Fractals/Iterations in the complex plane/Discrete Lagrangian Descriptors

Lagranian descriptor for continoust-time dynamical systems ( Lagrangian way to describe the flow)C. Mendoza, A. M. Mancho. The hidden geometry of ocean flows. Physical Review Letters 105 (2010), 3, 038501-1-038501-4. A. M. Mancho, S. Wiggins, J. Curbelo, C. Mendoza. Lagrangian Descriptors: A Method for Revealing Phase Space Structures of General Time Dependent Dynamical Systems. Communications in Nonlinear Science and Numerical Simulation. 18 (2013) 3530-3557 C. Lopesino, F. Balibrea-Iniesta, V. J. García-Garrido, S. Wiggins, A. M. Mancho. A theoretical framework for Lagrangian descriptors. International Journal of Bifurcation and Chaos 27, 1730001 (2017). Lagrangian description of the flow field in wikipedia is a method for analyzing structure of the phase space. HereC. Lopesino, F. Balibrea, S. Wiggins, A.M. Mancho. Lagrangian Descriptors for Two Dimensional, Area Preserving Autonomous and Nonautonomous Maps. Communications in Nonlinear Science and Numerical Simulation 27 (1-3) (2015) 40-51. V. J . Garcia Garrido. Unveiling the Fractal Structure of Julia Sets with Lagrangian Descriptors. https://arxiv.org/abs/2001.08937 V. J. García-Garrido, F. Balibrea-Iniesta, S. Wiggins, A. M. Mancho, C. Lopesino. Detection of Phase Space Structures of the Cat Map with Lagrangian Descriptors. Regular and Chaotic Dynamics 23, (6) 751-766 (2018). G. G. Carlo and F. Borondo. Lagrangian descriptors for open maps Phys. Rev. E 101, 022208 (2020) this method is extended to discrete dynamical systems: open maps in the complex plane. Images Full source code is on the commons page ( click on the image) key words complex number complex map Map Riemann sphere Point of the Riemann sphere inverse stereographic projection thumb|right|3D illustration of a stereographic projection from the north pole onto a plane below the sphere inverse stereographic projection maps point of complex plane to point of Riemann sphere : so and p-norm The -norm (also called -norm) of vector is[[:w:Lp space#When_0_<_p_<_1|Lp space when 0<p <1 in wikipedia]] where number p is a real number . It is called a power. It affects the steepness of the gradient near singularities (fractal features like a Julia set) p-norm is used to measure the distance between sequential iterations of the mapping f on the Riemann sphere Discrete Lagrangian Descriptor = DLD The simple idea is to compute the p-norm version of Lagrangian descriptors, not for the points on the complex plane, but for their projections on the Riemann sphere in the extended complex plane. ... in the complex mappings that we consider in this work, the functions that define the dynamics are not invertible, and therefore we will only keep the forward part of the definition. DLD: is a scalar value accumulates the p-norm along the orbit (= has information about the history of the orbit) and therefore unveiles the structure of interior and exterior of Julia set sum summing is what the original paper does ( pauldelbrot) where: N is a fixed number of iterations is any initial condition selected on a bounded subset D of the complex plane is a point of Riemann sphere: Averaging "averaging keeps the coloring stable if maxiters is changed but can lead to low variation over the image" pauldelbrot Steps For each point z of complex plane compute DLD ( scalar value) color is proportional to DLD Substeps: To compute DLD of z : iterate point z under the map f = compute zn map