Bifurcation Delay Research Articles

Bifurcations and mixed mode oscillations in a bi-stable plasma model with slow parametric excitation.

In this study, considering a bi-stable plasma model with slow parametric excitation, the bifurcation of periodic and chaotic responses as well as the resulting fast-slow motions is discussed analytically and numerically. For a nonautonomous fast sub-system, the generalized harmonic balancing method is utilized to obtain an averaged system. Bifurcation analysis about the averaged system shows that the critical manifolds form a S-shape structure. Meanwhile, supercritical and subcritical period doubling (PD) occurs on the upper branch simultaneously. As the frequency of the external excitation changes, bifurcation points on the limit cycle manifolds can present different relative locations. Moreover, an additional bi-stable structure induced by Cusp bifurcation emanates from the upper branch. On the other hand, the existence of a chaotic attractor and the corresponding boundary crisis phenomenon are verified using the Melnikov method and the basin of attraction. The structures of the numerical bifurcation diagram show good agreements with the analytical results. Considering two cases of low-frequency excitation, the corresponding fast-slow dynamics are discussed. It is found that, when the fast-slow flow passing the subcritical PD point, a low frequency with different magnitudes will lead to two patterns of bifurcation delay, i.e., the typical one and the excessive delay, which suppress the PD. As for the boundary crisis point, the slow passage effects show no distinct influence. Thus, three transition mechanisms based on two cases of the bifurcation structure are explained, including "fold of cycle-fold of cycle" type, "fold of cycle-delayed subcritical PD" type, and "fold of cycle-boundary crisis" type.

Bifurcation delay in a network of nonlocally coupled slow-fast FitzHugh–Nagumo neurons

Bifurcation delay in a network of nonlocally coupled slow-fast FitzHugh–Nagumo neurons

Bifurcation delay and front propagation in the real Ginzburg-Landau equation on a time-dependent domain.

This work analyzes bifurcation delay and front propagation in the one-dimensional real Ginzburg-Landau equationwith periodic boundary conditions on isotropically growing or shrinking domains. First, we obtain closed-form expressions for the delay of primary bifurcations on a growing domain and show that the additional domain growth before the appearance of a pattern is independent of the growth time scale. We also quantify primary bifurcation delay on a shrinking domain; in contrast with a growing domain, the time scale of domain compression is reflected in the additional compression before the pattern decays. For secondary bifurcations such as the Eckhaus instability, we obtain a lower bound on the delay of phase slips due to a time-dependent domain. We also construct a heuristic model to classify regimes with arrested phase slips, i.e., phase slips that fail to develop. Then, we study how propagating fronts are influenced by a time-dependent domain. We identify three types of pulled fronts: homogeneous, pattern spreading, and Eckhaus fronts. By following the linear dynamics, we derive expressions for the velocity and profile of homogeneous fronts on a time-dependent domain. We also derive the natural "asymptotic" velocity and front profile and show that these deviate from predictions based on the marginal stability criterion familiar from fixed domain theory. This difference arises because the time dependence of the domain lifts the degeneracy of the spatial eigenvalues associated with speed selection and represents a fundamental distinction from the fixed domain theory that we verify using direct numerical simulations. The effect of a growing domain on pattern spreading and Eckhaus front velocities is inspected qualitatively and found to be similar to that of homogeneous fronts. These more complex fronts can also experience delayed onset. Lastly, we show that dilution-an effect present when the order parameter is conserved-increases bifurcation delay and amplifies changes in the homogeneous front velocity on time-dependent domains. The study provides general insight into the effects of domain growth on pattern onset, pattern transitions, and front propagation in systems across different scientific fields.

Chaotic bursting patterns induced by transient chaos in a smooth three-dimensional dynamic model

This paper aims to report the transient-chaos-induced chaotic bursting oscillations in a smooth three-dimensional dynamic model with a low-frequency excitation when an order gap exists between the excitation frequency and the natural frequency. Based on the Nosé-Hoover system, a periodically driven nonlinear system is modelled, in which the transient chaos induced by boundary crisis route can be observed. Through theoretical analysis and numerical simulation, the stability conditions of the equilibrium point and six transient-chaos-induced chaotic bifurcation structures are obtained. Six typical cases in which different routes of the trajectories pass through the transient chaos areas, leading to different patterns of bursting oscillations. It can be found that not only the bifurcation structures but also the bifurcation delay behavior can influence the bursting patterns. In addition, in the transient chaos areas, the bifurcation delay behavior can result in the disappearance of the periodic signals in the small periodic windows.

