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The pulse-shaped explosion (PSE), characterized by the pulse-shaped quantitative of system solutions varying dramatically, is a special route to bursting oscillations reported recently. This paper reports interesting dynamical behaviors related to the PSE of equilibria, and based on that, the complex bursting dynamics is investigated in a van der Pol–Mathieu–Duffing system with multiple-frequency slow-varying excitations. We find that bifurcations can be observed in a narrow parameter interval within PSE. We also show that two groups of bifurcations are symmetrically arranged on both sides of PSE, and each of which determines a different bursting part. Based on this, two compound bursting patterns, i.e. compound Hopf/Hopf bursting oscillation and compound subHopf/fold cycle bursting oscillation, and a novel type of relaxation oscillation (bursting oscillation of point/point) independent of bifurcations, are revealed. Our results enrich the knowledge of dynamical behaviors related to PSE as well as the possible routes to complex bursting dynamics. [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]

By introducing time-dependent power source, a periodically excited piecewise linear circuit with double-scroll is established. In the absence of the excitation, all possible equilibrium points as well as the stability conditions are presented. Analyzing the corresponding characteristic equations with perturbation method, Hopf bifurcation conditions associated with the equilibria are derived, which can be demonstrated by the numerical simulations. The Hopf bifurcations of the two symmetric equilibrium points may cause two symmetric periodic orbits, which lead to single-scroll chaotic attractors via sequences of period-doubling bifurcations with the variation of the parameters. The two chaotic attractors expand to interact with each other to form an enlarged chaotic attractor with double-scroll. The behaviors on the switching boundaries are investigated by the generalized Jacobian matrix. When periodic excitation is applied to work on the circuit, three periodic orbits with the frequency of the excitation may exist, which can be called generalized equilibrium points (GEPs) with the same characteristic polynomials as those of the corresponding equilibrium points for the autonomous case. It is shown that when the trajectories do not pass across the switching boundaries, the solutions are the same as the GEPs. However, when the trajectories pass across the switching boundaries, complicated behaviors will take place. Three forms of chaotic attractors via different bifurcations can be observed and the influence of the switching boundaries on the phase portraits is discussed to explore the mechanism of the dynamical evolution. [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 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]

Abstract: In this paper, a four-dimensional (4D) continuous autonomous hyperchaotic system is introduced and analyzed. This hyperchaotic system is constructed by adding a linear controller to the 3D autonomous chaotic system with a reverse butterfly-shape attractor. Some of its basic dynamical properties, such as Lyapunov exponents, Poincare section, bifurcation diagram and the periodic orbits evolving into chaotic, hyperchaotic dynamical behavior by varying parameter d are studied. Furthermore, the full state hybrid projective synchronization (FSHPS) of new hyperchaotic system with unknown parameters including the unknown coefficients of nonlinear terms is studied by using adaptive control. Numerical simulations are presented to show the effective of the proposed chaos synchronization scheme. [Copyright &y& Elsevier]

Abstract: The dynamics of two coupled chaotic electrical circuits has been investigated in this paper. Transition boundaries have been obtained to divide the parameter space into regions associated with different types of phase portraits. It is pointed out that two stable equilibrium points coexist for certain parameter conditions, which may evolve to different chaotic attractors via period-doubling bifurcations, respectively. [Copyright &y& Elsevier]

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]