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We study the influence of the coexisting steady states in high-dimensional systems on the dynamical evolution of the vector field when a slow-varying periodic excitation is introduced. The model under consideration is a coupled system of Bonhöffer–van der Pol (BVP) equations with a slow-varying periodic excitation. We apply the modified slow–fast analysis method to perform a detailed study on all the equilibrium branches and their bifurcations of the generalized autonomous system. According to different dynamical behaviors, we explore the dynamical evolution of existing attractors, which reveals the coexistence of a quasi-periodic attractor with diverse types of bursting attractors. Further investigation shows that the coexisting steady states may cause spiking oscillations to behave in combination of a 2D torus and a limit cycle. We also identify a period-2 cycle bursting attractor as well as a quasi-periodic attractor according to the period-2 limit cycle. [ABSTRACT FROM AUTHOR]

By taking a low-frequency excited modified Chua's circuit model with discontinuous vector field as an academic example, this paper introduces the periodic bursting patterns characterised by stick–slip motions as well as the underlying generation mechanism uniquely belonging to Filippov-type slow–fast dynamical systems (FSFDSs). Based on the tangencies and the visibilities, this paper is the first to clearly explain the unique transition routes to bursting phenomena involving stick–slip motions by introducing the subdivisions of the sliding region. The results indicate that such transition routes are heavily dependent on the local structures of pseudoequilibrium (PE), performing distinct non-conventional bifurcation schemes. Furthermore, two transition routes respectively corresponding to a stable node-type PE and a stable focus-type PE as well as the induced periodic bursting oscillations are further discussed in numerical simulations. Particularly, it should be pointed out that the second one is a novel route to bursting oscillation in FSFDSs. [ABSTRACT FROM AUTHOR]

Since the famous slow–fast dynamical system referred to as the Hodgkin–Huxley model was proposed to describe the threshold behaviors of neuronal axons, the study of various slow–fast dynamical behaviors and their generation mechanisms has remained a popular topic in modern nonlinear science. The primary purpose of this paper is to introduce a novel transition route induced by the comprehensive effect of special rest spike bistability and timescale difference rather than a common bifurcation via a modified Chua's circuit model with an external low-frequency excitation. In this paper, we attempt to explain the dynamical mechanism behind this novel transition route through quantitative calculations and qualitative analyses of the nonsmooth dynamics on the discontinuity boundary. Our work shows that the whole system responses may tend to be various and complicated when this transition route is triggered, exhibiting rich slow–fast dynamics behaviors even with a very slight change in excitation frequency, which is described well by using Poincaré maps in numerical simulations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper reports the complex dynamics of a class of two-dimensional maps containing hidden attractors via linear augmentation. Firstly, the method of linear augmentation for continuous dynamical systems is generalized to discrete dynamical systems. Then three cases of a class of two-dimensional maps that exhibit hidden dynamics, the maps with no fixed point and the maps with one stable fixed point, are studied. Our numerical simulations show the effectiveness of the linear augmentation method. As the coupling strength of the controller increases or decreases, hidden attractor can be annihilated or altered to be self-excited, and multistability of the map can be controlled to being bistable or monostable. [ABSTRACT FROM AUTHOR]

In this paper, mixed-mode oscillations and bifurcation mechanism for a Filippov-type system including two time-scales in the frequency domain are demonstrated. According to classic Chua's system, we investigate a non-smooth dynamical system including two time-scales. As there exists an order gap between the exciting frequency and the natural one, the whole external excitation term can be considered as a slow-changing parameter, which results in two smooth subsystems divided by the non-smooth boundary. In addition, the critical condition about fold bifurcation (FB) is studied, and by applying the Hopf bifurcation (HB) theorem, specific formulas for determining the existence of HBs are presented. By introducing an auxiliary parameter via differential inclusions theory, the non-smooth bifurcations on the boundary are discussed. Then, the equilibrium branches and the bifurcations are derived, and two typical cases associated with different bifurcations are considered. In light of the superposition between the bifurcation curve and the transformed phase portrait, the dynamical behaviours of the mixed-mode oscillations as well as sliding movement along the non-smooth boundary are obtained, which reveal the corresponding dynamical mechanism. [ABSTRACT FROM AUTHOR]

