Compound Relaxation Oscillations Based On Pulse-shaped Explosion

The multi-time-scale coupling system has a wide range of engineering background,and relaxation oscillations in it are widespread.Exploring the dynamic mechanism of multi-time-scale coupled systems,especially relaxation oscillations,has always been one of the frontier topics and key objects of attention in nonlinear disciplines.In recent years,exploring the possible path of relaxation oscillation has gradually become a research hotspot of scholars at home and abroad.Recently,a new type of dynamic mechanism that may induce relaxation oscillations,that is,the phenomenon of pulse-shaped explosions,has attracted attention from scholars.Previous scholars’ studies on pulse-shaped explosions have focused on the simple structure and single form of pulse-shaped explosions and related reports.However,they have not done in-depth research on the complex relaxation phenomena of pulse-shaped explosions.In order to further reveal the complex dynamics of impulse explosions,this topic takes the multi-frequency coupled Mathieu-Van der Pol-Duffing system as an example,and borrows many numerical simulations by borrowing tools such as fast-slow analysis and frequency conversion fast-slow analysis.Interesting results.Mainly include: impulse explosion phenomenon with multiple bifurcations,compound impulse explosion phenomenon with bi-stability under phase difference,new form of induced impulse explosion,compound PSE phenomenon under amplitude modulation excitation,and compound type of connection Relaxation oscillation.The main results of this subject are:(1)A pulse-shaped explosion phenomenon with multiple bifurcation behaviors is obtained,which provides a new idea for scholars to study the generation mechanism of relaxation oscillations.That is,in a system with multi-frequency coupled excitation,the presence of slowly varying parameters may lead to complex dynamics of the system.In addition,the impulse explosion of multiple bifurcation behaviors revealed that the trajectory transfers between different attractors,and then reveals the excitation effect of the compound relaxation oscillation.The trajectory changes sharply at the equilibrium point near the PSE,which provides new ideas for studying the mechanism of nonlinear dynamics.(2)The compound relaxation oscillation in a multi-frequency coupled excitation system with phase difference is studied.Previous scholars’ studies on pulse-shaped explosions have focused on the increase in the vector field caused by the frequency relationship between different excitations,so that the increase in the number of attractors eventually produces complex relaxation oscillations.The research in this article finds that the original system is divided into two vector fields even when the frequency is the same and the phase difference acts,thus showing a bi-stable structure.Eventually,the system produced two types of complex structures with multiple bifurcations,positive and negative bidirectional PSE.Although the point under the phase difference is for a specific system.However,it is reasonable to speculate that the compound PSE phenomenon and the bistable in the system may be reported in other systems when there is a phase difference in the multi-frequency coupling excitation system,which provides a new idea for the related research of fast and slow dynamics.(3)Discovered a new way to produce a pulse-shaped explosion.Previously,not only the research on pulse-shaped explosions and even the fast and slow dynamics have been carried out around periodic non-amplitude modulated excitation,but little consideration has been given to amplitude modulated excitation.The specially considered amplitude modulation excitation can also induce the PSE phenomenon of the system,and even the introduction of amplitude modulation excitation also causes three types of compound PSE phenomena,that is,there are multiple combinations of PSE behaviors in one exercise cycle.In addition,the introduction of amplitude modulation excitation will lead to several different complex dynamic behaviors due to the complexity of the system.The thinking of amplitude-modulated excitation provides a new idea for the action form of fast and slow dynamics excitation.