Sliding bursting oscillations related to transcritical bifurcation delay in an excited vector field with frequency switching

The purpose of this paper is to report sliding bursting dynamics related to transcritical bifurcation delay, a common route to bursting dynamics. As an example, a simple nonlinear oscillator with a slow parametric excitation is considered, in which bursting oscillations resulted from transcritical bifurcation delay can be observed. Typically, such oscillations exhibit prolonged rest phases, characterized by the oscillations that moves along the unstable origin of the system. We show that interesting dynamical characteristics can be observed in the bursting oscillations if the excitation frequency switches between two different values. In particular, additional rest phases, i.e., the newly created non-zero sliding behaviors, characterized by that the trajectory slides along the frequency switching threshold for some time, can be exhibited in the presence of frequency switching. This forms the so-called sliding bursting oscillations. Then, we investigate the dynamical mechanisms of these oscillations and explore their transitions in relation to the variation of frequency switching threshold. As a result, several different patterns of sliding bursting oscillations are obtained. In particular, we find that sliding bursting oscillations may diverge to infinity for some threshold windows, and this can be understood by the analysis of attracting basins.

Bursting Oscillations in General Coupled Systems: A Review

In this paper, the bursting oscillation phenomenon in coupled systems with two time scales is introduced. Firstly, several types of bifurcation are briefly introduced: fold bifurcation, Hopf bifurcation, fold limit cycle bifurcation, homoclinic bifurcation, etc. The bursting oscillations of the system with two excitation terms and the bifurcation delay in the bursting oscillations are considered. Secondly, some simple bursting oscillations are introduced, such as fold/fold bursting, fold/supHopf bursting, subHopf/subHopf bursting, fold/LPC bursting, Hopf/LPC bursting, fold/homoclinic bursting, Hopf/homoclinic bursting, etc. At the same time, the system also has some complex bursting oscillations, such as asymmetric bursting, delayed bursting, bursting with hysteresis loop, etc. Finally, the practical applications of bursting oscillations, such as dynamic vibration absorbers and nonlinear vibration energy harvesting technology, are introduced.

THE EFFECT OF AN ADDITIVE NOISE ON SOME SLOW-FAST EQUATION NEAR A TRANSCRITICAL POINT

We consider the effect of small additive noise with intensity $\sigma$ on trajectories of a slow-fast system with small parameter $\varepsilon$ which admits bifurcation delay at a transcritical point. We estimate the probability that the perturbed stochastic paths stay in some tubular neighborhood of the deterministic path to show that small but not exponentially small noise destroys the bifurcation delay caused by transcritical point and obtain a noise intensity threshold value $N(\varepsilon)$ of order $\varepsilon^{\frac{3}{4}}$. When <i>e</i>^{-1/$\varepsilon$}《<i>σ</i>＜$N(\varepsilon)$, the paths are likely to leave the neighborhood of the corresponding determinate path before some time of order $\sqrt{\varepsilon|log\sigma|}$. When $\sigma>N(\varepsilon) $, the paths are likely to leave before some time of order $\sigma^{\frac{2}{3}}$.

Bursting oscillations, bifurcation delay and multi-stability in complex nonlinear systems

This special issue presents 32 contributions on bifurcation, chaos, and bursting oscillations. Some of them are devoted to dynamical behaviors, some to modeling and stability analysis. Bifurcation and chaos are important in the research of nonlinear dynamical behaviors, and bursting oscillations at multiple time scales also have a significant influence on these systems. This leads to many conclusions that can be used to guide practical engineering. A large number of new methods have been proposed for practical model-building and analysis, including scaling, adaptive control, and deep learning methods.

