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OdeFactory Images and Annotations

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View/Sys/Gal: EMap "AGF: 1 complex line and a real line, EMap" in "AGFractals."
Range: (vMax,vMin) = (6.500,-3.500), (hMin,hMax) = (-4.000,6.500)
VFld: AGF

This AGF is generated by the lines:

L[1]: (x-1)+(y-1) = 0, m = -1, λ1 = 1

I[1]: x+i*y = 0, λ1 = i

The degree of the RHS of the system is: 2.

Shift-click on the complex line and on the real line to open slider-based parameter controllers. Vary the 12 control parameters to find interesting EMap variations.

In general a 2D polynomial system has 2*(3+2+1) = 12 control parameters. The 2D Mandelbot system, z <- z^2+c, has only 2 control parameters.

Image 1: Ode view.

Image 2: EMap view.

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View/Sys/Gal: EMap "AGF: 1 complex line and a real line, ver 2, EMap" in "AGFractals."
Range: (vMax,vMin) = (3.919,-3.826), (hMin,hMax) = (-2.484,4.823)
VFld: AGF

This AGF is generated by the lines:

L[1]: (x-1.2)+0.10753*(y-1) = 0, m = -9.3, λ1 = 0.9

I[1]: x+i*y = 0, λ1 = (0.14+i)

The degree of the RHS of the system is: 2.

Image 1: Ode view.

Image 2: EMap view.

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View/Sys/Gal: EMap "AGF: 2 complex lines, EMap" in "AGFractals."
Range: (vMax,vMin) = (1.455,-3.545), (hMin,hMax) = (-2.094,2.906)
VFld: AGF

This AGF is generated by the lines:

I[1]: (x-1)+i*y = 0, λ1 = i

I[2]: (x+1)+i*y = 0, λ2 = (1-i)

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

Image 1: Ode R2+ view.

Image 2: EMap view.

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View/Sys/Gal: EMap "AGF: 2 complex lines, in dx/dt, dy/dt form, EMap" in "AGFractals."
Range: (vMax,vMin) = (1.450,-3.550), (hMin,hMax) = (-2.090,2.910)
VFld: ((1-x^2+x^3+y^2+x*(-1+4 *y+y^2))/4,(x^2*(-2+y)-2*x*y+(2+y) *(1+y^2))/4)

This system of odes is defined by:

dx/dt = (1-x^2+x^3+y^2+x*(-1+4*y+y^2))/4,

dy/dt = (x^2*(-2+y)-2*x*y+(2+y)*(1+y^2))/4.

It is the same as the previous AGF but it is has been rewritten in the traditional dx/dt, dy/dt form.

Image 1: EMap view.

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View/Sys/Gal: EMap "AGF: 2 complex lines, ver 2, EMap" in "AGFractals."
Range: (vMax,vMin) = (1.443,-4.145), (hMin,hMax) = (-2.845,2.428)
VFld: AGF

This system is the result of adjusting the 8 parameters for I[1] in system:

        AGF: 2 complex lines, EMap

The AGF is generated by the lines:

I[1]: (1.1+0.01*i)*(x-0.9)+(0.12+0.8*i)*(y-0.01) = 0, λ1 = (0.03+1.2*i)

I[2]: (x+1)+i*y = 0, λ2 = (1-i)

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

Image 1: Ode view.

Image 2: EMap view.

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View/Sys/Gal: EMap "AGF: 2 complex lines, ver 3, EMap" in "AGFractals."
Range: (vMax,vMin) = (2.440,-1.800), (hMin,hMax) = (-1.950,2.410)
VFld: AGF

This AGF is generated by the lines:

I[1]: 1.43*(x-1)+1.2*i*y = 0, λ1 = (1.3-0.4*i)

I[2]: 1.1*(x+1)+i*y = 0, λ2 = (-1.56+0.99*i)

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

Image 1: Ode view.

Image 2: EMap view.

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View/Sys/Gal: EMap "AGF: 2 complex lines, ver 4, EMap" in "AGFractals."
Range: (vMax,vMin) = (0.532,-0.872), (hMin,hMax) = (-1.205,0.013)
VFld: AGF

This AGF is generated by the lines:

I[1]: 1.43*(x-1)+1.2*i*y = 0, λ1 = (1.3-0.4*i)

I[2]: 1.1*(x+1)+i*y = 0, λ2 = (-1.56+0.99*i)

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

Image 1: Ode R2+ view.

