Search

Slow–fast dynamics such as bursting behaviors are common in many physical and engineering systems. In the previous study, some focused on the bursting behavior caused by the codimension-1 bifurcations, and others focus on the bursting behavior due to the particular structures. However, systems under critical conditions may exhibit many complicated dynamics due to the high co-dimensional bifurcations. Our research aims to investigate the bursting oscillations near a triple zero eigenvalues singularity. A perturbation model of the double pendulum system with external excitation is taken as an example and investigated the dynamical mechanism of bursting behaviors. Because of the order gap between the exciting frequency and natural frequency, the perturbation model with the external excitation can be regarded as a generalized autonomous system. By overlapping the transformed phase portrait and the equilibrium branch, four types of bursting oscillations are determined: fold/fold type, zero-Hopf/zero-Hopf type, symmetrical zero-Hopf/sup-Hopf/fold-cycle type, and symmetrical zero-Hopf/sup-Hopf/fold-cycle/sub-Hopf type. Primarily, we find that due to the singularity of compound fold conditions, many bifurcations of the limit cycle occur, which cause many complex dynamics such as 2-D tori bursting, breaking of symmetric structure, and chaotic bursting. These results play an essential role in understanding the stability of a system with high co-dimensional bifurcation conditions. They can be expected to provide a theoretical basis for formulating a control strategy. [ABSTRACT FROM AUTHOR]

This paper investigates bursting dynamics of a Rayleigh oscillator with multiple-frequency slow excitations, in which two different bursting patterns related to the bistable pulse-shaped explosion (PSE) are obtained. Typically, the PSE, a novel sharp transition behavior reported recently, can be observed in the Rayleigh oscillator. We show that if there is an initial phase difference - π 2 between the slow excitations, two coexisting solution branches exhibiting PSE, which we call bistable PSE, may be created in the fast subsystem. Then, the route to bursting by the bistable PSE is analyzed, and two different bursting patterns, i.e., bursting of point–point type and bursting of cycle–cycle type, are obtained. Our findings show that the initial phase difference of excitations may have great effects on PSE, which thus plays an important role in transitions to different attractors and complex bursting dynamics. [ABSTRACT FROM AUTHOR]

This paper investigates the low-velocity impact response of functionally graded multilayer nanocomposite plates reinforced with a low content of graphene nanoplatelets (GPLs) in which GPLs are randomly oriented and uniformly dispersed in the polymer matrix within each individual layer with GPL weight fraction following a layer-wise variation along the plate thickness. The micromechanics-based Halpin–Tsai model is used to evaluate the effective material properties of the GPL-reinforced composite (GPLRC), and the modified nonlinear Hertz contact theory is utilized to define the contact force between the spherical impactor and the GPLRC target plate. The equations of motion of the plate are derived within the framework of the first-order shear deformation plate theory and von Kármán-type nonlinear kinematics and are solved by a two-step perturbation technique. The present analysis is validated through a direct comparison with those in the open literature. A parametric study is then performed to study the effects of GPL distribution pattern, weight fraction, geometry and size, temperature variation as well as the radius and initial velocity of the impactor on the low-velocity impact response of functionally graded GPLRC plates. [ABSTRACT FROM AUTHOR]

This paper addresses the modified function projective bursting synchronization (MFPBS) for a class of fast-slow systems with uncertainties and external disturbances. By using an active sliding mode control method (SMC), bursting synchronization can occur between two different chaotic systems. An active SMC law is established to guarantee uncertainties and disturbances rejection and realize the MFPBS. The sufficient condition is drawn for the stability of error dynamics, where the controllers are designed by using the relevant variables in fast-slow systems. More interestingly, more complex bursting oscillations such as chaotic bursting, as well as chaotic or periodic spiking can be obtained in slave systems with free scales when the MFPBS is maintained. And the corresponding numerical simulation is provided to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to demonstrate that bursting oscillations not only occur in slow-fast systems, which have been presented in many publications, but also may take place in periodic excited systems with order gap between the excited frequency and natural frequency. Furthermore, a typical procedure to analyze the bifurcation mechanism of the bursting oscillations in periodic excited dynamical systems is presented, in which the conceptions of generalized autonomous system (GAS) and transformed phase portrait are introduced. For the periodic external excited dynamo oscillator, different types of bursting phenomena can be obtained, the mechanism of which is presented based on the bifurcation analysis of the corresponding GAS. From the evolution of the dynamics with the variation of the parameters, it is found that different forms of bifurcations, occurring at the transitions between the quiescent state and spiking state, may cause different patterns of bursting oscillations. [ABSTRACT FROM AUTHOR]

