both unimodal and bimodal frequency distributions. We also deduce the Hopf, pitchfork, and saddle-node bifurcation curves from the evolution equations for the complex order parameters mediating the dynamical transitions. Simulation results of the original discrete set of equations of the generalized Kuramoto model agree well with the analytical bifurcation curves.

Abrupt desynchronization and abrupt transition to [formula omitted]-state in globally coupled oscillator simplexes with contrarians and conformists

We study the phase transitions in a generalized Kuramoto model with contrarians and conformists along with a higher order coupling encoded simplex. Incoherent state persists for a large fraction of contrarians, while there is a continuous transition from the former to π-state and traveling wave state as the fraction of conformists increases under the high proportion of the pair-wise coupling and for a higher coupling strength of the conformists than contrarians. Hong and Strogatz (2011) showed the abrupt transition to the π-state when the contrarians coupling strength supersede in a purely pair-wise coupling. A large proportion of higher order coupling induces the abrupt transition to the π-state even when the contrarians coupling strength is feeble than that of the conformists, destabilizes the traveling wave state, and stabilizes the incoherent state in the entire phase diagram despite the presence of a large fraction of the conformists, which coexists with the π-state. Consequently, one can observe an abrupt desynchronization transition to the incoherent state during the backward trace, while the system remains in the incoherent state during the forward trace. The range of the bistable region between the incoherent state and the π-state increases with the proportion of the higher order coupling to a maximum until the abrupt transition to the π-state ceases to exist. We deduce the pitchfork and saddle–node bifurcation curves across which the incoherent state and the π-state lose their stability from the evolution equation for the low-dimensional dynamics corresponding to the macroscopic order parameters.

Fast–slow variable dissection with two slow variables related to calcium concentrations: a case study to bursting in a neural pacemaker model

Neuronal bursting is an electrophysiological behavior participating in physiological or pathological functions and a complex nonlinear behavior alternating between burst and quiescent state modulated by slow variables. Identification of dynamics of bursting modulated by two slow variables is still an open problem. In the present paper, a novel fast–slow variable dissection method with two slow variables is proposed to analyze the complex bursting simulated in a four-dimensional neuronal model to describe firing associated with pathological pain. The lumenal (\(C_\mathrm{lum}\)) and intracellular (\(C_\mathrm{in}\)) calcium concentrations are the slowest variables, respectively, in the quiescent state and burst duration. Questions encountered when the traditional method with one slow variable is used. When \(C_\mathrm{lum}\) is taken as slow variable, the burst is successfully identified to terminate near the saddle-homoclinic bifurcation point of the fast subsystem and begin not from the saddle-node bifurcation. With \(C_\mathrm{in}\) chosen as slow variable, \(C_\mathrm{lum}\) value of the initiation point of burst is far from the saddle-node bifurcation point, due to \(C_\mathrm{lum}\) not contained in the equation of the membrane potential. To overcome this problem, both \(C_\mathrm{in}\) and \(C_\mathrm{lum}\) are regarded as slow variables; the two-dimensional fast subsystem exhibits a saddle-node bifurcation point, which is extended to a saddle-node bifurcation curve by introducing the \(C_\mathrm{lum}\) dimension. Then, the initial point of burst is successfully identified to be near the saddle-node bifurcation curve. The results present a feasible method for fast–slow variable dissection and a deep understanding of the complex bursting behavior with two slow variables, which is helpful for the modulation to pathological pain.