The main purpose of the paper is to explore the influence of the coupling of two scales on the dynamics of a nonsmooth dynamical system. Based on a typical Chua's circuit, by introducing a nonlinear resistor with piecewise characteristics as well as a harmonically changed electric source, a modified nonsmooth model is established, in which the coupling of two scales in frequency domain exists. Different types of bursting oscillations, appearing in the combination of large-amplitude oscillations, called spiking oscillations (S P), and small-amplitude oscillations or at rest, denoted by quiescent states (Q S), can be observed with the variation of the exciting amplitude. When the exciting frequency is relatively small, by regarding the whole exciting term as a slow-varying parameter, the original system can be transformed into a generalized autonomous system. The phase space can be divided into three regions by the nonsmooth boundaries, in which the trajectory is governed by three different subsystems, respectively. Based on the analysis of the three subsystems as well as the behaviors on the nonsmooth boundaries, all the equilibrium branches and their bifurcations can be obtained, which can be employed to investigate the mechanism of the bursting oscillations. It is found that, for relatively small exciting amplitude, since no bifurcation on the equilibrium branches can be realized with the variation of the slow-varying parameter, the system behaves in periodic movement, which may evolve to bursting oscillations when a pair of fold bifurcations occurs with the increase of the exciting amplitude. Further increase of the exciting amplitude may lead to more complicated bursting oscillations, which may bifurcate into two coexisted asymmetric bursting attractors via symmetric breaking. Interaction between the two attractors may result in an enlarged symmetric bursting attractor, in which more forms of bifurcations at the transitions between the quiescent states and repetitive spiking states can be observed. [ABSTRACT FROM AUTHOR]

The main purpose of the paper is to investigate the effect of multiple scales in frequency domain on the complicated oscillations of Filippov system with discontinuous right-hand side. A relatively simple model based on the Chua's circuit with periodic excitation is introduced as an example. When the exciting frequency is far less than the natural frequency, implying that an order gap between the exciting frequency and the natural frequency exists, the whole exciting term can be considered as a slow-varying parameter, based on which the bifurcations of the two subsystems in different regions divided by the nonsmooth boundary are presented. Two typical cases are considered, which correspond to different distributions of equilibrium branches as well as the related bifurcations. In the first case, periodic symmetric Hopf/Hopf-fold-sliding bursting oscillations can be obtained, in which Hopf bifurcations may cause the alternations between the quiescent states and the spiking states, while fold bifurcations connect the two quiescent states moving along the stable equilibrium branches and sliding along the nonsmooth boundary, respectively. While the second case is the periodic symmetric fold/fold-fold-sliding bursting, where the fold bifurcations not only lead to the alternations between the quiescent states and the spiking states, but also connect the two quiescent states moving along the stable equilibrium branches and sliding along the nonsmooth boundary, respectively. It is pointed out that, different from the bursting oscillations in smooth dynamical systems in which the bifurcations may cause the alternations between quiescent states and spiking states, in the nonsmooth system, bifurcations may not only lead to the alternations, but also connect different forms of quiescent states. Furthermore, in the Filippov system, sliding movement along the nonsmooth boundary can be observed, the mechanism of which is presented based on the analysis of the two subsystems in different regions. [ABSTRACT FROM AUTHOR]

• Bursting oscillations of the vector field with a Double Hopf bifurcation are observed. • Bifurcation mechanism of bursting oscillations is presented. • Different forms of bursting oscillations with both single mode and mixed modes are obtained. 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. [ABSTRACT FROM AUTHOR]

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. • Bursting oscillations of the vector field with a Double Hopf bifurcation are observed. • Bifurcation mechanism of bursting oscillations is presented. • Different forms of bursting oscillations with both single mode and mixed modes are obtained. [ABSTRACT FROM AUTHOR]

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. • The transient chaos induced by the boundary crisis route can be observed. • Six different chaotic bifurcation structures can be observed. • Chaotic bursting oscillations are investigated by the modified fast/slow analysis method. • The sensitivity of system parameters on the dynamic evolutions is analyzed. [ABSTRACT FROM AUTHOR]

Time delays are many sources of complex behavior in dynamical systems. Yet its relationship with bursting dynamics needs to be further explored, particularly when the strength of feedback is a nonlinear function of delay. In this paper, we analyze the dynamics of the van der Pol–Duffing fast-slow oscillator controlled by the parametric delay feedback, where the strength of feedback control is a function exponential varying with the time delay. The system may exhibit a unique equilibrium point and three ones for the different parameters by employing the pitchfork bifurcation. Next, the stability-switches and the Hopf bifurcation curves are presented as the delay varies, which leads to the occurrence of novel bursting phenomena. Some weak resonant or non-resonant double Hopf bursting oscillations are presented due to the vanishing of real parts of two pairs of characteristic roots. Not only the magnitude of the time delay itself but also the strength of feedback control may influence the dynamical evolution process of bursting behaviors in the delayed system. Such fast-slow forms about bursting dynamics, as well as classifications about local dynamics are investigated. Furthermore, periodic and quasi-periodic bursting motions are verified in both theoretical and numerical ways. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the CH-γ equation is investigated by employing the bifurcation theory and the method of phase portraits analysis. The dynamical behavior of equilibrium points and the bifurcations of phase portraits of the traveling wave system corresponding to this equation are discussed. Under some parameter conditions, some bounded traveling wave solutions such as solitary waves, peakons and periodic cusp waves are presented. Furthermore, based on the auxiliary equation, various new traveling wave solutions of parametric form are given. The previous results for this equation are extended. [Copyright &y& Elsevier]