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Relaxation oscillations are ubiquitous in various fields of natural science and engineering technology. Exploring possible routes to relaxation oscillations is one of the important issues in the study of relaxation oscillations. Recently, the pulse-shaped explosion (PSE), a novel mechanism which can lead to relaxation oscillations, has been reported. The PSE means pulse-shaped sharp quantitative changes related the variation of system parameters in branches of equilibrium points and limit cycles, which leads the system’s trajectory to undertake sharp transitions and further induces relaxation oscillations. Regarding externally and parametrically excited nonlinear systems with different frequency ratios, some work on PSE has been reported. The present paper focuses on the PSE and the related relaxation oscillations in a externally and parametrically excited Mathieu-van der Pol-Duffing system. We show that if there is an initial phase difference between the slow excitations with the same frequency ratio, the fast subsystem may compose of two parts with different expressions, each of which determines a different vector field. As a result, the bistable behaviors are observed in the system. In particular, one of the vector fields exhibits two groups of bifurcation behaviors, which are symmetric with respect to the positive and negative PSE, and each can induce a cluster in the relaxation oscillations. Based on this, we present several routes to compound relaxation oscillations, and obtain new types of relaxation oscillations connected by pulse-shaped explosion, which we call compound “subHopf/fold-cycle” relaxation oscillations and compound “supHopf/supHopf” relaxation oscillations, respectively. Our results show that the two clusters in the resultant relaxation oscillations are connected by the PSE, and the initial phase difference plays an important role in transitions to different attractors and the generation of relaxation oscillations. Although the research in this paper is based on a specific nonlinear system, we would like to point out that the bistable behaviors, the PSE and the resultant compound relaxation oscillations can also be observed in other dynamical systems. The reason is that the fast subsystem composes of two different vector fields induced by the initial phase difference, which dose not rely on a specific system. The results of this paper deepen the research about PSE as well as the complex dynamics of relaxation oscillations.

Using hourly averaged data of > 2 MeV electron fluxes measured with GOES 7/9/10/11 satellites during 1988–2010, this paper investigates statistically conditions of solar wind and geomagnetic activities during relativistic electrons flux (Fe) enhancements, and describes the dependence of relativistic electrons at geosynchronous orbit (GEO) on local time, magnetic storms. The results are as follows: (1) The local time dependence of GEO relativistic electrons is influenced by solar wind and geomagnetic activities. The noon/midnight electron flux ratio grows with increasing solar wind velocity (Vsw). On conditions of the Dst index above −50 nT, relativistic electrons show regular variation with local time, while during storms with Dst Femax ≥ 103 pfu exhibited no such solar cycle and seasonal dependences. For most intense events, relativistic electron fluxes begin to increase during main phases of magnetic storms, while for weaker events, relativistic electron fluxes tend to increase during recovery phases. (3) The solar wind density shows as a good indicator of subsequent relativistic electron activities. For most electron flux enhancement events, relativistic electron flux decreased to its minimum after 0∼1 days of the maximum solar wind density, Nmax, while approaching the maximum after 0∼2 days of the minimum solar wind density, Nmin. (4) Above 90% of relativistic electron events occurred on conditions of high-speed solar wind and geomagnetic disturbances, which are as follows: Vswmax > 516 km/s, Dstmin 14.1 cm−3, Bzmin 698 nT. (5) For all magnetic storms, occurrence probability of daily maximum electron flux above 103 pfu after Dstmin, is about 53%, the percent of intense, weak events being 36%, 64%, respectively, which is hardly affected by intensities of storms. During storms, solar wind velocity, density, and the AE index are important indicators for subsequent relativistic electron activities.

A photon imaging system with single photon detecting and imaging capabilities is constructed and a detection method based on pipeline filtering is proposed to detect the stationary point target under a photon limited condition.The stationary point target detection method based on pipeline filtering is studied on the characteristics of position and noise for stationary point targets in photon images.The diameter of pipeline filter is optimized to reduce the false alarm rate.The relationships of detection performance with the number of signal photons and noise photons,the pipeline length and the detection threshold are explored through photon image sequences.For the image sequence with the average incidences of signal and noise of 0.4 and 5.215 respectively,the detection rate can be more than 0.9 and the false alarm rate less than 0.08 when the pipeline length is 9 and detection threshold is 2.The conclusion from experimental results is that the detection rate is mainly affected by the number of signal photons and the false alarm rate is dependent on the number of noise photons.

