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Here are 3 spirals

x = radius^t*cos(t)
y = radius^t*sin(t)

where angle t in radians is changing from tMin to tMax.

Difference between spirals :

• radius r > 1 ( escapes to infinity )
• radius r = 1 ( rotation on unit circle )
• radius r < 1 ( attracts to center z = 0 )

It can be a model^[2] of dynamics under complex quadratic polynomial ${\displaystyle f_{0}(z)=z^{2}}$

kill(all)$
remvalue(all)$

/* for starting point = e^t use tMin = 1 */

GiveParametric(radius, tMin,tMax) :=
parametric(radius^t*cos(t),radius^t*sin(t),t,tMin,tMax)$

compile(all);

exterior:GiveParametric(1.05, 1, 6*%pi)$
interior:GiveParametric(0.90, 1, 12*%pi)$
boundary:GiveParametric(1.0, 1, 1+2*%pi)$

draw2d(file_name = "expospirals",
        pic_width=1000, 
        pic_height= 1000,
     terminal  = 'svg,

             title         = "Exponential Spirals",
             user_preamble = "set grid polar ; set size square; set size square;set key out;set key top right ;set xtics 0.5; set mxtics 0.5 ",
             nticks        = 200,
             xrange        = [-2,2],
             yrange        = [-2,2],

             color         = blue,
             line_width    = 3,
             key = "radius <1",
             interior,
             key = "radius > 1",
             color = green,
             exterior,
             
             key = "radius = 1",
             color = red,
            
             boundary

 )$

1. ↑ Powers of complex numbers by Tetyana Butler
2. ↑ Complex numbers by David E Joyce