Transformed Phase Portraits Research Articles

Multiple modes of bursting phenomena in a vector field of triple Hopf bifurcation with two time scales

For the development of the theoretical framework of slow–fast dynamics, it is essential to study the mechanism of bursting oscillations with high co-dimensional bifurcations in the high-dimensional vector field. This paper is aimed to explore the bursting oscillations in the normal form up to the third order of a vector field with triple Hopf bifurcation at the origin. We establish an eight-dimension system with the slow–fast effect, in which two time scales exist. The fast subsystem is represented as a six-dimensional vector field with triple Hopf bifurcation at the origin, while the slow subsystem is adopted to adjust the variation of parameters in the fast subsystem, which may lead to abundant dynamic behaviors. To better understand the qualitatively different behaviors in the fast subsystem, the trivial solution, single-mode, two-mode, and three-mode movements are introduced based on the projections of trajectory on the three sub-planes. As a result, several types of bursting phenomena, such as Hopf/Hopf bursting oscillations with single-mode, Hopf/Hopf/fold bursting oscillations with single-mode, Hopf/Hopf bursting oscillations with two-mode have been revealed, the mechanism of which is obtained upon the superposition of equilibrium branches together with bifurcations in the fast subsystem as well as transformed phase portrait. In addition, for bursting oscillations on the whole phase space, the amplitude of spiking oscillations may change according to two different limit cycles, which leads to an abrupt change of the oscillating amplitude on the sub-plane, corresponding to different projections of Poincaré. It can be observed the state variables on the different sub-planes alternate between synchronized and non-synchronized states in the fast subsystem, which may be accounted for by the bifurcations.

All possible bursting oscillations in neighborhood of a Hopf bifurcation point induced by low-frequency excitation

Hopf bifurcation may be one of most common bifurcations, which results in the transitions between quiescence and spiking states in a bursting attractor, since it causes the alternation between an equilibrium point and a limit cycle. Though a lot of bursting attractors related to Hopf bifurcation have been obtained, while how many types of bursting attractors may appear in the neighborhood of a Hopf bifurcation point still remains an open problem. Here we try to answer the question based on the normal form up to the seventh order of a vector field with Hopf bifurcation at the origin. By introducing a low-frequency external excitation to the normal form, we derive the equilibrium branches and their bifurcations of the fast subsystem with the variation of the slow-varying excitation, based on which four typical cases corresponding to qualitatively different evolution of the behaviors are presented. Accordingly, four types of quasi-periodic bursting oscillations, i.e. Hopf/Hopf bursting attractors with (1) and (2) without saddle on the limit cycle bifurcation in the spiking oscillations, respectively, (3) fold/Hopf/fold/Hopf bursting attractors, and (4) bursting attractor with saddle on limit cycle bifurcation at the transition, are obtained with the increase of the exciting amplitude, the mechanism of which is obtained by overlapping equilibrium branches together with bifurcations in generalized autonomous system and transformed phase portrait. It is found that when two coexisted stable attractors are separated by an unstable attractor on the phase plane in generalized autonomous system, the influence of one of the stable attractors on the dynamics of full system may disappear. Furthermore, when the window for a stable attractor between two bifurcation points along slow-varying parameter axis is relatively short, the effect of the attractor as well as the bifurcation can only appear when the external excitation changes slowly enough. The amplitude of the repetitive spiking oscillations may vary according to two different limit cycles separated by saddle on limit cycle bifurcation, at which a sudden change of the oscillating amplitude can be observed.

Bursting oscillations with multiple modes in a vector field with triple Hopf bifurcation at origin

The mechanism of bursting oscillations in high dimensional systems with the coupling of two scales is an open problem in nonlinear dynamics, since there may coexist stable attractors in the fast subsystem with different bifurcations. Here we consider the bursting oscillations in the normal form up to the third order of a vector field with triple Hopf bifurcation at origin. When slow-varying external excitation is introduced, with the increase of the exciting amplitude, Hopf bifurcations in the three sub-planes occur in turn, which leads to three stable limit cycles. The dynamics may evolve from a 2-D torus bursting oscillations with two-mode and three-mode, respectively, the mechanism of which can be presented by overlapping the transformed phase portrait and the equilibrium states as well as the bifurcations of the fast subsystem. Synchronization and non-synchronization between the state variables in different sub-planes can be observed, which is explained by the bifurcations. An interesting phenomenon is found that the three-mode bursting oscillations still behave in quasi-periodic type instead of high dimensional torus. The projections of the trajectory in the sub-planes alternate between quiescence and repetitive spiking state with the oscillating frequency approximated at the frequency of the corresponding limit cycle.

