1 / 12

Julia Sets

Julia Sets. Consider the map: x n+1 =x n 2 +c with c=-3/4 Choose some values of x and iterate many times - what happens ? See that if initial x lies between the values -1.5 and 1.5 it will always remain so.

tex
Download Presentation

Julia Sets

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Julia Sets • Consider the map: xn+1=xn2+c with c=-3/4 • Choose some values of x and iterate many times - what happens ? • See that if initial x lies between the values -1.5 and 1.5 it will always remain so. • But if the initial x lies outside that range the value of x will blow up.

  2. Complex Maps • Now can consider a more general set of numbers than just the decimals - complex numbers • x --> z • Maps of complex numbers: zn+1=zn2+c • Can ask what is the set of complex numbers that do not blow up under iteration.

  3. Drawing complex numbers • Can use the real and imaginary parts as a pair of (x,y) coordinates in plane • i.e 2+3i is plotted at the point (2,3) • -4i is plotted at point (0,-4) • Usual real numbers lie along the x-axis of this plot.

  4. Complex numbers • Arise when we want to generalize the notion of square root. • What is the number which multiplied by itself gives -1 ? • There is no ordinary, decimal (real) number which has this property. • Define a new number called i i2=-1

  5. More complex numbers • Now I can square root any negative number • eg. -4=4*-1 sqrt(-4)=sqrt(4)i =2i • The number 2i is an example of an imaginary number • In general a complex number is the sum of a real number and an imaginary number z=x+iy

  6. Adding complex numbers • Suppose I have 2 complex numbers z1=c+id and z2=a+ib - how do I add them ? z=z1+z2 • Just add real and imaginary parts separately z=(a+c)+i(b+d)

  7. Multiplying ... • z1=a+ib, z2=c+id z= z1*z2 ? z=(a+ib)*(c+id) = (a*c+ib*c+id*a+i2b*d) = (a*c-b*d)+i(b*c+d*a) • Absolute value of z=x+iy |z|2=(x+iy)*(x-iy) • |z|=sqrt(x*x+y*y)

  8. Back to Maps ... • OK, so now we have a new sort of number to play with • Consider Maps znew=z2+c (z and c are now complex) • Choose some c eg (-3/4,0) • Can now ask what is the set of z which do not blow up under iteration ? (absolute values that do not get infinite )

  9. Drawing them ... • This set of z’s which remain bounded make up the Julia Set associated with the map. • Can draw these points on the plane - it is a fractal! • Black points - values of z which do not get infinite. • Red get big fastest • Yellow get big next fastest • Green next • Blue are slowest to get big.

  10. Different Julia Sets • Choosing different c yields different fractals! • All have structure at all length scales and possess a fractional dimension.

  11. Mandelbrot Set • For the map znew=z2+c • Choose initial z=(0,0). • Choose a (complex) c. • Does the iteration diverge ? • Compute the set of c’s where the iterates never get big this set is again a fractal -- the famous Mandelbrot set

  12. Summary • Generalize the concept of ordinary (real) number to complex numbers z • Consider nonlinear maps of z • Set of z’s which do not diverge under iteration - Julia set and when plotted is seen to have a fractal structure (once again!) • Motion of z on boundary of this set is again chaotic.

More Related