Bogdanov\u2013Takens bifurcation of a Holling IV prey\u2013predator model with constant-effort harvesting

A prey–predator model with constant-effort harvesting on the prey and predators is investigated in this paper. First, we discuss the number and type of the equilibria by analyzing the equations of equilibria and the distribution of eigenvalues. Second, with the rescaled harvesting efforts as bifurcation parameters, a subcritical Hopf bifurcation is exhibited near the multiple focus and a Bogdanov–Takens bifurcation is also displayed near the BT singularity by analyzing the versal unfolding of the model. With the variation of bifurcation parameters, the system shows multi-stable structure, and the attractive domains for different attractors are constituted by the stable and unstable manifolds of saddles and the limit cycles bifurcated from Hopf and Bogdanov–Takens bifurcations. Finally, a cusp point and two generalized Hopf points are found on the saddle-node bifurcation curve and the Hopf bifurcation curves, respectively. Several phase diagrams for parameters near one of the generalized Hopf points are exhibited through the generalized Hopf bifurcation.

Bifurcation of multi-stable behaviors in a two-parameter plane for a non-smooth nonlinear system with time-varying parameters

To obtain the correlation between multiple parameters, multi-initial values and multi-stable behaviors for a non-smooth nonlinear system with time-varying parameters, a new method of calculating the bifurcation of multi-stable behaviors in the parametric plane is first proposed based on Poincare mapping theory, Lyapunov theory and Floquot theory. The bifurcation and distribution of multi-stable behaviors of a nonlinear gear system with time-varying meshing stiffness in a two-parameter plane are studied by using the proposed method. Various multi-stable behaviors and potential hidden bifurcation curves are fully revealed. Double-bifurcation points formed by the intersection of two different bifurcation curves are further investigated. The probability of occurrence of hidden bifurcation curve is calculated and analyzed based on statistical theory. Results indicate that saddle-node bifurcation curves are sensitive to the initial value and change both the type of multi-stable behaviors and the topology of the attraction basin. However, period-doubling bifurcation curves are not sensitive to the initial value, and only change the type of multi-stable behavior, but do not greatly change the topology of the attraction basin. Four different types of multi-stable behaviors are observed around double-bifurcation points. Multi-stable behaviors and bifurcation curves are easily hidden in the parametric plane due to their small occurring probabilities.

Dynamic characteristics of a mechanical impact oscillator with a clearance

A two-degree-of-freedom mechanical impact oscillator with a clearance is considered. Pattern types, occurrence and stability domains and bifurcation characteristics of periodic motions are investigated through two-parameter co-simulation analysis. With varying of the bifurcation parameters in two directions, forward and backward, two-parameter grazing, period-doubling and saddle-node bifurcations of impactless, fundamental and subharmonic motions are calculated and analyzed. There exist two types of transition zones, hysteresis and liguliform zones, in mutual transition between neighboring motions p/1 and (p + 1)/1 (p ≥ 0). Dynamics in the hysteresis and liguliform zones is introduced in detail. Attracting domains and Poincaré mapping diagrams of coexisting motions in the neighborhood of grazing bifurcations are discussed. Pattern types and bifurcations of subharmonic motions in liguliform zones show regularity. In the two-parameter plane, two grazing bifurcation curves of p/1 (p ≥ 0) motion and period-doubling and saddle-node bifurcation curves of (np + 1)/n (n ≥ 1) motion intersect at a singular point along the upper boundary of liguliform zones LZp/1∩(p + 1)/1. Consequently, the singular point is a point of double-grazing bifurcation of p/1 motion and codimension-2 flip-fold bifurcation of (np + 1)/n motion. The grazing bifurcation of neighboring motions is continuous and reversible only at the singular points, and displacement amplitudes of the impact oscillator vary continuously when the varying parameter passes through the grazing bifurcation boundary.