Bursting oscillations induced by multiple coexisting attractors in a modified 3D van der Pol-Duffing system

In this paper, a novel three-dimensional modified van der Pol-Duffing circuit with a quintic nonlinear resistor is proposed, and the multiple stable attractors are observed. The complex bursting patterns are investigated, various patterns of bursting oscillations are obtained under slowly changing frequency, and the transition mechanism of which can be revealed via the modified slow–fast analysis as well as the transformed phase portrait. It can be found that for the system with mono-stability, only one bursting pattern exists, while for the case with multi-stability, the trajectory can enter into different attraction areas in the transition region, leading to complex forms of bursting oscillations. Furthermore, the multi-valued characteristic of the system are observed by the domain of attraction.

Bursting dynamics and the zero-Hopf bifurcation of simple jerk system

We characterize the zero-Hopf bifurcation at a singular point of a parameter jerk system. By employing the second order averaging theory, we demonstrate that up to three periodic orbits generated as disturbance parameters tend to zero. Again, both the bifurcation mechanism and bursting dynamics of the 3D jerk system with external periodic excitation are systematically explored. While an order difference exists between the frequency of external excitation and the average frequency of the system, the system exhibits bursting oscillations. The mechanisms of different bursting oscillations are investigated by means of the equilibrium point curve and the transformed phase portraits. As the amplitudes of the excitation change, the system displays “delayed symmetric pitchfork/point, delayed symmetric pitchfork/supHopf, and delayed pitchfork/supHopf/homoclinic connection bursting”. Finally, an analog circuit is designed to verify the complex bursting phenomena of the system.

A memristive non-smooth dynamical system with coexistence of bimodule periodic oscillation

In order to explore the burstingoscillations and the formation mechanism of memristive non-smooth systems, a third-order memristor model and an external periodic excitation are introduced into a non-smooth dynamical system, and a novel 4D memristive non-smooth system withtwo-timescale is established. The system is divided into two different subsystems by a non-smooth interface, which can be used to simulate the scenario where a memristor encounters a non-smooth circuit in practical application circuits. Three different bursting patterns and bifurcation mechanisms are analyzed with the time series, the corresponding phase portraits, the equilibrium bifurcation diagrams, and the transformed phase portraits. Itispointedthatnotonly the stability of the equilibrium trajectory butalso the non-smooth interface mayinfluence the bursting phenomenon, resulting in the sudden jumping of the trajectory and non-smooth bifurcation at the non-smooth interface. In particular, the coexistence of bimodule periodic oscillations at the non-smooth interface can be observed in this system. Finally, the correctness of the theoretical analysis is well verified by the numerical simulation and Multisim circuit simulation. This paper is of great significance for the future analysis and engineering application of the memristor in non-smooth circuits.

On occurrence of mixed-torus bursting oscillations induced by non-smoothness

The paper devotes to the slow–fast behaviors of a higher-dimensional non-smooth system with the coupling of two scales. Some novel bursting attractors and interesting phenomena are presented, especially the so-called mixed-torus bursting oscillations, in which the trajectory moves along different tori in turn, though the bursting attractor still behaves in quasi-periodic form. Based on a 4-D hyper-chaotic model with two scrolls, a modified non-smooth slow–fast version is established. Upon the smooth and non-smooth bifurcations analysis, the coexisted attractors with the variation of the slow-varying parameter are derived. With the increase in the exciting amplitude, bursting oscillations may change from periodic to quasi-periodic attractor, the mechanism of which is obtained by employing the overlap of the transformed phase portrait and coexisted attractors as well as the bifurcations. Sliding along the boundary on the trajectory can be observed on the bursting attractor, the mechanism of which can be accounted for by employing the non-smooth theory or by the in-turn influence of two pseudo-attractors in different regions. Super-cone defined by an unstable focus may lead to the period-adding or period-decreasing phenomena in the spiking oscillations, while fold limit cycle bifurcation may result in the dramatical change of the spiking amplitude. Different limit cycles may involve the full vector field, resulting in the so-called mixed-torus bursting oscillations, in which the quiescent states seem to be pivots of the oscillations.