Bursting behaviors induced by the bifurcation delay in a generalized parametrically forced van der Pol-Duffing system

Bursting behaviors induced by the bifurcation delay in a generalized parametrically forced van der Pol-Duffing system

Complex mixed-mode oscillations based on a modified Rayleigh-Duffing oscillator driven by low-frequency excitations

The complex mixed-mode oscillation patterns are proposed and analyzed in a modified Rayleigh-Duffing oscillator based on the bifurcation theory in this paper. Four mixed-mode oscillations, namely “Homoclinic/Homoclinic-Homoclinic/Homoclinic” intermittent type, “fold/Homoclinic-Homoclinic/supHopf” intermittent type, “fold/Homoclinic-supHopf/supHopf” intermittent type and “fold/Homoclinic” type, are discussed in detail. Considering the low frequency excitations as slow-changing state variables, a modified autonomous system is obtained. The bifurcation characteristics of the fast subsystem are presented briefly by using the bifurcation theory. Then, we investigate the generation principle of the bifurcation delay phenomenon that performs a critical role in the production of two mixed-mode oscillations. This paper presents a fact that the dynamical behaviors are sensitive to the values of the system parameters and the parameters determine different forms of the repetitive spiking states that leads to different patterns of the mixed-mode oscillations. In addition, the theoretical analysis and numerical simulations are compared to illustrate the correctness of this paper.

Driven Force Induced Bifurcation Delay on the Chaotic Financial System

To understand the variations in the financial characteristics, we examine the dynamical behaviors by considering the chaotic financial model with external force. First, the dynamical characteristics are analyzed by introducing the external driven force in the price index with commodity demand. We discover that the presence of an external force causes the alternate occurrence of oscillatory and steady states as a function of time. Interestingly, we find the existence of bifurcation delay (BD) during the transition from oscillatory (OS) to steady state (SS) or vice versa. Bifurcation delay is a phenomenon in which the bifurcation does not occur at the actual bifurcation point but rather at a later time, which is referred to as bifurcation delay. To confirm the delay in bifurcation, we estimate the actual bifurcation point and compare it to the observed bifurcation transition. Furthermore, to understand the variations in the bifurcation delay, we estimate the delay time between each consecutive cycle and find random fluctuations in the BD. Following that, the BD is virtualized via a transformed phase portrait. In addition, we show decreasing the value of average BD while increasing the frequency of external forcing. Second, the presence of BD is explored by incorporating external forces into the investment demand with unit investment cost. We discover the existence of a similar phenomenon with a constant bifurcation delay.

On Stability Switches and Bifurcation of the Modified Autonomous Van der Pol–Duffing Equations via Delayed State Feedback Control

This paper considers the Modified Autonomous Van der Pol–Duffing equation subjected to dynamic state feedback, which can well characterize the dynamic behaviors of the nonlinear dynamical systems. Both the issues of local stability switches and the Hopf bifurcation versus time delay are investigated. Associating with the τ decomposition strategy and the center manifold theory, the delay stable intervals and the direction and stability of the Hopf bifurcation are all determined. Specifically, the computation of purely imaginary roots (symmetry to the real axis), the positive real root formula for cubic equation and the sophisticated bilinear form of adjoint operators are proposed, which make the calculations mentioned in our discussion unified and simple. Finally, the typical numerical examples are shown to illustrate the correctness and effectiveness of the practical technique.

Complex mixed-mode vibration types triggered by the pitchfork bifurcation delay in a driven van der Pol-Duffing oscillator

The intent of this work is to explore the generation mechanism of four mixed-mode vibration types triggered by the pitchfork bifurcation delay phenomenon in a driven van der Pol-Duffing oscillator. The four mixed-mode vibration types can be named “delayed sup-pitchfork/sup-Hopf” type for Homoclinic connection, symmetric “delayed sup-pitchfork/sup-Hopf” type, “point/point” type for delayed sup-pitchfork and “delayed sup-pitchfork/delayed Homoclinic” type, respectively. Our study proves that the pitchfork bifurcation delay phenomenon deserves a prominent role in the occurrence of the four mixed-mode vibration types, which enriches the dynamical behaviors of the bifurcation delay phenomenon as well as the possible routes to mixed-mode vibrations.