The main purpose of this study is to investigate the characteristics as well as the bifurcation mechanisms of the bursting oscillations in the asymmetrical dynamical system with two scales in the frequency domain. Since the slow-fast Hodgkin-Huxley model was established to successfully reproduce the activities of neuron, the complicated dynamics of the system with multiple time scales has become a hot research topic due to the wide engineering background. The dynamical system with multiple scales often presents periodic oscillations coupled by large-amplitude oscillations at spiking states and small-amplitude oscillations at quiescent states, which are connected by bifurcations. Up to now, most of the reports concentrate on bursting oscillations in the symmetric systems, in which there exists only one form of spiking oscillations and quiescence, respectively. Here we explore some typical forms of bursting behavior in an asymmetrical dynamical system with periodic excitation, in which there exists an order gap between the exciting frequency and the natural frequency. As an example, based on the typical Chua's oscillator, by introducing an asymmetrical controller and a periodically changed current source, and choosing suitable parameter values, we establish an asymmetrical dynamical system with two scales in the frequency domain. Since the exciting frequency is much smaller than the natural frequency, the whole periodic exciting term can be regarded as a slowly-varying parameter, leading to the fast subsystem in autonomous form. Since all the equilibrium curves and relevant bifurcations are presented in the form related to the slowly-varying parameter, the transformed phase portraits describing the evolution relationship between the state variables and the slowly-varying parameter are employed to account for the mechanism of the bursting oscillations. With the variation of the slowly-varying parameter, different equilibrium states and relevant bifurcations in the fast subsystem are presented. It is found that for different parameter values, multiple balance curves of the fast subsystem may coexist, which affect the structure of the bursting attractor. For the other parameters fixed to certain values, the balance curve with the variation of the slowly-varying parameter is presented. Three typical cases with different exciting amplitudes are considered, corresponding to different situations of coexistence of equilibrium states in the fast subsystem. In the first case, there exist at most three stable equilibrium points in the fast subsystem. Bursting attractor that oscillates around the three points can be observed, in which fold and Hopf bifurcations lead to the alternations between spiking states and quiescent states, while in the second case, saddle on the limit cycle bifurcation may cause the repetitive spiking oscillations to jump to the equilibrium curve. In the third case with relatively large exciting amplitude, only two equilibrium curves may involve the bursting oscillations, in which fold bifurcations lead to the alternation between the quiescent states and spiking states. Unlike the structures of bursting oscillations in the symmetric system, different forms of asymmetrical bursting oscillations with different periodic exciting amplitudes can be observed, the mechanisms of which are presented. It is pointed out that the change of the external exciting amplitude, does not only cause the variation of the attracting basins corresponding to different stable equilibrium branches, but also leads to the change of the temporal intervals when the trajectory passes different bifurcation points, respectively, which results in different patterns of bursting oscillations. Furthermore, since the slowly-varying parameter determined by the whole exciting term changes between two extreme values determined by the amplitude, the trajectory of the bursting oscillations of the transformed phase portrait returns at the two extreme values. The properties of equilibrium branches between the two extreme values determine the forms of the moving attractors.

Since most of the work relevant to the control of Chen system is based on the same time scale, and the results associated with the coupled systems with two time scales are mainly for the cases with only one slow variable, in this paper we investigate the dynamical evolution of the Chen system with the controller described by Duffing oscillator on a slow time scale, which implies two slow variables may be involved in the coupled vector field. Different types of bursting oscillations such as the symmetric fold/fold bursting, symmetric fold/Hopf bursting, symmetric homoclinic/homoclinic bursting, and mechanism are presented, revealing that the mutual influence between the two slow variables may cause the bursting behaviors similar to those in the periodic excited systems.

Based on the classical Chua's circuit, a four-dimensional Generalized Chua's circuit with multiple interfaces is established by introducing feedback elements. For the appropriate condition, there exists a difference in order of magnitude between the variables of state and a fast-slow coupled system, thereby forming a fast- and slow-coupled system at time scale. Analyzing the equilibrium points and the characteristics of the fast subsystems, and combining the theory of Clarke differential inclusions, the singularities on the non-smooth boundaries are explored. Two types of periodic bursting phenomena for different conditions are presented. Fast-slow analysis is employed to explore the special cluster phenomenon while the system trajectory passes across multiple interfaces. The coexisting different bursting mechanisms for the case with multiple attractors are explored in detail, while the influence of non-smooth bifurcations on bursting behavior is revealed.

The dynamical evolution process of the coupled system connecting two nonlinear electrical circuits with suitable circuit is investigated. The bifurcation behavior as well as the ways to chaos of the two subsystems is presented. It is pointed out when both of the two subsystems behave as periodic, the coupled system may also be led to chaos via cascading of period-doubling bifurcations. Meanwhile, in the chaotic region, critical increase of period as well as period-adding bifurcation can be observed. As to the interaction between periodic movement and the chaotic oscillation, the original periodic subsystem may chaotically oscillate around the original orbit. The amplitude associated with the oscillation increases rapidly, resulting in the obvious chaotic characteristics. On the contrary, the periodic subsystem may not only cause the instability of the chaotic subsystem, but also lead to change of the chaotic structures.