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Range: (vMax,vMin) = (2.500,-2.500), (hMin,hMax) = (-2.500,2.500)
VFld (0)

This gallery is a collection of 2D systems resulting from AGFs defined by two complex lines.

Each complex line has 8 parameters. Consequently each AGF has 16 adjustable parameters. By varying the parameters, each complex line can generate a:

center,

spiral or

star.

For any choice of the 16 parameters the AGFs can be converted to the "dx/dt, dy/dt" form, without parameters. In this gallery, the resulting systems are designated by using "AGFv" in the system names. The parameterless AGFv systems can be made more interesting by adding a few parameters.

The gallery begins with a discussion of AGFs defined by one complex line. These AGFs can produce: points, straight lines, circles, squares and spirals in various views.

The last 4 systems in the gallery demonstrate connections between AGFs defined by 2 complex generators and the Mandelbrot fractals.

Defining complex lines at (-a,-b) and (a,b) and setting λ2 = - λ1 will produce all of the Mandelbrot fractals. Gallery AGFs&Mandelbrot contains a detailed discussion of how to find (a,b) and λ1. Breaking the symmetry gives interesting new extensions of the Mandelbrot fractals.  View/Sys/Gal: Ode "(0) AGF: center ccw" in "AGFEMapExs."
Range: (vMax,vMin) = (5.000,-5.000), (hMin,hMax) = (-5.000,5.000)
Algebraically Generated Flow

This AGF is generated by the lines:

I: x+i*y = 0, λ1 = -i

The degree of the RHS of the system is: 1.

Complex "lines" are used in complex conjugate pairs as generators in AGFs. The other complex line in this example is conj(I): x-i*y = 0.

ICs: (x,y) = (1,1)

re( λ1) determines the rate of the radial motion and sign(re( λ1)) determines the direction of radial motion (+ = out, - = in)

im( λ1) determines the rate of the angular motion and sign(im( λ1) determines the direction of angular motion (+ = cw, - = ccw).

This system has 8 parameters. If we restrict parameter changes to changes in λ only the bifurcation diagram, in the complex plane, is:

real axis -> stars,

complex axis -> centers,

origin -> fixed point

re( λ)

im( λ)regiontype

1

0+x axisstar out

1

0

1+y axiscw center

-1

-1

0-x axisstar in

-1

0

-1-y axisccw center

1

Note that the center and star states are not stable in that a small change will cause either to become a spiral.

Image 1: Ode view, all trajectories are circles w/period 2* π.

Image 2: IMap view, all orbits are squares w/period 4.

The traditional cartesian coordinate "dx/dt, dy/dt" form of the system is:

dx/dt = -y, dy/dt = x.

To get to the corresponding polar coordinate form use:

r' = (x*x'+y*y')/r = 0 and

θ' = (x*y'-y*x')/r^2 = 1

(constant ccw motion in angular direction)

or, in OdeFactory notation, the system is

"dx/dt = 0, dy/dt = 1"

in polar coordinates with ICs

(x,y) = (sqrt(2),pi/4) = (1.4141,.7854)  View/Sys/Gal: IMap "(0b) AGFv: center ccw, polar coordinates" in "AGFEMapExs."
Range: (vMax,vMin) = (5.000,-5.000), (hMin,hMax) = (-5.000,5.000)
VFld (0,1)

This system of odes is defined by the equations:

dx/dt = 0,

dy/dt = 1.

ICs: (1.4141,.7854)

Image 1: Ode view, of what was a center in cartesian coordinates polar coordinates.

The circular trajectories become straight lines.

Image 2: IMap view, all orbits go to the fixed point (0,1) in one iteration.

Think of x as r and y as θ.

"r is constant and θ increases" as expected.    View/Sys/Gal: EMap "(0c) AGF: spiral out ccw" in "AGFEMapExs."
Range: (vMax,vMin) = (1.250,-1.250), (hMin,hMax) = (-1.250,1.250)
Algebraically Generated Flow

This AGF is generated by the lines:

I: x+i*y = 0, λ1 = (1-i)

The degree of the RHS of the system is: 1.