Effect of Ankle-Pivot Misalignment and Upward Ankle Vertical Displacement on Stability and Equilibrium Location for an Ankle-Hip Model of Balance on a Balance Board

AbstractDuring individual training on a balance board (BB), misalignment between the ankle joint and the BB axis of rotation may exist. The ankle-pivot misalignment influences the dynamics of human balance and more importantly, the stability properties of the equilibrium positions of the human-BB dynamical system. Similarly, ankle displacement in the upward direction with respect to the BB pivot also plays a critical role in the stability properties of the human-BB system. This paper investigates these effects through bifurcation analyses performed to the ankle-hip model of balance on a BB developed in our previous work. By using local bifurcation analyses, we have obtained the stability regions of the upright posture (UP) of a human-BB model of balance in different parameter spaces. The stability regions are delimited by Hopf, pitchfork, and saddle-node bifurcation curves in some selected parameter spaces. Results show that ankle-pivot misalignment has an impact on the location of the Hopf and unfolds the pitchfork bifurcation curves (found in the aligned case) into saddle-node bifurcation curves. Moreover, ankle-pivot misalignment breaks the mirror symmetry of upright static equilibrium positions and induces the establishment of equilibrium positions away from the vertical UP. With respect to the ankle vertical displacement, it has a minimal impact on the location of the pitchfork bifurcation curves but has a large impact on the location of the Hopf bifurcation curves, especially when combined with large BB time delay. This suggests that a larger ankle vertical displacement may result in sway oscillations of larger amplitude. The analyses also provide insight into different mechanisms of stability that can be found in the ankle-hip model of balance on a BB, namely, limit cycle oscillations and leaning postures. If an individual suffers from neuropathy, results from this study can be useful for researchers and clinicians in understanding what types of instabilities might be encountered, and during rehabilitation how to position the subjects carefully to avoid inadvertent instabilities.

Bogdanov-Takens bifurcation in a SIRS epidemic model with a generalized nonmonotone incidence rate

In this paper, we study a SIRS epidemic model with a generalized nonmonotone incidence rate. It is shown that the model undergoes two different topological types of Bogdanov-Takens bifurcations, i.e., repelling and attracting Bogdanov-Takens bifurcations, for general parameter conditions. The approximate expressions for saddle-node, Homoclinic and Hopf bifurcation curves are calculated up to second order. Furthermore, some numerical simulations, including bifurcations diagrams and corresponding phase portraits, are given to illustrate the theoretical results.

Route to shrimps: Dissipation driven formation of shrimp-shaped domains

Abstract In this paper, two scenarios for the formation of shrimp-shaped domains [1] are presented. The employed test model is the Keller–Miksis equation that is a second order, harmonically forced nonlinear oscillator describing the dynamics of a single spherical gas bubble placed in a liquid domain. The results have shown that with an increasing dissipation rate (liquid viscosity), shrimp-shaped domains are evolved within the complex structure of each subharmonic resonances in the amplitude-frequency parameter plane of the external forcing. The mechanism is the coalescence and interaction of two pairs of a period-doubling and a saddle-node codimension-two bifurcation curves.

Non-linear stability analysis of supercritical carbon dioxide flow in inclined heated channel

In this paper, the nonlinear stability analysis of supercritical carbon dioxide (SC−CO2) has been carried out using a reduced order nodalized model for a single inclined heated channel. The primary objective of the present study is to portray the generic view of linear as well as nonlinear stability characteristics of SC−CO2 channel along with the prospect of different types of bifurcation phenomena. The linear stability analysis is carried out with the help of eigenvalues of the Jacobian at steady state conditions, and stability boundary is shown in the parameter plane of pseudo-sub-cooling (Nspc) and pseudo-phase-change numbers (Ntpc). The non-linear analysis includes the detailed study of dynamic and static instabilities. Different types of bifurcation phenomena namely; sub-critical, super-critical and Generalized Hopf are observed; indicating various features of the dynamic instabilities. The first Lyapunov coefficient has been calculated to distinguish between sub-critical and super-critical Hopf bifurcations. Whereas in static instability; Ledinegg excursive phenomena, which is characterized as a saddle-node bifurcation is observed. Additionally, on saddle-node bifurcation curve, Bogdanov-Takens bifurcation points (as interaction with Hopf bifurcation) appear. These bifurcations lead to complex dynamics in the system therefore, several numerical simulations have been carried out around the stability threshold. This type of non-linear analysis is rarely reported for SC−CO2 in the existing literature. To extend this analysis, the dependence of various system design parameters on the bifurcation curve has been investigated along with the shifting of Generalized Hopf (GH) bifurcation point. Furthermore, the effect of the inclination angle of the channel on the stability threshold in parametric space is investigated.

