Computational Design Blog


My name is Matouš Stieber and I am fractal enthusiast. The blog was created with the initial intention of an architectural portfolio. The blog outlines my interest in computational design, 3d modelling, scripting and photorealistic visualizations. The blog includes recent research of evolutionary algorithms, optimization, fractals, attractors, etc. The research is inspired either by biological processes or mathematics which also represents my interest for architecture.

Since 2013 I’ve been engaging in computational design including evolutionary algorithms, optimization, recursion, fractals, agents behaviour, tensegrity, etc. The blog particularly outlines my skills with Rhino, Grasshopper and Python. Although I have developed an advanced knowledge about computational design while studying architecture, I’m constantly improving my skills and broadening my knowledge about scripting, mathematics. Apart from computational design, photorealistic visualization and drawing, I really enioy training of judo.

Currently I´m looking for a job or an internship which would improve my skills and broaden my knowledge about computational design or photorealistic visualizations.

Please do not hesitate to contact me for further cooperation. 

 

 

 

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Featured post

#chimpanzee3d


Chimpanzee 0.1 release

Chimpanzee plug-in is suitable for fractal enthuasists.

Grasshopper plug-in for Rhino 6 written in ghPython includes currently 33 components which focus on fractals, maps, strange attractors, iterated function systems, …

ChimpanzeeBanner2

#chimpanzee3d

I would appreciate any feedback or reports of bugs.

food4Rhino

McNeel Forum

 

Chimpanzee 0.2 coming soon

ScreenShot_20190715085830

Further development may include hyperchaotic systems, Mandelbulb, Quaternion Julia set, etc.

 

Chimpanzee changelog

Nov 11, 2018 – Chimpanzee 0.1 initial release

Chimpanzee component list

  • Mandelbrot set
  • Julia set
  • Burning ship

Iterated function systems

  • Barnsley fern
  • Sierpinski carpet
  • Sierpinski triangle
  • Maple leaf
  • Dragon
  • Tree

Maps

  • Duffing map
  • Gumowski-Mira map
  • Multifold Henon map
  • MacMillan map
  • Hopalong map
  • Tinkerbell map

Strange attractors

  • Bouali system
  • Chen-Lee system
  • Coupled Lorenz system
  • Dadras-Momemi system
  • Hadley system
  • Lorenz system
  • Newton-Leipnik system
  • Qi-Chen system
  • Rabinovich-Fabrikant system
  • Rössler system
  • Sprott A system
  • Nosé-Hoover system
  • Thomas´s cyclically symmetric system
  • Three-scroll unified chaotic system

Featured post

Galery


#3dsmax #photoshop #rhino3d #grasshopper3d

Featured post

Workshop Printed Phenomena & Folded Spaces


#rhino3d #grasshopper3d #arduino #tudresden

The workshop “Printed Phenomena and Folded Spaces” included cooperation of Junior Professorship for Knowledge Architecture; Junior Professorship for Industrial Design Engineering; Professorship for Media Design; Professorship for Communication Acoustics; supported by Saxon State and University Library Dresden.

Wissens.Werkstatt „Printed Phenomena“ und „Folded Spaces“

The workshop “Printed Phenomena and Folded Spaces” responds to the recent interest in “makerspace” and the asset for the libraries. The initial thought of the workshop has been inspired by the discussion about the development of the learning process; knowledge sharing and creativity support. The workshop offered the participants knowledge about assorted tools including 3D scanning; printing; laser cutting; rapid prototyping; Grasshopper; Arduino; etc.

Responsive Acoustic Surface

The main parameter influencing acoustic performance is reverberation time; yet this phenomena of acoustic experience may not necessarily be perceived in different types of reconfigurable spaces, which may emerge along the rise of the cyber-physical systems in architecture.

Author Adam Urban

The thought of using a rigid origami tessellation as the basic shape reinforces the main functional necessity of the structure; combining reflective and absorptive surfaces on interior surfaces in order to transform the acoustic performance. The triangular and square inter-combined surfaces can dynamically alter their form to expose or hide the varying conditions to modify the acoustic performance. The system is controlled by a central electrical panel using an Arduino micro controller. The surfaces should be able to learn, possibly using computer simulations, to predicatively arrange the model into the optimal variation, set by parameters.

The rigid origami was chosen due to its ability to change between the main stages; a flat, reflective and the folded, absorptive tessellation. The functional diversity is being supported by the constructive characteristics of the surfaces; the square reflective surfaces and the triangular containing of porous diffusive materials. Regarding the shape of the tessellation itself which develops through the mechanical movement, cavities between the surfaces emerge, which might be used as Helmholtz Resonators to reduce undesirable low frequency sounds.

