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Abstract: In this paper we consider the Hirota transformation of the Caudrey–Dodd–Gibbon equation (CDGE) from another point of view. As a result, the local equivalence between the CDGE and its bilinear equation is established, and a new type of Bäcklund transformation, which is defined by a second-order ODE along with the appropriate initial values, is presented to construct new solutions for the bilinear CDGE from the seed solutions of original CDGE. [Copyright &y& Elsevier]

Abstract: In this paper, the Fornberg–Whitham equation with linear dispersion term is investigated by employing the bifurcation method of dynamical systems. As a result, the existence of smooth and non-smooth traveling wave solutions is obtained. And the analytic expressions of solitary wave solutions, periodic cusp wave solutions and peakons are given under some parameter conditions. [Copyright &y& Elsevier]

• 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]

Abstract: The influence of initial phase difference as well as the amplitude of parametric excitation on the dynamics of a compound oscillator has been investigated in this paper. Based on the multiple time scale method, bifurcation sets of the averaged equations have been derived to divide the parameter space into regions associated with two types of phase trajectories. Imperfect cascading of period-doubling bifurcations have been observed, which may lead to different chaotic attractors. It is presented that, the three frequencies evolved in the vector field, though in resonance, may cause modulated effect on the structures of chaotic attractors. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: The dynamics of a 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. [Copyright &y& Elsevier]

The dynamical behavior of two coupled parametrically excited van der pol oscillators is investigated in this paper. Based on the averaged equations, the transition boundaries are sought to divide the parameter space into a set of regions, which correspond to different types of solutions. Two types of periodic solutions may bifurcate from the initial equilibrium. The periodic solutions may lose their stabilities via a generalized static bifurcation, which leads to stable quasi-periodic solutions, or via a generalized Hopf bifurcation, which leads to stable 3D tori. The instabilities of both the quasi-periodic solutions and the 3D tori may directly lead to chaos with the variation of the parameters. Two symmetric chaotic attractors are observed and for certain values of the parameters, the two attractors may interact with each other to form another enlarged chaotic attractor. [Copyright &y& Elsevier]

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]

The using of an elliptical nozzle has been considered as a potential method for improving the air-fuel mixing and atomization quality. The present paper focuses on the transverse jet breakup behaviors and droplet size distribution for circular and elliptical orifices in supersonic crossflow on a pulse wind tunnel test platform, and the laser based pulsed laser background imaging method and high-speed spray particle size analyzer were adopted. The results show that the jet penetration depth of elliptical 4 (Aspect ratio of 4) orifice is always smaller than that of the circular orifice. Another interesting finding is that the difference of the jet penetration depth between the circular and elliptical 4 orifices decreases with the increase of injection pressure. Moreover, the jet surface wave number of the elliptical orifice is more than that of the circular jet, and the elliptical jet surface wave length is smaller than that of circular one. Additionally, the spray Sauter Mean Diameter (SMD) of elliptical 4 nozzle is 14.5% smaller than the circular spray at injection pressure of 5 MPa, and the elliptical 4 spray also has the smallest Dv50 and Dv90 among other orifices. Because elliptical 4 jet held larger windward surface area, which is helpful to increase the shear action of supersonic crossflow. Furthermore, the elliptical 4 spray produces the smallest droplet diameter at the largest volume fraction, and also the elliptical 4 spray has the smallest span of droplet diameter distribution than other orifices. • Transverse jet behaviors and droplet size were studied at supersonic crossflow. • The large windward area of elliptical jet resulted in short penetration depth. • The elliptical transverse jet has smaller SMD than that of circular jet. • The application of elliptical orifice improves mixing and atomization quality. [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]

In this paper, a switched system that switches between two Chen systems based on the parameters altered periodically is established. To investigate the mechanism of switching oscillation of the whole system, Poincaré map of the whole system is defined by the switching scheme and the solutions of the subsystems. Different types of periodic switching oscillations revolved around the unstable equilibrium points and their transitions to chaotic oscillations can be observed. We conclude that bifurcation scenarios, such as period-doubling bifurcation, pitchfork bifurcation and saddle-note bifurcation, play an important role in the generation of various (symmetric or asymmetric) periodic oscillations and chaos in the switched system. [ABSTRACT FROM AUTHOR]

The mechanism for the action of time delay in a non-autonomous system with two time scales is investigated in this paper. The original mathematical model under consideration is a shape memory alloy oscillator with external forcing. The delayed system is obtained by adding both linear and nonlinear time delayed position feedbacks to the original system. Typical bursting patterns can be presented, including symmetric fold/supHopf, double-fold/supHopf and supHopf/supHopf bursting when periodic forcing changes slowly. The time delay is taken as a variable parameter to investigate its effect on the dynamics of the system such as the stability and bifurcation. We calculate the conditions of fold bifurcation and Hopf bifurcation as well as its stability with the aid of the normal form theory and center manifold theorem. Through bifurcation analysis, we can identify that the occurrence and evolution of bursting dynamic depends on the magnitude of the delay itself and the strength of time delayed coupling in the model. Furthermore, we use phase space analysis to explore the associated mechanisms for the oscillator with multiple coexisting attractors. Numerical simulations are also included to illustrate the validity of our study. [ABSTRACT FROM AUTHOR]

• Characteristics for asymmetric coalescence process of charged conical droplet were experimentally studied in various conditions. • Charge distribution and transportation were analyzed during conical coalescence process. • Liquid bridge expansion and retraction characteristics were quantitatively discussed. • A normalized phase diagram for conical coalescence and recoil of charged droplets is illustrated with the critical transition boundary. The dynamic behaviors of charged droplet coalescence in electric field has drawn much attention over the past decades due to its complex underlying physics and potential industrial application. This work experimentally investigated asymmetric coalescence dynamics of conical droplets in non-uniform electric field. With the effect of electric field, the cone apexes of deformed droplets merged with each other to form a liquid bridge that initially expanded but then retracted back gradually. It was found that the bridge length increased fast at initial conical coalescence and shrank slowly at retraction stage in various conditions. In addition, the effect of electric charge distribution and transportation on liquid bridge evolution behaviors was analyzed. The characteristics of cone angle and contact distance during droplet conical coalescence and recoil processes were quantitatively discussed with consideration of different parameters. Besides, the phase diagram for determining droplet coalescence and recoil was illustrated based on contact distance and electric Bond number, where the critical transition boundary was figured out with a linear increasing trend. [ABSTRACT FROM AUTHOR]