File:Bifurcation1-2.png
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[edit]DescriptionBifurcation1-2.png |
English: Bifurcation of periodic points from period 1 to 2 for fc(z)=z*z +c. Parabolic parameter c = -3/4 and fixed point z = 1/2 |
Date | |
Source | Own work |
Author | Adam majewski |
Other versions |
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Summary
[edit]This image shows some features of the discrete dynamical system
based on complex quadratic polynomial :
.
When coefficient goes from c=0.25 to c=-2 along horizontal axis ( imaginary part of c is zero and it is a 3D diagram of function which gets real input and gives complex output)
then limit cycle is changing from fixed point ( period 1) to period 2 cycle. This qualitative change is called bifurcation.
This path is inside Mandelbrot set ( escape route). It s also first of period doubling bifurcation.
Note that :
- there are fixed points for all c values, but they change from attracting to indifferent( in parabolic point, root point) and repelling
- there are 2 period 2 points for all c values. They also change from from attracting to indifferent( in parabolic point, root point) and repelling.
- in bifurcation point ( root, parabolic) all period 2 values and fixed point have the same value and the same (=1) stability index .
- before and after bifurcation point period 2 points creates 3D parabolas, which are rotated ( 90 degrees) with respect to themselves
stability index of period 1 points | period 1 points on dynamic plane | period 1 points on parameter plane |
---|---|---|
changes from attractive through indifferent to repelling | moves from interior of Kc to its boundary | moves from interior of component of M-set to its boundary |
Please check demo 2 page 3 from program Mandel by Wolf Jung to see another visualisation of this bifurcation.
dynamics
[edit]parameter c | location of c | Julia set | interior | type of critical orbit dynamics | critical point | fixed points | stability of alfa |
---|---|---|---|---|---|---|---|
c = -3/4 | boundary, root point | connected | exist | parabolic | attracted to alfa fixed point | alfa fixed point equal to beta fixed point, both are parabolic | r = 1 |
0 < x < -3/4 | internla ray 1/2 | connected | exist | attracting | attracted to alfa fixed point | 0 < r < 1.0 | |
c = 0 | center, interior | connected = Circle Julia set | exist | superattracting | attracted to alfa fixed point | fixed critical point equal to alfa fixed point, alfa is superattracting, beta is repelling | r = 0 |
0<c<1/4 | internal ray 0, interior | connected | exist | attracting | attracted to alfa fixed point | alfa is attracting, beta is repelling | 0 < r < 1.0 |
c = 1/4 | cusp, boundary | connected = cauliflower | exist | parabolic | equal to alfa fixed point | alfa fixed point equal to beta fixed point, both are parabolic | r = 1 |
c>1/4 | external ray 0, exterior | disconnected = imploded cauliflower | disappears | repelling | repelling to infinity | both finite fixed points are repelling | r > 1 |
Stability r is absolute value of multiplier m at fixed point alfa :
c = 0.0000000000000000+0.0000000000000000*I m(c) = 0.0000000000000000+0.0000000000000000*I r(m) = 0.0000000000000000 t(m) = 0.0000000000000000 period = 1 c = 0.0250000000000000+0.0000000000000000*I m(c) = 0.0513167019494862+0.0000000000000000*I r(m) = 0.0513167019494862 t(m) = 0.0000000000000000 period = 1 c = 0.0500000000000000+0.0000000000000000*I m(c) = 0.1055728090000841+0.0000000000000000*I r(m) = 0.1055728090000841 t(m) = 0.0000000000000000 period = 1 c = 0.0750000000000000+0.0000000000000000*I m(c) = 0.1633399734659244+0.0000000000000000*I r(m) = 0.1633399734659244 t(m) = 0.0000000000000000 period = 1 c = 