ICs: (x,y) = (1,1)

Image 1: Ode view, all trajectories are ccw spirals-out.

Image 2: Ode in 3D/(t,x) view.

Image 3: IMap view, all orbits are ccw spirals-out.

Image 4: EMap view, zoomed in a bit, w/CT 2.

The traditional cartesian coordinate "dx/dt, dy/dt" form of the system is:

dx/dt = x-y, dy/dt = x+y

To get to the corresponding polar coordinate form use:

r' = (x*x'+y*y')/r = r and

θ'= (x*y'-y*x')/r^2 = 1

(angular motion is ccw)

or, in OdeFactory, notation, the system is

"dx/dt = x, dy/dt = 1"

in polar coordinates with ICs

(x,y) = (sqrt(2),pi/4) = (1.4141,.7854)     View/Sys/Gal: IMap "(0d) AGFv: spiral out ccw, polar coordinates" in "AGFEMapExs."
Range: (vMax,vMin) = (5.000,-5.000), (hMin,hMax) = (-5.000,5.000)
VFld (x,1)

This system of odes is defined by the equations:

dx/dt = x,

dy/dt = 1.

ICs: (1.4141,.7854)

It shows a "spiral" in polar coordinates. Spiral trajectories become

x(t)= x(0)*e^t, y(t)=y(0)+1.

Image 1: Ode (x,y) view.

Image 2: Ode R2+ view w/fixed point at (u,v) = (1,0).

Image 3: 3D/(t,x) Ode view.

Image 4: 3D/(t,y) Ode view.

Image 5: IMap view, all orbits go to the fixed points (x,1).

Think of x as r and y as θ.

"r increases as r(0)*e^t and θ increases as θ(0)+t" as expected.

See the 3D/(t,x) and 3D/(t,y) views.  View/Sys/Gal: IMap "(0e) AGF: star out" in "AGFEMapExs."
Range: (vMax,vMin) = (5.000,-5.000), (hMin,hMax) = (-5.000,5.000)
Algebraically Generated Flow

This AGF is generated by the lines:

I: x+i*y = 0, λ1 = 1

The degree of the RHS of the system is: 1.

ICs: (x,y) = (1,1)

Image 1: Ode view, all trajectories are straight lines.

Image 2: IMap view w/"Show 2D IMap Orbit Sequence" on, all seeds are fixed points.

The traditional cartesian coordinate "dx/dt, dy/dt" form of the system is:

dx/dt = x, dy/dt = y

To get to the corresponding polar coordinate form use:

r' = (x*x'+y*y')/r = r and

θ' = (x*y'-y*x')/r^2 = 0

(no angular motion)

or, in OdeFactory notation, the system is

"dx/dt = x, dy/dt = 0"

in polar coordinates with ICs

(x,y) = (sqrt(2),pi/4) = (1.4141,.7854)  View/Sys/Gal: IMap "(0f) AGFv: star out, polar coordinates" in "AGFEMapExs."
Range: (vMax,vMin) = (5.000,-5.000), (hMin,hMax) = (-5.000,5.000)
VFld (x,0)

This system of odes is defined by the equations:

dx/dt = x,

dy/dt = 0.

Ics: (1.4141,.7854)

Image 1: Ode view, all trajectories are straight horizontal lines. The y axis consists of Ode fixed points.

Image 2: IMap view, all seeds (x,y) go to (x,0). The x axis consists of IMap fixed points.

"r increases and θ stays constant" as expected.

It shows a "straight radial line" in polar coordinates. Radial trajectories in cartesian coordinates become straight lines parallel to the horizontal axis in polar coordinates.

Think of x as r and y as θ.

"r increases as r(0)*e^t and θ increases as θ(0)+t" as expected.

See the 3D/(t,x) and 3D/(t,y) views.   View/Sys/Gal: IMap "(1) AGF two centers" in "AGFEMapExs."
Range: (vMax,vMin) = (0.625,-0.625), (hMin,hMax) = (-0.625,0.625)
Algebraically Generated Flow

This AGF is generated by the lines:

I: x+i*y = 0, λ1 = i

I: (x-1)+i*(y-1) = 0, λ2 = -i

The first is a cw center at (0,0) and the second is a ccw center at (1,1). The two green dots are I and I. Clicking on a line opens the line's parameter controller.