All Possible Bursting Attractors in the Neighborhood of Hopf Bifurcation Point Under Periodic Excitation

For a vector field with Hopf bifurcation at the origin, when periodic excitation is introduced in which the frequency of the excitation is far less than the natural frequency, bursting oscillations, characterized by the alternations between the large-amplitude and the small-amplitude oscillations, can often be observed. The paper tries to answer the common question of how many types of bursting attractors may appear in such a vector field. Based on the normal form of Hopf bifurcation up to the third order, by regarding the whole exciting term as a slow-varying bifurcation parameter, all possible equilibrium branches of the generalized autonomous system are derived, the bifurcation sets of which divide the parameter space into regions associated with different types of dynamical behaviors. It is found that, there exist totally four possible combinations of the regions that the trajectory of the full system may visit as the slow-varying parameter varies. Accordingly, four qualitatively different behaviors, i.e. periodic symmetric oscillations, periodic symmetric mixed-mode oscillations, fold/Hopf/Hopf/fold and fold-Hopf/fold-Hopf bursting attractors, are presented, the mechanism of which is obtained by overlapping the transformed phase portrait and the equilibrium branches in the generalized autonomous system. Furthermore, the dynamics of the generalized autonomous system in different regions in the parameter space may influence the structures of the bursting oscillations when the trajectory enters the corresponding regions, while the bifurcations at the boundaries of the regions may lead to changes between the quiescent states and spiking states in the bursting attractors.

Influence of the Coexisting Attractors on the Slow–Fast Behaviors in the Fast Subsystem

This paper is devoted to exploring the effect of the coexisting attractors in the fast subsystem of a full nonlinear system on the complex dynamics. As an example, a modified 3D van der Pol–Duffing circuit is discussed, in which the coexistence of five attractors including a hidden one was reported in 2020. By introducing a parametric excitation, when the exciting frequency is far less than the natural frequency, a slow–fast model with the two-scale coupling is established in the frequency domain. Regarding the exciting term as a slow-varying parameter, all the attractors as well as the bifurcations of the fast subsystem are derived. The coexistence conditions as well as the coexisting attractors, including the equilibrium points and limit cycles, are presented. With the variation of the exciting amplitude, different types of bursting oscillations are observed, the mechanism of which is revealed using a modified slow–fast analysis method, by combining the transformed phase portraits and the attractors as well as the bifurcations. It is found that the coexisting attractors of the fast subsystem may lead to coexisting bursting solutions in the full system, in which a trajectory may visit one of the attracting basins of the attractors, or result in a merged bursting motion, which may visit the attracting basins in turn. The influence of some attractors and the associated bifurcations may vanish in the case that the trajectory passes directly across the parameter interval corresponding to the attractors because of the inertia of the motion; this may also occur in the case that the trajectory does not visit the associated attracting basins, but it starts initially from the basins. It should be pointed out that, when the exciting frequency is small enough, the disappeared effect of stable attractors may reoccur. Furthermore, if the parameter interval between two types of codimension-1 bifurcation points is short enough, the trajectory may behave in a combination of two bifurcations, which is somewhat similar to that caused by a codimension-2 bifurcation.

