Dynamics Of Spiral Waves And Turbulent States Under Feedback Derived From A Line

Recently,the formation of spatiotemporal patterns away from equilibrium has been extensively studied both theoretically and experimentally.One of the most frequently encountered structures is the spiral wave.Spiral waves can exist in many forms,such as rigidly rotating spiral wave,the roaming spiral wave,superstructure spiral wave,anti-spiral wave,segmentation spiral waves,the spiral state in the frozen structure and the small spiral waves in the defect turbulence.In practical systems such as heart-brain,spiral waves and the turbulence through spontaneous instabilities of spiral waves usually predict the emergence of some diseases.It has important practical significance to prevent their emergence and control them.In addition to experimental research,more research is based on the mathematical model of the spiral wave and turbulence,there is also a clearer understanding of some of their dynamic properties.For example,the mechanism of the spiral wave formation and fragmentation,the transition from the rigidly rotating to the roaming spiral wave,and the dynamic behavior of spiral waves in an field of polarization.Under the action of external forces,spiral waves will show complex dynamic behavior and appear new patterns.such as spiral wave strange states.It has important implications to study them in both theory and practice.In this paper,we will study the dynamics of the spiral wave and the state of turbulence under the line feedback in CGLE systems,and also analyze the statistical properties of the turbulence.The specific work is mainly divided into the following three parts:The first part: We study the dynamic behavior of spiral waves in the CGLE system under the line feedback.It is pointed out that this feedback can drive original the tip of the spiral wave to drift and form various shapes of attractors.There are some change rules of attractors with some parameters that include feedback strength,the delay time,the ratio of the real and imaginary parts,position and the length of the line.We find that the increased feedback strength leads to the formation of superstructure spiral waves and the fragmentation and elimination of the tip of spiral waves.With large feedback gain,the tip of the spiral wave drift parallel to the feedback measurement line for some time,superstructure spiral wave appear in the system.The appearance of the superstructure spiral wave can lead to the fragmentation of the wave arm away from the center.The appearance of the superstructure spiral wave can lead to the fragmentation of the spiral arm away from the center.This fragmentation will continuously advance towards the central region.Eventually,the system be divided into two regions with different phases.Line feedback can lead to the appearance of a square attractor.When the measurement line is relatively short,the square attractor is degraded to a circular attractor.When the measurement line ends extend to the boundary,the trajectory mutated from a square attractor to short lines.The initial drift of the tip of the spiral wave changes regularly with the position of the measurement line.When the delay time exceeds the critical value,the square attractor changes to a short line,a new circular attractor and other traces.The second part: We study the dynamic behavior of the spiral state in the frozen structure in the CGLE system under the line feedback.It is found that line feedback will make the defect in the wall into small spiral wave and lead to the elimination and breakage of the original wall,pointed out that the line feedback will lead to defect turbulence,small spiral waves coexist with the turbulence and the emergence of new frozen structures.There are some change rules number of defects with the feedback strength and length of the line.Some statistical laws of the number of defects over time are given.When the measurement line is short,the average of the number of small modes after stabilization changes with the feedback gain can divide into four dynamic regions.When the feedback gain is in the range of0.001-0.026,the number of points increases linearly with the increase of the feedback gain.The turbulent sea appears in the real space,and turbulence expands with the increasing feedback gain.When the feedback gain is in the range of 0.027-0.037,the number of modes is the most,and the number after stabilization satisfies the Gaussian distribution.When the feedback gain is in the range of 0.038-0.091,points decrease linearly with the feedback gain.When the feedback gain is greater than or equal to 0.092,the frozen structure is eliminated and the number of small modes in the system is 0.When the measurement line is long,the average value of the number of small modes presents a linear relationship with the feedback gain.Except for the individual feedback gain,the system is always in the coexistence state of small spiral waves and turbulent sea.The turbulent sea expands as the feedback gain increases.The third part: We study the dynamic behavior of turbulent states in the CGLE model and an improved FHN model.The statistical properties of defect turbulence are different from those of phase turbulence.When the parameters of defect turbulence are taken close to the value of phase turbulence,the number of small modes in the system is relatively small and satisfying the Gaussian distribution.In the generation stage of turbulence,the generation of small modes in defect turbulence is faster.In the improved FHN model,the number of small modes in the system changes significantly with changing the system size.However,the distribution of the number of small modes and the wave number shows stationarity.