[en] The traveling waves of a generalized BBM equation with positive exponents have been investigated in this Letter. The influence of the parameters on the number as well as the characteristics of the equilibrium points have been discussed in detail. Qualitatively different kinds of phase portraits have been explored with the variations of the parameters. All the possible bounded waves have been obtained, including periodic waves, kink-waves and solitons. For the special parameter conditions, the exact explicit expressions of waves have been presented in the end

[en] The dynamics of coupled Duffing's oscillators with initial phase difference is investigated in this Letter. For the averaged equations, different equilibrium points can be observed, the number of which may vary with the parameters. The stable equilibrium points, corresponding to the periodic motion of the original coupled oscillators, may coexist with different patterns of dynamics, including chaos. Furthermore, two different chaotic attractors associated with different attracting basin coexist for certain parameter conditions, which may interact with each other to form an enlarged chaotic attractor. Several new dynamical phenomena such as boundary chaos crises have been predicted as the initial phase difference varies

[en] The evolution of wave patterns of a generalized KdV equation is investigated in this Letter. For the topological vector filed, a singular line can be observed which separates the phase trajectories on the phase plane into two parts. All possible waves including compactons, solitary waves, smooth periodic waves and non-smooth periodic waves with peaks as well as the existence conditions have been presented. Specially, a new type of waves called as kink-compactons which combine both half structures of compacton and kink-wave can be found

[en] The dynamics of a 1+1 unidirectional non-linear wave equation which combines the linear dispersion of the Korteweg-de Vries (KdV) equation with the non-linear/non-local dispersion of the Camassa-Holm (CH) equation is explored in this Letter. Phase plane analysis is employed to investigate the bounded traveling-wave solutions. By considering the properties of the equilibrium points and the relative position of the singular line, transition boundaries have been derived to divide the parameter space into regions in which different types of phase trajectories can be observed. The explicit expressions of different types of solutions have been presented, which contain both the KdV solitons and the CH peakons as limiting cases

[en] We present new types of singular wave solutions with peaks in this paper. When a heteroclinic orbit connecting two saddle points intersects with the singular curve on the topological phase plane for a generalized KdV equation, it may be divided into segments. Different combinations of these segments may lead to different singular wave solutions, while at the intersection points, peaks on the waves can be observed. It is shown for the first time that there coexist different types of singular waves corresponding to one heteroclinic orbit

[en] We explore the complicated bursting oscillations as well as the mechanism in a high-dimensional dynamical system. By introducing a periodically changed electrical power source in a coupled BVP oscillator, a fifth-order vector field with two scales in frequency domain is established when an order gap exists between the natural frequency and the exciting frequency. Upon the analysis of the generalized autonomous system, bifurcation sets are derived, which divide the parameter space into several regions associated with different types of dynamical behaviors. Two typical cases are focused on as examples, in which different types of bursting oscillations such as subHopf/subHopf burster, subHopf/fold-cycle burster, and double-fold/fold burster can be observed. By employing the transformed phase portraits, the bifurcation mechanism of the bursting oscillations is presented, which reveals that different bifurcations occurring at the transition between the quiescent states (QSs) and the repetitive spiking states (SPs) may result in different forms of bursting oscillations. Furthermore, because of the inertia of the movement, delay may exist between the locations of the bifurcation points on the trajectory and the bifurcation points obtained theoretically. (paper)

[en] Different from what we have expected before, when a homoclinic orbit intersects with a quadratic singular curve on the topological phase plane derived from a generalized KdV equation, corresponding to the homoclinic orbit, there exist a few types of solitary waves, including peakons and antipeakons as well as periodic waves and furthermore, new types of solitary waves with peaks. The investigation shows that, when a trajectory along the homoclinic orbit moves at the intersection points between the homoclinic orbit and the singular curve, it may jump between these intersection points, which forms peaks on the waves, implying the nonsmooth solitary waves occur

[en] 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. (paper)

[en] The main purpose of this paper is to explore the mechanism of the non-smooth bursting oscillations in a Filippov neuronal model with the coupling of two scales and to try to explain some special phenomena appearing on the attractors. Based on a typical Hindmarsh–Rose neuronal model, when the recovery variable of the slow current is replaced by a slow-varying periodic excitation, which means the exciting frequency is far less than the natural frequency, the coupling of two scales in frequency exists, leading to the non-smooth bursting oscillations. By regarding the whole exciting term as a slow-varying parameter, we can define the full subsystem as Filippov type, which appears in generalised autonomous form. Equilibrium branches and their bifurcations of the fast subsystem can be derived by varying the slow-varying parameter. With the increase of the exciting amplitude, different types of equilibrium branches and the bifurcations may involve the slow–fast vector field, which may cause qualitative change of the bursting attractors, resulting in several types of periodic non-smooth bursting oscillations. By employing the modified slow–fast analysis method, the mechanism of the bursting oscillations is presented upon overlapping the transformed phase and the equilibrium branches as well as their bifurcations of the generalised autonomous system. The sliding phenomenon in the bursting oscillations may occur since the governing system with different stable attractors may alternate between two subsystems located in two neighbouring regions divided by the boundary. Furthermore, the inertia of the movement along an equilibrium branch increases with the increase of the exciting amplitude, leading to the disappearance of the influence of the associated bifurcations on the attractors. (author)

[en] A new type of wave solutions, called as multiple-mode waves, which can be expressed in the superposition forms of more than two types of single-mode waves of Vakhnenko equation have been investigated in this Letter. A new general method for obtaining the multiple-mode waves is proposed, based on which four cases of the possible forms of wave solutions with two-mode have been derived. The explicit expressions of the two-mode waves as well as the existence conditions have been presented, which may be the nonlinear combinations between periodic waves, solitons, compactons, etc., with different wave speeds, respectively. It is pointed out that more complicated multiple-mode waves with more than three single-mode waves can be derived accordingly, which can be used to reveal the evolution of interactions between different types of waves, especially between various solitons