0.1000000000000000+0.0000000000000000*I m(c) = 0.2254033307585166+0.0000000000000000*I r(m) = 0.2254033307585166 t(m) = 0.0000000000000000 period = 1 c = 0.1250000000000000+0.0000000000000000*I m(c) = 0.2928932188134524+0.0000000000000000*I r(m) = 0.2928932188134524 t(m) = 0.0000000000000000 period = 1 c = 0.1500000000000000+0.0000000000000000*I m(c) = 0.3675444679663241+0.0000000000000000*I r(m) = 0.3675444679663241 t(m) = 0.0000000000000000 period = 1 c = 0.1750000000000000+0.0000000000000000*I m(c) = 0.4522774424948338+0.0000000000000000*I r(m) = 0.4522774424948338 t(m) = 0.0000000000000000 period = 1 c = 0.2000000000000000+0.0000000000000000*I m(c) = 0.5527864045000419+0.0000000000000000*I r(m) = 0.5527864045000419 t(m) = 0.0000000000000000 period = 1 c = 0.2250000000000000+0.0000000000000000*I m(c) = 0.6837722339831620+0.0000000000000000*I r(m) = 0.6837722339831620 t(m) = 0.0000000000000000 period = 1 c = 0.2500000000000000+0.0000000000000000*I m(c) = 0.9999999894632878+0.0000000000000000*I r(m) = 0.9999999894632878 t(m) = 0.0000000000000000 period = 1 c = 0.2750000000000000+0.0000000000000000*I m(c) = 1.0000000000000000+0.3162277660168377*I r(m) = 1.0488088481701514 t(m) = 0.0487455572605341 period = 1 c = 0.3000000000000000+0.0000000000000000*I m(c) = 1.0000000000000000+0.4472135954999579*I r(m) = 1.0954451150103321 t(m) = 0.0669301182003075 period = 1 c = 0.3250000000000000+0.0000000000000000*I m(c) = 1.0000000000000000+0.5477225575051662*I r(m) = 1.1401754250991381 t(m) = 0.0797514300099943 period = 1 c = 0.3500000000000000+0.0000000000000000*I m(c) = 1.0000000000000000+0.6324555320336760*I r(m) = 1.1832159566199232 t(m) = 0.0897542589928440 period = 1 ==Maxima CAS src code== <pre> GiveRoots_bf(g):= block( [cc:bfallroots(expand(g)=0)], cc:map(rhs,cc),/* remove string "c=" */ return(cc) )$ /* functions for computing periodic points ; */ give_beta(_c):= (1+sqrt(abs(1-4*_c)))/2 $ give_alfa(_c):= (1-sqrt(abs(1-4*_c)))/2 $ give_2(c):= block( [eq,rr], eq:z*z +z +c +1, rr:GiveRoots_bf(eq), return(float(rr)) ); xMax:0; xMin:-1.39; yMin:-2; yMax:2; iXmax:1000; dx:(xMax-xMin)/iXmax; /* points */ p_pts:[ [-0.75,-0.5,0] ]; p1_beta:[]; p1_alfa_r:[]; p1_alfa_a:[]; p2_r:[]; /* period 2 repelling */ p2_a:[]; /* period 2 attracting */ /* -------------------- main ----------------------- */ for c:xMin step dx thru xMax do ( alfa:give_alfa(c), if cabs(2*alfa)>1 then p1_alfa_r:cons([c,realpart(alfa),imagpart(alfa)],p1_alfa_r) else p1_alfa_a:cons([c,realpart(alfa),imagpart(beta)],p1_alfa_a), roots:allroots(z*z +z +c +1=0), z2:rhs(roots[1]), if cabs(float(4*z2*(z2*z2+c)))>1 /* multiplier */ then for z in roots do p2_r:cons([c,realpart(rhs(z)),imagpart(rhs(z))],p2_r) else for z in roots do p2_a:cons([c,realpart(rhs(z)),imagpart(rhs(z))],p2_a) ); load(cpoly); /* for bfallroots */ load(draw); draw3d( terminal = screen, pic_height= iXmax, title = "periodic z-points for c along horizontal axis for fc(z)= z*z +c ", ylabel = "Re(z)", zlabel ="Im(z)", xlabel = "c-coefficient", yrange = [yMin,yMax], point_type = filled_circle, point_size = 0.2, points_joined = true, /* period 1 */ key = " alfa repelling", color = dark-blue, points(p1_alfa_r), key = " alfa attracting", color = light-blue, points(p1_alfa_a), /* period 2 */ points_joined = false, key = " period 2 attracting", color = dark-green, points(p2_a), key = " period 2 repelling", color = light-green, points(p2_r), /* grid and tics */ xtics = {-3/4}, /* -2,root points,centers, 0 */ /*xtics_axis = true, plot tics on x-axis */ xtics_rotate = true, ytics = {-0.5}, ztics = {-1,0,1}, grid = true, /* draw grid*/ /* special points */ point_size = 0.7, color = red, key = "bifurcation", points(p_pts) )$
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