Novel bursting patterns and the bifurcation mechanism in a piecewise smooth Chua’s circuit with two scales

The aim of this paper is to investigate the influence of the coupling of two scales on the dynamics of a piecewise smooth dynamical system. A relatively simple model with two switching boundaries is taken as an example by introducing a nonlinear piecewise smooth resistor and a harmonically changed electric source into a typical Chua’s circuit. Taking suitable values of the parameters, four different types of bursting oscillations are observed corresponding to different values of the exciting amplitude. Regarding the periodic excitation as a slow-varying parameter, equilibrium branches of the fast subsystem as well as the related bifurcations, such as fold bifurcation, Hopf bifurcation, period doubling bifurcation, nonsmooth Hopf bifurcation and nonsmooth fold limit cycle bifurcation, are explored with theoretical and numerical methods. With the help of the overlap of the transformed phase portrait and the equilibrium branches, the mechanism of the bursting oscillations can be analyzed in detail. It is found that for relatively small exciting amplitude, since the trajectory is governed by a smooth subsystem, only conventional bifurcations take place, leading to the transitions between the spiking states and quiescent states. However, with an increase of the exciting amplitude so that the trajectory passes across the switching boundaries, nonsmooth bifurcations occurring at the boundaries may involve the structures of attractors, leading to complicated bursting oscillations. Further increasing the exciting amplitude, the number of the spiking states decreases although more bifurcations take place, which can be explained by the delay effect of bifurcation.

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.

Bursting Oscillation and Its Mechanism of a Generalized Duffing–Van der Pol System with Periodic Excitation

The complex bursting oscillation and bifurcation mechanisms in coupling systems of different scales have been a hot spot domestically and overseas. In this paper, we analyze the bursting oscillation of a generalized Duffing–Van der Pol system with periodic excitation. Regarding this periodic excitation as a slow-varying parameter, the system can possess two time scales and the equilibrium curves and bifurcation analysis of the fast subsystem with slow-varying parameters are given. Through numerical simulations, we obtain four kinds of typical bursting oscillations, namely, symmetric fold/fold bursting, symmetric fold/supHopf bursting, symmetric subHopf/fold cycle bursting, and symmetric subHopf/subHopf bursting. It is found that these four kinds of bursting oscillations are symmetric. Combining the transformed phase portrait with bifurcation analysis, we can observe bursting oscillations obviously and further reveal bifurcation mechanisms of these four kinds of bursting oscillations.

Bursting Oscillations and Experimental Verification of a Rucklidge System

Bursting oscillations are ubiquitous in multi-time scale systems and have attracted widespread attention in recent years. However, research on experimental demonstration of the bursting oscillations induced by delayed bifurcation is very rarely reported. In this paper, a parametrically driven Rucklidge system is introduced and a distinct delayed behavior is observed when the time-varying parameter passes through the pitchfork bifurcation point. Different bursting patterns induced by such a delayed behavior are numerically investigated under different excitation amplitudes based on the fast–slow analysis method. Furthermore, in order to reproduce the bursting electronic signals and explore the underlying formation mechanisms experimentally, a real physical circuit of the parametrically driven Rucklidge system is developed by using off-the-shelf electronic devices. The real-time measurement results such as time series, phase portraits and transformed phase portraits are in good qualitative agreement with those obtained from the numerical computations. The experimental evidence to verify bursting oscillations induced by delayed pitchfork bifurcation is thus provided in this study.

Slow–Fast Behaviors and Their Mechanism in a Periodically Excited Dynamical System with Double Hopf Bifurcations

This paper focuses on the influence of two scales in the frequency domain on the behaviors of a typical dynamical system with a double Hopf bifurcation. By introducing an external periodic excitation to the normal form of the vector field with double Hopf bifurcation at the origin and taking the exciting frequency far less than the natural frequency, a theoretical model with two scales in the frequency domain is established. Regarding the whole exciting term as a slow-varying parameter leads to a generalized autonomous system, in which the equilibrium branches and their bifurcations with the variation of the slow-varying parameter can be derived. With the increase of the exciting amplitude, different types of bifurcations may be involved in the generalized autonomous system, resulting in several qualitatively different forms of bursting attractors, the mechanism of which is presented by overlapping the transformed phase portraits and the bifurcations of the equilibrium branches. It is found that the single mode 2D torus may evolve to the bursting attractors with mixed modes, in which the trajectory alternates between the single mode oscillations and the mixed mode oscillations. Furthermore, the transitions between the quiescent states and the spiking states may not occur exactly at the bifurcation points because of the slow passage effect, while Hopf bifurcations may cause different forms of repetitive spiking oscillations.