Bifurcation and evolution of a forced and damped Duffing system in two-parameter plane

A general method to calculate multi-parameter bifurcation diagram in the parameter space is designed based on top Lyapunov exponent and Floquet multiplier to study the effect of different combinations of system parameters on the system’s dynamics. Bifurcation and chaos of the forced and damped Duffing system in two-parameter plane are investigated by using the method designed in this work. The correlation and matching laws of the Duffing system between dynamic performance and system parameters are analyzed. The effect of different types of bifurcation curves on the bifurcating of coexisting attractors is investigated according to basins of attraction, bifurcation diagrams, top Lyapunov exponent spectrums, phase portraits, Poincare maps, and Floquet multipliers. The evolution of various bifurcation curves and codimension-two bifurcation in the parametric plane is studied as well. Coexisting attractors are found in the parameter plane. The results indicate that the different bifurcating curves are selective for the bifurcation of coexisting attractors. Both the pitchfork bifurcation curve and the period-doubling bifurcation curve just change the stability of some of the coexisting attractors, but have no effect on the stability of the other part of the attractors. The saddle-node bifurcation curve has an effect on the stability of all the coexisting attractors. A series of period-doubling bifurcation curves and codimension-two bifurcation points lead to chaos existence region in two-parameter plane. The special evolution of bifurcation points and bifurcation curves in two-parameter plane with the change of the system parameter is observed. The codimension-two bifurcation points and bifurcation curves play an important role in understanding nonlinear dynamics of the system in the parametric plane. The work in this study emphasizes the importance of the different combinations of system parameters on the system dynamics.

Remarkable similarities of two pairs of stable and saddle canards in a van der Pol oscillator under extremely weak periodic perturbation

Canards are interesting nonlinear phenomena that have generated intense research interest since their discovery in the late 20th century. We are interested here in how canard-generating dynamics are influenced by extremely weak periodic perturbations that cause the formation of saddle-node bifurcations in the fundamental harmonic entrainment region. In a previous study, we discovered that another entrainment region exists within the fundamental harmonic entrainment region surrounded by the second saddle-node bifurcation curves. We found that two pairs of stable and saddle canards coexist in this second entrainment region under such weak periodic perturbation. Moreover, the stable and saddle canards are matched pairwise; i.e., each stable canard quite closely resembles a corresponding saddle canard. Calculation of the correlation coefficients of the four canards revealed two similar solutions on the order of 0.9999⋯ between the two pairs of similar canards. In contrast, the correlation coefficients of the dissimilar canards differ from unity in proportion to the difference between the given bifurcation parameter value and the parameter values at the saddle-node bifurcation points. Approximately, they take values from 0.998 to 0.975. These contrasts are noteworthy. Similar bifurcation phenomena were observed in the 1/2-subharmonic entrainment region. We hypothesize that the two pairs of stable and saddle canards are invariant with respect to a slight shift of time at the saddle-node bifurcation points, and we numerically prove that such a property approximately holds at the bifurcation points.

Effects of different dispersal patterns on the presence-absence of multiple species

