Dynamics Of Spiral Waves Driven By Rectangular Waves And Double Periodic Force

Pattern dynamics is an important branch of nonlinear science.The study contents to physics,chemistry,mathematics,biology and other fields.Spiral is an significant type of nonequilibrium Pattern. The dynamics behaviours of spiral driven by external force is the foreword topics.In this paper,based F-H-N modle, employed numerical calculation ,the dynamics behaviours of spiral driven double periodic force rectangular force and double periodic force are considered.The following works are done:Firstly , dynamics of spiral under rectangular modulation of excitability parameter are considered .we introduce the rectangular modulation into the excitability parameter and investigate the dynamical changes of spiral with modulate amplitude and period varied .More modulate freqency components in spiral tip can occur duo to the modulation of excitability parameter. There are two certain rules among freqency components,some modulate freqency components and spiral basic freqency determine a curvatuer period which is a multiple of modulate period, while others is a linear combination of freqency components.With modulate period changed, modulate freqency move in freqency axis.In some modulate period scale , the relationship between modulate period and curvatuer period is maintained,which formate entrainment band.In this part,we study systematicly entrainment band length and distribution with modulate amplitude and systematic excitability vaied.Spiral tip trajectories exradius increase with modulate period increase in major entrainment band. Spiral tip can disappear as result of rectangular modulation of excitability parameter in entrainment band end. Particularly, dependency of spiral tip disappear to modulate amplitude and systematic excitability vaied is studied.Secondly , dynamics of spiral driven directly by retangular force are investigated. There are some important differences between meader spiral and rotating spiral when system is acted by retangular force.For meader spiral , entrainment band and some particular tip trajectories are observed .Compared to retangular modulation of excitability parameter, tip trajectories are more complex ,entrainment band types diverse ,which due to the modulate freqency components are more complex. In major entrainment band end , retangular force can eliminate the spiral too,which is different from eliminate process of retangular modulation of excitability parameter,in the eliminate process of tip moving to boundary ,spiral core region can be excited periodly by retangular force and core region become more large,which accelerate tip eliminate.This cheme is very important in actual system for tip eliminate.Effect of retangular force amplitude is studied in this party,in amplitude-period two dimrnsions parameter space,Anold tongues is provided,we found,for the particular amplitude,length of tougues is maximum.For rotating spiral , retangular force lead to different dynamics.When retangular force period is lesser , retangular force can maintain multiple arm spiral periodly.Analysis shows that ,reason of multiple arm spiral emerge is the wave length of rotating spiral is longer than meader spiral ,recover region is more wider,there is a bidirection regionin recover region.There is no bidirection regionin,so multiple arm spiral doesn't appear.Thirdly,spiral dynamics driven by double periodic force.With the period of double periodic force varied,we found entrainment band and disappearency of spiral in the end of entrainment band,too.The eliminate process is same as rectangular modulation of excitability parameter.The period scale eliminated spiral is given ,with the periodic force amplitude.For double periodic force, amplitudes are A1 , A freqencies areω1 ,ω.when A is a constant,andω1 = 2ω,with A1 increase, minimum in the period scale eliminated spiral decrease. when A is a constant,and12ω=ω,if A= A1, the period scale eliminated spiral is maximum,and minimum in the period scale eliminated spiral is least.This scheme of eliminated spiral is simple in actual system.