The main purpose of the paper was to explore the structures of bursting oscillations as well as the mechanism of non-smooth dynamic system with two timescales. Based on the typical Chua's system, a non-smooth dynamic system with two timescales is established, which can be considered as the coupling of slow and fast subsystems. Possible bifurcations as well as the critical conditions related to the equilibrium points of the fast subsystem in different regions divided by the switching boundaries are derived. The distribution of the eigenvalues related to the generalized Jacobian matrix reveals that multiple crossing bifurcations may occur at the switching boundaries, which may influence the dynamics of system. Periodic bursting oscillations can be observed, in which the fold-type non-smooth bifurcations connect the quiescent state and spiking state, leading to the symmetric focus/focus-fold/fold bursting attractors. Different behaviors can be observed at the intersection points between the trajectories and the switching boundaries, which can be explained by the characteristics of the equilibrium points in two regions beside the boundaries. Furthermore, it is found that the real parts of the pair of complex conjugate eigenvalues related to the foci may dramatically depend on certain parameters, which is employed to account for the change in the oscillation patterns related to the spiking state. [ABSTRACT FROM AUTHOR]

Dynamical model of a typical centrifugal flywheel governor system with periodic switches between two forms of external torque has been established, in which the state variables may be governed by two subsystems corresponding to autonomous and non-autonomous systems with periodic excitation, respectively. Two types of bifurcation can be observed in the autonomous subsystem, where the Hopf bifurcation may lead to the occurrence of periodic oscillation, while the fold bifurcation may result in the disappearance of the equilibrium points. The dynamics often behaves in periodic oscillations with the same frequency as the periodic switch, in which the switching points may divide the trajectory of the orbit into two groups of segments governed by the two subsystems, respectively. Cascading of period-doubling bifurcations related to the Floquet multiplier passing across the unit cycle can be observed, which cause the system to evolve to chaotic oscillations. Furthermore, because of the symmetry of the vector field, two attractors symmetric to each other can be obtained, which may interact with each other to form an enlarged attractor with the variation of the parameters. Generalized Hopf bifurcation may lead to the alternation of the behaviors between periodic movements and quasi-periodic oscillations. [ABSTRACT FROM AUTHOR]

This paper investigates the emergence of mixed-mode oscillations (MMOs) in the forced Van der Pol's equation. It is found that the MMOs studied here can be classified as a slow passage through canard explosion, which is different from the usual fast-slow bursters. We first consider the external forcing as a control parameter and study its influence on the Van der Pol's equation with constant forcing (VPCF). Then we briefly discuss the famous canard phenomenon in VPCF. The results of these analysis, together with the 'transformed phase diagram,' are applied to the forced Van der Pol's equation, which shows that the canard explosion and the external forcing plays an important role in the generation of MMOs, that is, the MMOs are created since the external forcing slowly and periodically visits the rest and spiking areas of VPCF. [ABSTRACT FROM AUTHOR]

Impulsively coupled oscillators which are assumed to interact with each other only at discrete times have many applications in practice. In this paper, we introduce the concept of partial contraction theory of impulsive systems, which is used to investigate the synchronization problem of impulsively coupled oscillators. Contraction analysis of two impulsively coupled oscillators and networked impulsively coupled oscillators is provided, respectively. Very simple but very general results for synchronization of impulsively coupled oscillators are derived. Numerical simulations show the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]