Bifurcation delay, travelling waves and chimera-like states in a network of coupled oscillators

We report novel bifurcation delays and chimera-like states in a network of driven FitzHugh-Nagumo (FHN) oscillators, where each oscillator can rotate either clockwise or anticlockwise. Slow variation of the time-dependent parameter of FHN oscillator near the bifurcation point leads to a delay in the bifurcation. The time delay in the bifurcation is independent of the direction of rotation of the system (clockwise or anticlockwise rotation). When the FHN oscillators are coupled via dissimilar variables, then bifurcation delay in the anticlockwise rotating oscillator changes, creating chimera-like structures and also travelling waves at certain coupling strength. Bifurcation preponement is also observed for other range of coupling strengths. Similar results are also observed in networks of van der Pol and Landau-Stuart oscillators suggesting that the phenomenon is general rather than specific to coupled FHN systems.

Complex aperiodic mixed mode oscillations induced by crisis and transient chaos in a nonlinear system with slow parametric excitation

For dynamic systems with multiple time scales, its long-term behaviors usually present as mixed mode oscillations (MMOs). The formation of MMOs is closely related to the fast transitions evoked by the bifurcations of the fast sub-system. Up till now, most of the results reported in deterministic systems focus on periodic MMOs, namely the holistic structure of waveform shows periodicity, no matter whether the slow manifolds that the fast–slow flow follows are periodic or not. Moreover, since the motion of the fast–slow flow is modulated by the critical manifolds, qualitative influences of transient behaviors on the holistic structure of MMOs are less concerned. While in this paper, three patterns of aperiodic MMOs are analyzed based on a three-dimensional nonautonomous system under slow parametric excitation. According to the types of manifolds among which the fast transitions take place, these MMOs can be named as aperiodic “cycle-chaos-cycle” MMOs, aperiodic “cycle-chaos-cycle” MMOs with additional chaotic bursters, and aperiodic “chaos-cycle-point” MMOs. By combining the fast–slow analysis on certain time intervals and the probe into the transient chaos near boundary crisis, generation mechanisms of these MMOs are investigated. Our results show that the relative location between the boundary crisis value and the minimum of slow variable will qualitatively change the pattern of large amplitude oscillations (LAOs); thus, the composition of bursters can be decided by the intensity of transient chaos. On the other hand, the small amplitude oscillations (SAOs) evoked by interior crisis may contain inner structures related to bifurcation delay and sharding phenomenon. Particularly, interior crisis can lead to merging phenomena of chaotic attractors. Thus, when the chaotic LAOs stages are over, the fast–slow flow can transit to upper or lower branch in a irregular way, which results in the uncertain locations of SAOs.

Effect of processing delay on bifurcation delay in a network of slow-fast oscillators.

Bifurcation delay or slow passage effect occurs in dynamical systems with slow-fast time-varying parameters. In this work, we report the impact of processing delay on bifurcation delay in a network of locally coupled slow-fast systems with self-feedback delay. We report that the network exhibits coexisting coherent (synchronized) and incoherent (desynchronized) states among the oscillators as a function of various parameters like self-feedback delay, processing delay, and amplitude of the external current. In particular, we show the decrease of the synchronized region (control of synchronization) for (i) a fixed value of processing delay with varying self-feedback delay and (ii) fixed self-feedback delay with increasing processing delay. In contrast, we observe the increase of the synchronized region (control of desynchronization) for fixed processing delay and self-feedback delay while varying the amplitude of the external current. Finally, we have also analyzed the effect of processing delay on bifurcation delay with the presence of noise and we report that the inhomogeneity in the additional noise does not affect the asymmetry in a bifurcation delay.

Bursting oscillations in Sprott B system with multi-frequency slow excitations: two novel “Hopf/Hopf”-hysteresis-induced bursting and complex AMB rhythms