Bursting behaviors as well as the mechanism of controlled coupled oscillators in a system with double Hopf bifurcations

Up to now, most of the work related to the slow-fast dynamical systems are based on the low dimensional systems with only codimension-one bifurcations at the transitions between the quiescent states and the spiking states, while there usually only exists one slow variable in the system. Since the coupling effect of two scales on the behaviors of high dimensional systems with two or more slow variables as well as high co-dimensional bifurcations lacks exploration, in this paper, we investigate the slow-fast effect on a 6-dimension system, which can be regarded as the coupling of two subsystems. The fast subsystem is expressed as the normal form of the vector field with a double Hopf bifurcation at the origin, while the slow-varying parameter in the fast subsystem is controlled by the slow subsystem. For the fast subsystem, the trajectory is projected onto two independent sub-planes for better observation, and the movement of bursting oscillators can be synthesized with information from each dimension. In the sub-plane, the mechanism of bursting oscillations is derived by employing the overlap of the transformed phase portrait and the equilibrium branches as well as the bifurcations. It is found that, Hopf bifurcation sets may cause the fast subsystem to transform between the single mode and the mixed-mode or between different single modes, leading to bursting oscillations in certain sub-planes or the whole four dimensions of the fast subsystem. Furthermore, interesting phenomena may be caught, such as the trajectory jumping in the process of oscillation as well as transforming between different oscillation modes of the fast subsystem by the way of jump as a result of the fold bifurcation set.

Bursting oscillations in a slow-varying periodically excited vector field with Bogdanov–Takens bifurcation

The main purpose of the article is to classify all the possible bursting oscillations in a vector field with Bogdanov–Takens bifurcation at the origin. Based on the universal unfolding of the normal form of the vector field, the topological structure in the neighborhood of the bifurcation point on the unfolding parameters is presented. Replacing one of the unfolding parameters by a slow-varying periodic exciting term, the coupling of two scales in frequency domain involves the vector field, which may lead to the bursting oscillations. According to the bifurcation analysis, we focus on three typical cases to investigate the dynamical evolution with the increase of the exciting amplitude. By introducing the transformed phase portrait, the mechanism of bursting oscillations can be presented. Three types of bifurcations, that is, fold, Hopf, and saddle on the limit cycle bifurcations may cause the alternations of the trajectory between the quiescent states and the spiking states, different combinations of which may result in different bursting attractors. Furthermore, the inertia of the movement may result in the delay effect of the bifurcation, which may lead to the disappearance of the bifurcation influence and the corresponding spiking oscillations.

On occurrence of bursting oscillations in a dynamical system with a double Hopf bifurcation

The slow-fast effect in a vector field with a codimension-two double Hopf bifurcation at the origin is investigated in the paper. To explore the typical evolution of the dynamics, the universal unfolding of the normal form of the vector field is taken into consideration. When the parametric excitation is introduced, the frequency of which is far less than the two natural frequencies, slow-fast behaviors may appear. Regarding the whole parametric excitation term as a slow-varying parameter, the equilibrium branches and the bifurcations of the generalized autonomous fast subsystem can be derived. With the variation of the exciting amplitude, different types of bursting oscillations caused by the coupling of two scales in frequency domain may appear, the mechanism of which are obtained by employing the overlap of the transformed phase portraits and the equilibrium branches as well as the bifurcations of the fast subsystem. It is found that, for relatively small exciting amplitude, no bifurcation of the fast subsystem occurs, and the system behaves in quasi-periodic oscillations. With the increase of the exciting amplitude, different types of bifurcations may take place, leading to the single-mode bursting oscillations, which may evolve to mixed-mode bursting oscillations. Symmetric breaking bifurcations result in the transitions between a symmetric attractor and two co-existed asymmetric attractors in the system. Furthermore, for the mixed-mode bursting oscillations, the trajectory may alternate between the single-mode oscillations and the mixed-mode oscillations via bifurcations of the equilibrium branches, causing the synchronization and non-synchronization between different state variables.

