Dynamics Of Spiral Waves In The Complex Ginzburg-landau Equation Subjected To External Forces

The complex Gingsburg-Landau equation(CGLE)is a typical model of oscillation systems,and it has a wide range of applications in various fields,such as superconductivity and fluid mechanics,etc.This equation can also describe the dynamics of the reaction-diffusion system near the Hopf bifurcation point,depending on the kinetic parameters,the system can show many spatiotemporal patterns such as traveling wave,target wave,spiral wave,superstructure spiral wave,frozen structure and state of turbulence,and there are also many nonlinear mechanisms such as long-wave instability,loss stability,Benjamin-Feir instability and Eckhaus instability,which is one of the important research objects of nonlinear science.The study of the dynamic behavior of the spatiotemporal pattern driven by external forces has important practical significance,for example,in myocardial tissue,spiral wave and the state of turbulence after the instability of spiral wave are closely related to heart diseases such as arrhythmia and ventricular fibrillation,and some external drives can be used to control these spiral waves and state of turbulence,provide a theoretical reference for the treatment of heart disease.In this paper,the effects of different forms of external forces on the dynamics of spiral waves in CGLE systems will be studied,and the main work is divided into the following three parts:The first part: the influence of global constant external forces on the dynamics of spiral waves in the CGLE system is studied,and the variation law that dynamics of spiral waves with the strength of external forces and the auxiliary angle of complex external forces is given,and it is pointed out that external forces can lead to the emergence of superstructure waves.The position that the tip of the initial spiral wave is fixed,the tip will drift after the external force is started,and eventually follow on a circular attractor,with the increase that the intensity of the external force,drifting speed of the tip,moving speed of the tip on the attractor and the radius of the circular attractor increase linearly.At the same time,spiral waves begin to appear in the module space of the complex variable,and the forms of superstructure spiral wave appear in the real space of the complex variable.The external force intensity is relatively small,the superstructure spiral wave can remain stable,the larger intensity of the external force drive will lead to line defects near the boundary,the spiral wave will be broken near the boundary,the broken area begins to invade inward with the increase that intensity of the external force,when the crushing area expands to a certain extent,its interior begins to allow the formation of many small spiral waves,and coexists with the reserved part of the original spiral wave.The difference in the auxiliary angle of the complex external force will lead to a difference in the drift direction and the position of the circular attractor,and will show regular changes.The second part: The influence of global time-varying external forces on the dynamics of spiral waves in the CGLE system is studied,and it is found that when the external force period is equal to or similar to the oscillation period on the system space point,the tip of spiral wave will move to the boundary along the straight-line or small curvature drift path,and the law that radius of modulation circle and direction of initial drift with changing of external force parameters is given.Under the drive of segmented linear periodic external forces,the tip of spiral wave begins to move in a circular rolling manner,there are two frequency values in the spectrum of the time series,and there will be a linear roll parameter value on the parameter axis of the external force amplitude or period,and on both sides of the parameter value,the rolling mode corresponding to the tip movement is different,usually divided into inward rolling and outward rolling.In the case of straight-line rolling,the direction that drift of the spiral wave is linear with the initial phase of the external force.When the periodic external force unit is a nonlinear function,the motion of tip will become complicated and more frequency components will appear.The third part: The influence of global constant external forces and global time-varying external forces on the freezing structure of spiral wave is studied,and the changes of average number that oscillating points of low-amplitude in the system space with the intensity of external forces and the amplitude of periodic time-varying external forces are given,and dynamic behaviors of several different type are found.Driven by a constant external force,the average number that oscillating points of low-amplitude in the relatively stable system space can be divided into three cases as a result of the change that the intensity of the external force:the average number that region of low-intensity increases nearly linearly(case I);the average number of region in the larger-intensity decreases linearly until it stabilizes(case III);and the average number of points in the middle part of remains relatively stable(case II).In case I,the spiral waves in the system first break at the intersection of impacting lines after the external force drives,and then the retained part of the respective spiral wave will squeeze part of the state of turbulence,but the retained part of the spiral wave can only limit its size in the process of increasing that the strength of external force;In case II,III,the instability first appears near the tip of respective spiral waves,and the state of spiral wave in the final system can basically control the state of turbulence.Driven by a time-varying external force,the average number that oscillating points of low-amplitude in the system space after relative stabilization can be divided into three cases with the change that the amplitude of external force: case I and case II are roughly the same as the constant external force,and the average number that oscillating points of low-amplitude in a area of larger amplitude increases linearly until it stabilizes(case III).