Image 2: EMap view.

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View/Sys/Gal: EMap "AGF: 2 complex lines, ver 5, EMap" in "AGFractals."
Range: (vMax,vMin) = (1.760,-1.206), (hMin,hMax) = (-0.193,1.682)
VFld: AGF

This AGF is generated by the lines:

I[1]: (1.1+0.01*i)*(x-1)+(0.01+0.8*i)*y = 0,

         λ1 = (0.01+1.1*i)

I[2]: (x+1.4)+(-0.01+0.7*i)*y = 0,

         λ2 = (1-1.1*i)

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

Image 1: Ode R2+ view.

Image 2: EMap view.

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View/Sys/Gal: EMap "AGF: 2 complex lines, ver 6, EMap" in "AGFractals."
Range: (vMax,vMin) = (0.641,-0.641), (hMin,hMax) = (0.407,1.600)
VFld: AGF

This AGF is generated by the lines:

I[1]: (x-1)+i*y = 0, λ1 = (0.005+i)

I[2]: (x+1)+i*y = 0, λ2 = (1-i)

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

Image 1: Ode R2+ view.

Image 2: EMap view.

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View/Sys/Gal: EMap "AGF: 2 complex lines, ver 7, EMap" in "AGFractals."
Range: (vMax,vMin) = (2.705,-2.295), (hMin,hMax) = (-2.557,2.443)
VFld: AGF

This AGF is generated by the lines:

        I[1]: 1.4*(x-1)+1.2*i*y = 0,

         λ1 = (1.9-0.07*i)

        I[2]: 1.1*(x+1)+i*y = 0,

         λ2 = (-1.56+0.99*i)

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

Image 1: Ode view.

Image 2: EMap view.

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View/Sys/Gal: EMap "AGF: 3 complex lines" in "AGFractals."
Range: (vMax,vMin) = (3.000,-2.000), (hMin,hMax) = (-2.500,2.500)
VFld: AGF

This AGF is generated by the lines:

        I[1]: x+i*(y-1) = 0, λ1 = i

        I[2]: (x-1)+i*y = 0, λ2 = i

        I[3]: (x+1)+i*y = 0, λ3 = i

The degree of the RHS of the system is: 5.

This is an interesting Ode but it is not so interesting in the IMap or EMap views.

Image 1: Ode view.

Image 2: EMap view.

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View/Sys/Gal: EMap "AGF: 3 complex lines, 1 real line" in "AGFractals."
Range: (vMax,vMin) = (2.500,-2.500), (hMin,hMax) = (-2.500,2.500)
VFld: AGF

This AGF is generated by the lines:

        L[1]: x-y = 0, m = 1, λ1 = 0.01

        I[1]: x+i*(y-1) = 0, λ1 = (-0.1+i)

        I[2]: (x-1)+i*y = 0, λ2 = (-0.1+i)

        I[3]: (x+1)+i*y = 0, λ3 = (0.1+i)

The degree of the RHS of the system is: 6.

Another interesting Ode.

Image 1: Ode view.

Image 2: EMap view.

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View/Sys/Gal: EMap "AGF: 3 complex lines, 1 real line, ver 2" in "AGFractals."
Range: (vMax,vMin) = (1.801,-0.699), (hMin,hMax) = (-1.337,1.163)
VFld: AGF

This AGF is generated by the lines:

        L[1]: x-y = 0, m = 1, λ1 = -0.01

        I[1]: x+i*(y-1) = 0, λ1 = (-0.1+i)

I[2]: (1-0.3*i)*(x+0.3)+i*(y-0.82) = 0,

         λ2 = (-0.1-1.1*i)

        I[3]: (x+1)+i*y = 0, λ3 = (0.1+i)

The degree of the RHS of the system is: 6.

An interesting Ode and EMap.

Image 1: Ode view.

Image 2: EMap view.

