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Discrete Dynamical Systems: Stability, Cobweb Diagram, Time Series Diagram, Phase Diagram, Bifurcation Diagram

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####################### #Parameter Curve 2D #by Sinan Kapcak #2018 #www.k-interact.net/dds ####################### x,y,a=var('x y a') @interact def _( f=input_box(label='$f(x,y)$',default=a*x+y,width=40), g=input_box(label='$g(x,y)$',default=a^3*x+(a-1)*y,width=40), fp=input_box(label='$P^*$',default=(0,0),width=40), param1=input_box(label='Slider interval for paramater $a$',default=(-1,2),width=20) ): #assume(a,'real') #assume(param1[0] < a < param1[1]) if bool(fp[0] == f(x=fp[0],y=fp[1]))==False or bool(fp[1] == g(x=fp[0],y=fp[1] )) == False: show (' $P=$' , fp , ' is not a fixed point.') else: region = polygon([[-2,1], [2,1], [0,-1]], color='orange',fill=False, thickness=.8) parabol=plot(1/4*x^2,x,-2,2,thickness=1,linestyle='dotted',color='grey') F=(f,g) j=jacobian(F, (x,y)) j_fp=j.substitute(x=fp[0],y=fp[1]) func_trace=j_fp.trace() func_det=j_fp.determinant() parametrika=parametric_plot( (func_trace, func_det), (a, param1[0], param1[1]),color='black') t1=text("Neimark-Sacker Bifurcation", (0,1.1),fontsize=9, alpha=0.6, color='black') t2=text("Fold Bifurcation", (1.2,0), rotation=45.0,fontsize=9, alpha=0.6,horizontal_alignment='center', vertical_alignment='center', color='black') t3=text("Flip Bifurcation", (-1.2,0), rotation=-45.0, fontsize=9,alpha=0.6,horizontal_alignment='center', vertical_alignment='center', color='black') @interact def _(parameter0=slider(param1[0] , param1[1], .01, default= (5*param1[0]+param1[1])/6 ,label='Parameter $a$' ),zoom=slider(.1 , 1, .1, default= .6 ,label='Zoom +/-' )): parapoint=point( (func_trace(a=parameter0),func_det(a=parameter0) ) ,color='red',pointsize=30) show(t1+t2+t3+parametrika+region+parabol+parapoint,frame=True,axes=False,axes_labels=['Trace','Determinant'],ymin=-1.2/zoom, ymax=1.2/zoom,xmin=-2.2/zoom,xmax=2.2/zoom) ##################### ###### Code Ends #### #####################
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