Bursting oscillations in systems with multi-frequency slow excitations reflect multi-timescale dynamics. As there exist many kinds of multiple periodic excitations in actual dynamical systems, the analysis of dynamical behavior of such systems has received much attention. In this article, we first investigate the bursting in Sprott B system with a single excitation. We show that Hopf bifurcation delay may exhibit due to the effect of slow passage through the supercritical Hopf bifurcation. Interestingly, such delay behavior leads to a pinched “Hopf/Hopf”-hysteresis bursting when delay time is long enough. Then, we focus on the effects of the additional excitation $${\beta _2}{\mathrm{cos(}}{\omega _2}{\mathrm{t)}}$$ on the zero-crossing pinched “Hopf/Hopf”-hysteresis bursting. We find that, under the effects of $${\beta _2}{\mathrm{cos(}}{\omega _2}{\mathrm{t)}}$$ , “Hopf/Hopf”-bifurcation-delay-induced hysteresis behaviors may exhibit very interesting dynamical characteristics. In particular, each equilibrium branch may become the one with twists and turns. Then, the novel bursting pattern, turnover-of-zero-crossing pinched-“Hopf/Hopf”-hysteresis-induced bursting, is disclosed accordingly. In addition, we show that the equilibrium branches may cross each other spirally, leading to multiple intersection points under certain parameter conditions. There are two Hopf bifurcation points perching separately upon either side of the intersection points. This creates multiple delay-induced hysteresis behavior, and the other novel bursting pattern, cascaded pinched “Hopf/Hopf”-hysteresis bursting, is thus created. Finally, we investigate the complex amplitude-modulated bursting (CAMB) rhythms in Sprott B system with multi-frequency slow excitations.

BURSTING OSCILLATIONS AS WELL AS THE DELAYED PITCHFORK BIFURCATION BEHAVIORS IN A CLASS OF CHAOTIC SYSTEM

Due to wide existence of multiple-time-scale problems in practical engineering, the complicated dynamic behaviors and their generation mechanism have become one of the hot topics at home and abroad. The systems with multiple time scales can often exhibit bursting oscillations with the bifurcation delay phenomenon. In order to investigate the bifurcation mechanism of bursting oscillations caused by bifurcation delay in a nonlinear system, a parametric excitation is introduced in a novel three-dimensional chaotic system. When the exciting frequency is far less than the natural frequency, the coupling of two time scales involves the vector field, which leads to the bursting oscillations. By considering the whole exciting term as a slow-varying parameter, the original system can be considered as a generalized autonomous system, which can be regarded as the fast subsystem. Upon the analysis of equilibrium points and bifurcation conditions of the fast subsystem, combining with the transformed phase portraits, the bifurcation mechanisms of bursting oscillations is presented. Four typical cases with different parameter conditions are discussed to reveal the evolution of the bursting oscillations. It is found that when the slow-varying exciting term passes across the bifurcation points, the delayed behaviors of super-critical pitchfork bifurcation can be observed. With the increase of the exciting amplitude, the occurring needed for the bifurcation delay is increased gradually. When the delayed behaviors end in different parameter regions, different types of bursting oscillations which may surround different attractors such as equilibrium points and limit cycles appear.

Universality of miscible–immiscible phase separation dynamics in two-component Bose–Einstein condensates

We investigate the non-equilibrium dynamics across the miscible–immiscible phase separation in a binary mixture of Bose–Einstein condensates. The excitation spectra reveal that the Landau critical velocity vanishes at the critical point, where the superfluidity spontaneously breaks down. We analytically extract the dynamical critical exponent z = 2 and static correlation length critical exponent v = 1/2 from the Landau critical velocity. Moreover, by simulating the real-time dynamics across the critical point, we find the average domain number and the average bifurcation delay show universal scaling laws with respect to the quench time. We then numerically extract the static correlation length critical exponent v = 1/2 and the dynamical critical exponent z = 2 according to Kibble–Zurek mechanism. The scaling exponents (v = 1/2, z = 2) in the phase separation driven by quenching the atom–atom interaction are different from the ones (v = 1/2, z = 1) in the phase separation driven by quenching the Rabi coupling strength (2009 Phys. Rev. Lett. 102 070401; 2011 Phys. Rev. Lett. 107 230402). Our study explores the connections between the spontaneous superfluidity breakdown and the spontaneous defect formation in the phase separation dynamics.

Bifurcation delay in a network of locally coupled slow-fast systems.

We study the evolution of bifurcation delay in a network of locally coupled slow-fast systems. Our study reveals that a tiny perturbation even in a single node causes asymmetry in bifurcation delay. We investigate the evolution of bifurcation delay as a function of various parameters, such as feedback coupling strength, amplitude of external force, frequency of external force, and delay coupling strength. We show that a traveling wave is generated as the result of introducing local parameter mismatch, and the bifurcation delay shows a dip in the spatial profile. We believe that these spatiotemporal patterns in bifurcation delay shed light on the dynamics of neuronal networks.