Two bursting patterns induced by system solutions approaching infinity in a modified Rayleigh–Duffing oscillator

In this paper, the mechanism of system solutions approaching infinity is explored based on a modified Rayleigh–Duffing oscillator with two slow-varying periodic excitations. System solutions approaching infinity is a new novel route to bursting oscillation, and are not reported yet. The system can be separated into a fast subsystem and a slow subsystem according to the slow–fast analysis method. We find that there is a critical value for the fast subsystem, which limits the original region of the stable equilibrium point and the stable limit cycle, the right of which is the divergent region. When the control parameter slowly varies closely to the critical value $$\delta _{\mathrm{CR}} $$ , both the stable equilibrium point and the stable limit cycle quickly leave the original region and approach positive infinity. The mechanism of two different bursting forms called bursting oscillation of point/point and bursting oscillation of cycle/cycle induced by system solutions approaching infinity are explored. This paper provides a new possible route to bursting oscillation unrelated to bifurcations and deepens the comprehension of bursting dynamics behaviours. Lastly, the accuracy of our study is verified by overlapping the transformed phase portraits onto the bifurcation diagrams.

Bursting oscillations and bifurcation mechanism in a permanent magnet synchronous motor system with external load perturbation

Abstract This research aims to study the bursting oscillations and bifurcation mechanism in a permanent magnet synchronous motor (PMSM) system with external load perturbation. When an order gap exists between the external load perturbation frequency and the natural frequency of the PMSM system, the PMSM system exhibits bursting oscillation behaviors. First, the mathematical model of a PMSM with external load perturbation is established by considering the external load perturbation as a slow-varying parameter. The PMSM system is transformed into a generalized autonomous PMSM system. Second, the Fold and Hopf bifurcation conditions of the PMSM system are obtained through bifurcation analysis. Third, the bursting oscillations and bifurcation mechanism of the PMSM system are analyzed in detail by using equilibrium point distribution and bifurcation diagrams, transformed phase portraits, and time series diagrams. The PMSM system exhibits periodic symmetric Fold/Fold bursting oscillations and periodic delayed subHopf/subHopf bursting oscillations, as the parameters change. When the amplitude of the external load perturbation changes and the PMSM system parameters are fixed, the bursting oscillation processes in different ranges of amplitude are revealed.

On occurrence of bursting oscillations in a dynamical system with a double Hopf bifurcation and slow-varying parametric excitations

Slow-fast analysis has been extensively used in the past to study the occurrence of bursting oscillations. Most bursting oscillations studies are performed on the low dimensional autonomous systems where only codimension-1 bifurcations take place at the transitions of quiescent and spiking states. However, in high dimensional slow-fast dynamical systems, there exist higher co-dimensional bifurcations, which may lead to much more complicated mechanisms for bursting oscillations. To reveal and understand the complicated phenomena, the present paper investigates the normal form of a four-dimensional cubic order system with parametric excitation as a slow parameter. Parametric frequency is chosen far less than the natural frequencies to make sure the coupling of fast and slow scales. In the absence of parametric excitation, the system has a double Hopf bifurcation at the origin. Several equilibrium branches and mechanisms of bursting oscillations are obtained for different amplitudes of the parametric excitation. The trajectory of the system is projected onto two independent sub-planes, while the movement of bursting oscillators is synthesized by the properties in each sub-plane. In the sub-plane, the mechanism of bursting oscillations is derived by employing the overlap of the transformed phase portrait and the equilibrium branches as well as the bifurcations. The codimension-2 double Hopf bifurcation may result in the occurrence of the bursting oscillations in four dimensions. When the excitation amplitude is relatively small, the trajectory only follows the equilibrium branches of trivial and single-mode solutions. With the increase of the excitation amplitude, the trajectory switches between the equilibrium branches of single-mode and mixed-mode. Furthermore, a few interesting phenomena are presented.