The degree of the RHS of the system would in general be 3 but it is reduced to 2 in this case because of symmetry.

This AGF has interesting IMap and EMap views.

Image 1: Ode view centered at (.5,.5).

Image 2: EMap view, centered at (.5,.5) w/CT 4. This is a shifted Julia set fractal.

Image 3: IMap view, zoomed in, centered at (0,0). All orbits in the Julia set's prisoner set spiral in to (0,0). All other orbits go to infinity.   View/Sys/Gal: IMap "(1b) AGFv: two centers, p= 0, q=0" in "AGFEMapExs."
Range: (vMax,vMin) = (0.625,-0.625), (hMin,hMax) = (-0.625,0.625)
VFld (y+0.5*x^2-x*y-0.5*y^2+p,-x+0.5*x^2+x*y-0.5*y^2+q), p=0; q=0

This system of odes is defined by the equations:

dx/dt = y+0.5*x^2-x*y-0.5*y^2+p,

dy/dt = -x+0.5*x^2+x*y-0.5*y^2+q.

Parameters are:

p=0; q=0

It is the "dx/dt, dy/dt" form of (1) with parameters p and q added.

If p+q=0 then we get two centers otherwise we get two spirals.

Image 1: Ode view centered at (.5,.5).

Image 2: EMap view, centered at (.5,.5) w/CT 4. This is a shifted Julia set fractal. All orbits in the prisoner set (black region) spiral in to (0,0).

Image 3: IMap view, zoomed in, centered at (0,0).   View/Sys/Gal: Ode "(2) AGF center & spiral, as EMap" in "AGFEMapExs."
Range: (vMax,vMin) = (2.818,-1.614), (hMin,hMax) = (-1.560,2.621)
Algebraically Generated Flow

This AGF is generated by the lines:

I: x+i*y = 0, λ1 = (0.2+i), spiral

I: (x-1)+i*(y-1) = 0, λ2 = -i, center

The degree of the RHS of the system is: 3.

A center and a spiral will never produce a Julia set because the symmetry of z <- z^2+c is broken.

Image 1: EMap view. The symmetry is broken here, λ2 is not quite - λ1. So this is a variation of a Julia set.

Image 2: EMap view zoomed in.

Image 3: Ode view. The broken symmetry is very obvious here.  View/Sys/Gal: EMap "(3) AGF two spirals" in "AGFEMapExs."
Range: (vMax,vMin) = (5.000,-5.000), (hMin,hMax) = (-5.000,5.000)
Algebraically Generated Flow

This AGF is generated by the lines:

I: (x+1)+i*(y-1) = 0, λ1 = (2-i)

I: (x-1)+i*(y+1) = 0, λ2 = (-2+i)

The degree of the RHS of the system is 2 due to the symmetry of the system.

Image 1: Ode view.

Image 2: EMap view w/CT 5. Another Julia set.   View/Sys/Gal: Ode "(3b) AGF 2 spirals, as EMap" in "AGFEMapExs."
Range: (vMax,vMin) = (4.102,-2.424), (hMin,hMax) = (-2.412,2.824)
Algebraically Generated Flow

This AGF is generated by the lines:

I: (x+1)+i*(y-1) = 0, λ1 = (0.5-1.2*i)

I: (x-1)+i*(y+1) = 0, λ2 = (-2+0.7*i)

The degree of the RHS of the system is: 3.

Image 1: EMap view zoomed way in. Just an interesting image.

Image 2: EMap view zoomed out and centered. Not a fractal. Try other CTs.

Image 3: Ode view. No symmetry. View/Sys/Gal: EMap "(3c) AGFv two spirals with p =.80, q = -2, as EMap" in "AGFEMapExs."
Range: (vMax,vMin) = (0.625,-0.625), (hMin,hMax) = (-0.625,0.625)
VFld (x*(1.5*y-0.25*x)+0.25*y^2+p,.25*(q-3*x^2-2*x*y+3*y^2)), p = .80; q = -2.00;

This system of odes is defined by the equations:

dx/dt = x*(1.5*y-0.25*x)+0.25*y^2+p,

dy/dt = .25*(q-3*x^2-2*x*y+3*y^2)

Parameters are:

p = .80; q = -2.00;

This is a variation of the "dx/dt, dy/dt" form of system (3).