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View/Sys/Gal: EMap "AGF: 3 complex lines, ver 2, EMap" in "AGFractals."
Range: (vMax,vMin) = (2.200,-1.800), (hMin,hMax) = (-1.800,2.200)
VFld: AGF

This AGF is generated by the lines:

        I[1]: x+i*(y-1) = 0, λ1 = i

I[2]: (-2+0.5*i)*(x-1)+(0.3+2.01*i)*y = 0,

         λ2 = (-0.15+1.4*i)

        I[3]: (x+1)+i*y = 0, λ3 = i

The degree of the RHS of the system is: 5.

Sort of an interesting Ode but a more interesting EMap.

Image 1: Ode view.

Image 2: EMap view.

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View/Sys/Gal: EMap "AGF: 3 complex lines, ver 3, EMap" in "AGFractals."
Range: (vMax,vMin) = (2.469,-1.531), (hMin,hMax) = (-1.652,2.073)
VFld: AGF

This AGF is generated by the lines:

I[1]: (1.0001+1.2*i)*(x-0.001)+(0.5001+2*i)*(y-1.0001) = 0, λ1 = (-2.06-0.45*i)

        I[2]: (x-1)+i*y = 0, λ2 = i

        I[3]: (x+1)+i*y = 0, λ3 = i

The degree of the RHS of the system is: 5.

Image 1: Ode view.

Image 2: EMap view.

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View/Sys/Gal: EMap "AGF: 3 complex lines, ver 4, EMap" in "AGFractals."
Range: (vMax,vMin) = (1.800,-1.800), (hMin,hMax) = (-1.700,2.000)
VFld: AGF

This AGF is generated by the lines:

        I[1]: x+i*(y-1) = 0, λ1 = i

I[2]: (1-0.01*i)*(x-1)+(-0.04+i)*(y-0.03) = 0,

         λ2 = (-0.51-1.4*i)

        I[3]: (x+1)+i*y = 0, λ3 = i

The degree of the RHS of the system is: 5.

Image 1: Ode view.

Image 2: EMap view.

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View/Sys/Gal: EMap "AGF: 3 real lines" in "AGFractals."
Range: (vMax,vMin) = (10.000,-10.000), (hMin,hMax) = (-10.000,10.000)
VFld: AGF

This AGF is generated by the lines:

L[1]: x-(y+1) = 0, m = 1, λ1 = 1

L[2]: x+(y+1) = 0, m = -1, λ2 = 1

L[3]: (y-1) = 0, m = 0, λ3 = 1

The degree of the RHS of the system is: 2.

Image 1: Ode view.

Image 2: EMap view.

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View/Sys/Gal: EMap "AGF: 3 real lines, ver 2, EMap" in "AGFractals."
Range: (vMax,vMin) = (1.355,-0.542), (hMin,hMax) = (-0.432,1.107)
VFld: AGF

This AGF is generated by the lines:

L[1]: x-0.2381*(y+1) = 0, m = 4.2, λ1 = 1.2

L[2]: x+(y+1) = 0, m = -1, λ2 = -5

L[3]: (y-1) = 0, m = 0, λ3 = -1

The degree of the RHS of the system is: 2.

Image 1: Ode R2+ view.

Image 2: EMap view.

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View/Sys/Gal: EMap "AGFv: variation 1 of star-in & star-out, EMap" in "AGFractals."
Range: (vMax,vMin) = (5.400,-5.700), (hMin,hMax) = (-10.000,10.000)
VFld: (b*x^2-y^2+p,a*x*y%c+q), p = -.70; q = 1.00; a = 1.00; b = .30; c = 6.70;

This system of odes is defined by the equations:

        dx/dt = b*x^2-y^2+p,

        dy/dt = a*x*y%c+q

Parameters are:

        p = -.70; q = 1.00; a = 1.00;

        b = .30; c = 6.70;

Image 1: Ode view with colored V fld on.

Image 2: EMap view.

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View/Sys/Gal: EMap "AGFv: variation 2 of star-in & star-out, EMap" in "AGFractals."
Range: (vMax,vMin) = (1.250,-1.250), (hMin,hMax) = (-1.250,1.250)
VFld: (x^2-y^2+p,2*x%y+q), p = -.65; q = 1.00;

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 = -.65; q = 1.00;

Image 1: Ode view with colored V fld on.

Image 2: EMap view.