Predicting which species will be present (or absent) across a geographical region remains one of the key problems in ecology. Numerous studies have suggested several ecological factors that can determine species presence-absence: environmental factors (i.e. abiotic environments), interactions among species (i.e. biotic interactions) and dispersal process. While various ecological factors have been considered, less attention has been given to the problem of understanding how different dispersal patterns, in interaction with other factors, shape community assembly in the presence of priority effects (i.e. where relative initial abundances determine the long-term presence-absence of each species). By employing both local and non-local dispersal models, we investigate the consequences of different dispersal patterns on the occurrence of priority effects and coexistence in multi-species communities. In the case of non-local, but short-range dispersal, we observe agreement with the predictions of local models for weak and medium dispersal strength, but disagreement for relatively strong dispersal levels. Our analysis shows the existence of a threshold value in dispersal strength (i.e. saddle-node bifurcation) above which priority effects disappear. These results also reveal a co-dimension 2 point, corresponding to a degenerate transcritical bifurcation: at this point, the transcritical bifurcation changes from subcritical to supercritical with corresponding creation of a saddle-node bifurcation curve. We observe further contrasting effects of non-local dispersal as dispersal distance changes: while very long-range dispersal can lead to species extinctions, intermediate-range dispersal can permit more outcomes with multi-species coexistence than short-range dispersal (or purely local dispersal). Overall, our results show that priority effects are more pronounced in the non-local dispersal models than in the local dispersal models. Taken together, our findings highlight the profound delicacy in the mediation of priority effects by dispersal processes: “big steps” can have more influence than many “small steps”.

Incoherent chimera and glassy states in coupled oscillators with frustrated interactions.

We suggest a site disorder model that describes the population of identical oscillators with quenched random interactions for both the coupling strength and coupling phase. We obtain the reduced equations for the suborder parameters, on the basis of Ott-Antonsen ansatz theory, and present a complete bifurcation analysis of the reduced system. New effects include the appearance of the incoherent chimera and glassy state, both of which are caused by heterogeneity of the coupling phases. In the incoherent chimera state, the system displays an exotic symmetry-breaking behavior in spite of the apparent structural symmetry where the oscillators for both of the two subpopulations are in a frustrated state, while the phase distribution for each subpopulation approaches a steady state that differs from each other. When the incoherent chimera undergoes Hopf bifurcation, the system displays a breathing incoherent chimera. The glassy state that occurs on a surface of three-dimensional parameter space exhibits a continuum of metastable states with zero value of the global order parameter. Explicit formulas are derived for the system's Hopf, saddle-node, and transcritical bifurcation curves, as well as the codimension-2 crossing points, including the Takens-Bogdanov point.

The role of asymmetrical and repulsive coupling in the dynamics of two coupled van der Pol oscillators.

A system of two asymmetrically coupled van der Pol oscillators has been studied. We show that the introduction of a small asymmetry in coupling leads to the appearance of a "wideband synchronization channel" in the bifurcational structure of the parameter space. An increase of asymmetry and transition to repulsive interaction leads to the formation of multistability. As the result, the tip of the Arnold's tongue widens due to the formation of folds defined by saddle-node bifurcation curves for the limit cycles on the torus.

Qualitative analysis of a predator–prey system with double Allee effect in prey

Growing biological evidence from various ecosystems (Berec et al., 2007) suggests that the Allee effect generated by two or more mechanisms can act simultaneously on a single population. Surprisingly, prey–predator system incorporating multiple Allee effect is relatively poorly studied in literature. In this paper, we consider a ratio dependent prey–predator system with a double Allee effect in prey population growth. We have taken into account the case of the strong and the weak Allee effect separately and study the stability and complete bifurcation analysis. It is shown that the model exhibits the bi-stability and there exists separatrix curve(s) in the phase plane implying that dynamics of the system is very sensitive to the variation of the initial conditions. Generic normal forms at double zero singularity are derived by a rigorous mathematical analysis to show the different types of bifurcation of the proposed model system including Bogdanov–Takens bifurcation, saddle–node bifurcation curve, a Hopf bifurcation curve and a homoclinic bifurcation curve. The complete analysis of possible topological structures including elliptic, parabolic, or hyperbolic orbits, and any combination of them in a neighborhood of the complicated singular point (0,0) in the interior of the first quadrant is explored by using a blow up transformation. These structures have important implications for the global behavior of the model. Numerical simulation results are given to support our theoretical results. Finally, the paper concludes with a discussion of the ecological implications of our analytic and numerical findings.