Dynamical Behaviors Of The Two-dimensional Complex Ginzburg-Landau Equation Under Complex Forcing

The Ginzburg-Landau equation is one of the most studied nonlinear equations in physics.Because it describes the general dynamics of an extended system close to the Hopf bifurcation,it has been extensively studied in the context of spatiotemporal chaos.It describes various phenomena in field theory from qualitative and even quantitative perspectives,from nonlinear waves to second-order phase transitions,from superconductivity,superfluidity,Bose-Einstein condensation to liquid crystal chords.By changing the parameters in the equation,the Ginzburg-Landau equation exhibits rich dynamics,which includes spatiotemporal patterns such as spiral waves and turbulence.These spatiotemporal patterns have important value and research significance in biology and chemistry.In the formation of some spatiotemporal patterns,some control techniques play an important role such as feedback and coupling.These technologies can be designed in different ways to generate new spatiotemporal behavior patterns.In this thesis,by designing different feedback and coupling schemes,the dynamic behavior of the two-dimensional complex Ginzburg-Landau equation was studied under the effects of feedback and coupling.The thesis was divided into three chapters.The first chapter mainly introduced the reaction-diffusion system,patterns in the two-dimensional complex Ginzburg-Landau equation and some concepts.The second chapter studied the effect of feedback on the frozen structure in the two-dimensional complex Ginzburg-Landau equation.It was given that the change law of the initial frozen spiral wave structure with feedback gain and delay time.The parameter axis of the feedback gain was divided into four regions according to the evolution process and the final state of the system,and there were different characteristics in each region system.It was found that after the feedback was applied,the frozen state would be relieved through different instabilities.When the feedback gain was small,instability occurred in the areas ? and ? near the impact line and gradually defect turbulence emerged.When the area ? reached a relatively stable state,the system was divided into two different areas,including sequential spiral waves and external defect turbulence within a certain distance from the center of the spiral.With the increase of feedback gain,the stable spiral wave area in the region ? gradually decreased its relatively stable state and became the defectturbulence state.When the feedback gains further increased,the instability of region ? and ? occurred in the center of the spiral wave,the difference was that the frozen structure would reform after a period of transient instability in the area ? system,such as single arm,single arms,multiple arms,single target-like wave,and multiple target-like waves.And under some feedback gains,the double helical waves and target-like waves would be transformed into single-arm spiral waves.The frozen structure was eliminated in the IV region with greater feedback gain.In the process of changing the delay time,different types of frozen structures also appeared in the system,and the change of the delay time under positive feedback would change the spiral wave shape in the frozen structure.The third chapter mainly studied the dynamics behavior of the double-layer system under multiple locally coupled regions,and mainly investigated the effects of the coupling region width,the coupling region center distance and the coupling strength on the dynamics of the double-layer system.With the change of the width of the coupled zone and the center distance,there were asynchronous multiple spiral waves,and phenomenon of full synchronization of wave front position movement and full synchronization with unchanged wave front position in the double-layer system.When the center distance of the coupling area was large or the coupling strength was small,it was found that the wave fronts of the spiral waves of two two-dimensional layers move along attractors of different shapes and sizes.The synchronization function also had periodic asymptotic behavior,that was,phase-locking.Changing the coupling strength,when the coupling strength was large,it was found that the target waves and double helical waves were irregular in the double-layer system.And it was found that the wave fronts of the spiral waves that appeared separately would move along different attractors,and the synchronization function would also change periodically after the system was stable.