Image 1: EMap view. Variation of a Julia set. View/Sys/Gal: EMap "(3d) AGFv two spirals with p=.86, q=-2, as EMapCT1" in "AGFEMapExs."
Range: (vMax,vMin) = (0.230,0.206), (hMin,hMax) = (1.077,1.099)
VFld (x*(1.5*y-0.25*x)+0.25*y^2+p,.25*(q-3*x^2-2*x*y+3*y^2)), p = .86; q = -2.00;

This system of odes is defined by the equations:

dx/dt = x*(1.5*y-0.25*x)+0.25*y^2+p,

dy/dt = .25*(q-3*x^2-2*x*y+3*y^2)

Parameters are:

p = .86; q = -2.00;

Image 1: EMap view. View/Sys/Gal: EMap "(4) AGF two stars, as EMap" in "AGFEMapExs."
Range: (vMax,vMin) = (2.500,-2.500), (hMin,hMax) = (-2.500,2.500)
Algebraically Generated Flow

This AGF is generated by the lines:

I: (x+1)+i*y = 0, λ1 = -2

I: (x-1)+i*y = 0, λ2 = 2

The degree of the RHS of the system would be 3 in general but for these particular parameter values it reduces to 2.

Image 1: EMap view. Julia set.

The system in "dx/dt, dy/dt" form, with parameters p and q added, is

dx/dt = x^2-y^2+p

dy/dt = 2*x*y+q

p = -1, q = 0

which, for z = x+i*y, c = p+i*q is the Mandelbrot iteration

z <- z^2+c.

The "dx/dt, dy/dt" form of this system is discussed in system (4c). View/Sys/Gal: EMap "(4b) AGF two stars, as EMap" in "AGFEMapExs."
Range: (vMax,vMin) = (0.800,-0.800), (hMin,hMax) = (-2.000,2.000)
Algebraically Generated Flow

This AGF is generated by the lines:

I: (1-0.01*i)*(x+1)+0.9*i*(y-0.03) = 0, λ1 = (-2.5+0.13*i)

I: (x-1)+i*y = 0, λ2 = 2

The degree of the RHS of the system is: 3.

This is system (4) where the I parameters have been changed a bit. Notice that the EMap has changed a lot.

Image 1: EMap view.    View/Sys/Gal: IMap "(4c) AGFv two stars, Mandelbrot form, as EMap" in "AGFEMapExs."
Range: (vMax,vMin) = (2.500,-2.500), (hMin,hMax) = (-2.500,50.000)
VFld (x^2-y^2+p,2*x*y+q), p=-1; q=0

This system of odes is defined by the equations:

dx/dt = x^2-y^2+p,

dy/dt = 2*x*y+q.

Parameters are:

p = -1; q = 0

Two stars iff q = 0.

For p = -1, q = 0 all orbits in the prisoner set seem to have period 2. Furthermore they oscillate between x = -1 and x = 0.

Image 1: EMap view. Seven orbits have been started in the prisoner set (the black region).

Image 2: IMap view showing the seven orbits.

Image 3: IMap 3D/(t,x) view. The oscillation in the (t,x) plane.

Image 4: IMap 3D/(t,y) view. No oscillation in the (t,y) plane. View/Sys/Gal: EMap "(4d) AGFv two stars, Mandelbrot form, p = .40; q = .35; as EMapCT3" in "AGFEMapExs."
Range: (vMax,vMin) = (1.250,-1.250), (hMin,hMax) = (-1.250,1.250)
VFld (x^2-y^2+p,2*x*y+q), p = .40; q = .35;

This system of odes is defined by the equations:

dx/dt = x^2-y^2+p,

dy/dt = 2*x*y+q

Parameters are:

p = .40; q = .35;

Fixed points for the ode are approximately:

(-.256425,.682461) and (.256425,-.682461).

Image 1: IMap view. A nice Julia set.