each point zn from complex plane to the Riemann sphere ( Inverse Stereographic projection) for each zn compute summand ... ( to do ) Code UltraFractal DLD { ; Based on https://arxiv.org/pdf/2001.08937.pdf ; ucl file for UltraFractal by pauldelbrot init: float sum = 0.0 float lastx = 0.0 float lasty = 0.0 float lastz = 0.0 int i = 0 loop: float d = |#z| float dd = 1/(d + 1) ; Riemann sphere coordinates = (xx, yy,zz) float xx = 2*real(#z)*dd float yy = 2*imag(#z)*dd float zz = (d - 1)*dd : IF (i > 0) sum = sum + (xx - lastx)^@power + (yy - lasty)^@power + (zz - lastz)^@power ENDIF i = i + 1 lastx = xx lasty = yy lastz = zz final: #index = sum/(i - 1) default: title = "Discrete Langrangian Descriptors" param power caption = "Power" default = 0.25 hint = "Affects the steepness of the gradient near singularities (fractal features like a Julia set)" min = 0.0 endparam } " Here's the latest version I've been using in UF. It handles escaping points with no-bail formulae via the isInf/isNaN test (puts the Riemann sphere point at the north pole for those), allows averaging or summing (summing is what the original paper does, whereas averaging keeps the coloring stable if maxiters is changed but can lead to low variation over the image), and can use or not use absolute values on the differences being summed." pauldelbrot DLD { ; Based on https://arxiv.org/pdf/2001.08937.pdf ; ucl file for UltraFractal by pauldelbrot init: float sum = 0.0 float lastx = 0.0 float lasty = 0.0 float lastz = 0.0 int i = 0 loop: float d = |#z| float dd = 1/(d + 1) float xx = 2*real(#z)*dd float yy = 2*imag(#z)*dd float zz = (d - 1)*dd ; Riemann sphere coordinates IF (isInf(d) || isNaN(d)) ; Infinity, or thereabouts xx = 0 yy = 0 zz = 1 ENDIF IF (i > 0) IF(@qabs) sum = sum + abs(xx - lastx)^@power + abs(yy - lasty)^@power + abs(zz - lastz)^@power ELSE sum = sum + (xx - lastx)^@power + (yy - lasty)^@power + (zz - lastz)^@power ENDIF ENDIF i = i + 1 lastx = xx lasty = yy lastz = zz final: IF(@qsum) #index = sum ELSE #index = sum/(i - 1) ENDIF default: title = "Discrete Langrangian Descriptors" param power caption = "Power" default = 0.25 hint = "Affects the steepness of the gradient near singularities (fractal features like a Julia set)" min = 0.0 endparam param qsum caption = "Sum" default = false hint = "Averages if false, sums if true." endparam param qabs caption = "Abs differences" default = false endparam } Fragmentarium Code based on the UF code, modified and optimized for GLSL by 3Dickulusfractalforums.org: lagrangian-descriptors-fragment-code #include "Complex.frag" #include "MathUtils.frag" #include "Progressive2D.frag" #info Unveiling Fractal Structure with Lagrangian Descriptors #info https://fractalforums.org/fractal-mathematics-and-new-theories/28/unveiling-the-fractal-structure-of-julia-sets-with-lagrangian-descriptors/3376/msg20446#msg20446 #group Lagrangian // Number of iterations uniform int Iterations; slider[1,200,1000] uniform vec3 RGB; slider[(0,0,0),(0.0,0.4,0.7),(1,1,1)] uniform bool Julia; checkbox[false] uniform vec2 JuliaXY; slider[(-2,-2),(-0.6,1.3),(2,2)] uniform float p; slider[0,.6,1] /* partial pnorm input: z, c, p output ppn */ float ppnorm( vec2 z, vec2 c, float p){ vec3 s0,s1; // for 2 points on the Riemann sphere float d; // denominator float ds; // map from complex plane to riemann sphere // z d = z.x*z.x + z.y*z.y + 1.0; s0 = vec3(2.0*z,(d-2.0))/d; // zn d = c.x*c.x + c.y*c.y + 1.0; s1 = vec3(2.0*c,(d-2.0))/d; // sum vec3 ss = pow(abs(s1 - s0),vec3(p)); ds = ss.x+ss.y+ss.z; return ds; } // DLD = Discret Lagrangian Descriptior float lagrangian( vec2 z, vec2 c, float p ){ int i; // number of iteration float d = 0.0; // DLD = sum for (i=0; i<Iterations; ++i){ d += ppnorm(z, c, p); // sum z z = cMul(z,z) +c; // complex iteration if (cAbs(z) > 1e19 ) break; // exterior : upper limit of float type } d /= float(i); // averaging not summation return d; } vec3 color(vec2 c) { vec2 z = Julia ? c : vec2(0.,0.); if(Julia) c = JuliaXY; float co = lagrangian( z, c, p ); return .5+.5*cos(6.2831*co+RGB); } #preset Default Center = -0.724636541,0.025224931 Zoom = 0.64613535 EnableTransform = false RotateAngle = 0 StretchAngle = 0 StretchAmount = 0 Gamma = 2.2 ToneMapping = 1 Exposure = 1 Brightness = 1 Contrast = 1 Saturation = 1 AARange = 2 AAExp = 1 GaussianAA = true Iterations = 20 RGB = 0,0.4,0.7 p = 0.1444322 Julia = false JuliaXY = -1.05204872,0 Bailout = 1000 #endpreset #preset Basilica Center = -0.025346913,-0.013859176 Zoom = 0.561856826 EnableTransform = false RotateAngle = 0 StretchAngle = 0 StretchAmount = 0 Gamma = 2.2 ToneMapping = 1 Exposure = 1 Brightness = 1 Contrast = 1 Saturation = 1 AARange = 2 AAExp = 1 GaussianAA = true Iterations = 20 RGB = 0,0.4,0.7 p = 0.1444322 Julia = true JuliaXY = -1.05204872,0 Bailout = 1000 #endpreset c /* partial pnorm input: z , zn = f(z), p output ppn */ double ppnorm( complex double z, complex double zn, double p){ double s[2][3]; // array for 2 points on the Riemann sphere int j; double d; // denominator double x; double y; double ds; double ppn = 0.0; // map from complex plane to riemann sphere // z x = creal(z); y = cimag(z); d = x*x + y*y + 1.0; s[0][0] = (2.0*x)/d; s[0][1] = (2.0*y)/d; s[0][2] = (d-2.0)/d; // (x^2 + y^2 - 1)/d // zn x = creal(zn); y = cimag(zn); d = x*x + y*y + 1.0; s[1][0] = (2.0*x)/d; s[1][1] = (2.0*y)/d; s[1][2] = (d-2.0)/d; // (x^2 + y^2 - 1)/d // sum for (j=0; j <3; ++j){ ds = fabs(s[1][j] - s[0][j]); ppn += pow(ds,p); // |ds|^p } return ppn; } // DLD = Discret Lagrangian Descriptior double lagrangian( complex double z0, complex double c, int iMax, double p ){ int i; // number of iteration double d = 0.0; // DLD = sum double ppn; // partial pnorm complex double z = z0; complex double zn; // next z if (cabs(z) < AR || cabs(z +1)< AR) return 5.0; // for z= 0.0 d = inf for (i=0; i<iMax; ++i){ zn = z*z +c; // complex iteration ppn = ppnorm(z, zn, p); d += ppn; // sum // z = zn; if (cabs(z) > ER ) break; // exterior : big values produces NAN error in ppnorm computing if (cabs(z) < AR || cabs(z +1)< AR) { // interior d = -d; break; } } d = d/((double)i); // averaging not summation if (d<0.0) {// interior d = 2.5 - d; } return d; } unsigned char ComputeColorOfDLD(complex double z){ int iColor; double d; d = lagrangian(z,c, N,p); iColor = (int)(d*255) % 255; // color is proportional to d return (unsigned char) iColor; } Papers Unveiling the Fractal Structure of Julia Sets with Lagrangian Descriptors by Víctor J. García-Garrido in Communications in Nonlinear Science and Numerical Simulation, Volume 91, December 2020, 105417, a single anonymized review but paywalled through arxiv.org, free but not peer-reviewed preprint www fractalforums.org : unveiling-the-fractal-structure-of-julia-sets-with-lagrangian-descriptors/3376/msg19803 references V. J. García-Garrido. Unveiling the fractal structure of Julia sets with Lagrangian descriptors. Communications in Nonlinear Science and Numerical Simulation 91 (2020) 105417.