References:

  1. Wissens.Werkstatt „Printed Phenomena“ und „Folded Spaces“; Sommerschule des Dresden Design Hub der TU Dresden experimentierte im Makerspace der SLUB; Jens Krzywinski; BIS-Das Magazin der Bibliotheken in Sachsen; 2014
Featured post

3rd LIXIL International University Architectural Competition


#rhino3d #grasshopper3d

3rd LIXIL International University Architectural Competition, organized by the LIXIL JS Foundation, invited university research laboratories for proposals for sustainable architecture under the theme of “Retreat in Nature” in Taiki-cho, Japan.

http://www.archdaily.com/337021/3rd-lixil-international-university-architectural-competition

Retreat in Nature

In a world where there is no coherent growth, shared values become fragmented. More effort is required to forge a sense of happiness. Under such conditions, what kind of values could architecture offer? And how could one imagine a new affluence to contemporary life? The site in Taiki-cho, Hokkaido, is blessed by nature with a climate that varies considerably through the cold winters and hot summers. Here, qualities of life not found in the city could be apparent: like discovering plants budding through the snow or observing the seasons of the pastures. The landscape could even be fully constructed like the English garden. Also, the harsh climate does not necessarily mean it is desirable to resist it – its pleasure could be derived in the spring or the autumn. But thoughtful ideas should not deny the role of available technologies. It is important that the new values of living are clearly defined along with the proposals.

Tag_cloud

Tagcloud

Unfolding Community

“We believe that architecture should offer an interstitial environment on the base of interaction between each other and  nature in order to inspire creative activities. The method ist  to create eco-social micro-cells that are able to react on the use and disuse by its inhabitants.” 

Retreat in nature; Endabgabe-1

“In the springtime, the caterpillars in the cities search for places offering different opportunities for a valuable living. Being attracted by an image of a new kind of well-being, they are led out of the nature. The way leads them out of the cities into the nature.
At the ever flourishing tree of Taiki-Cho numerous buds sprout out. The newcomers choose the preferred ones and unfold them. Themselves metamorphosing to graceful and proud butterflies, they establish a symbiosis for the benefit of both.
When the butterflies return to their original environments, they use their newly acquired experiences of living. This experience then inspires even more caterpillars to move to Taiki-Cho.
With the increasing interest the community compresses and intensifies. Close and clear bonds form interstices of different qualities. Those rooms are well suited for social life, encouraging creativity and therefrom resulting fruits.
Just like the social relationships, the interstices are always on the move, and they may spread across the meadows creating various atmospheres of the entire site.
The lifecycle of the butterflies reacts to the seasonal transformations of their environment. When the fall is getting gradually colder, they create closed clusters.
Before retreating to the cities, the community prepares the meadow also for the next generation, leaving the shelters covered by snow.” 

「自然はかつて人間にとっておおもととなる環境であった.し かし都市化は自然に対して敬意を払わずに進められた.今回 のコンペでは,自然がかつて持っていた価値を取り戻すだけ でなく,次世代にも通じる自然との協調がもたらす価値を提 示したい. 」

“Nature was once an environment that was the essencials for humans, however urbanization was advanced without respect for nature. In this competition, I would like to restore/redefine the value that nature once had. I would like to illustrate the value brought about by the cooperation with nature that leads to the future generations.”

Matouš Stieber; http://www.facebook.com/LIXIL.IUAC

Members of TU Dresden research laboratory
Benjamin Herrnsdorf, Ramzi Krüger, Robert Megel, Matouš Stieber, Adam Urban

Participating universities
Aalto University / Delft University of Technology / Dresden University of Technology / Hanoi Architectural University / Harvard University / Hokkaido University / Kyoto University / National University of Singapore / Swiss Federal Institute of Technology / The Bartlett School of Architecture, University College London / Tongji University / Vienna University of Technology

Discussion for 3rd LIXIL International University Architectural Competition

JAU_Lixil_conv3

Jury
Kengo Kuma (Professor at the Graduate School of Architecture, The University of Tokyo)
Tomonari Yashiro (Professor at the Institute of Industrial Science, The University of Tokyo)
Darko Radovic (Professor of Architecture and Urban Design, Keio University)

 

References

  1. https://www.facebook.com/LIXIL.IUAC
  2. http://www.archdaily.com/337021/3rd-lixil-international-university-architectural-competition

Solar Access Analysis + Evolutionary Algorithms


#rhino3d #grasshopper3d #ecotect #maya #aftereffects 

The intention of the research engage in the application of evolutionary algorithms in architecture; in order to achieve optimized geometry in terms of sustainable architecture resulted from solar radiation data evaluation.

01

Evolutionary algorithms enable an evaluation of the solar radiation concept depending on the geometry by the application of the solar access analysis. The solar access analysis includes calculation of the distribution of incident solar radiation; which may be an option when heating is required; however may also be a disadvantage when heating is not intended; thus reducing the cooling; heating energy consumption. Thus in order to achieve optimized geometry in terms of sustainable architecture different results for heating and cooling seasons are required.

The research approach requires sustainable analysis software for the solar access analysis. The script was written in Grasshopper; algorithmic editor integrated in Rhinoceros; which exports the geometry including analysis setting to Autodesk Ecotect Analysis. The connection between Grasshopper interface and Autodesk Ecotect Analysis is enabled by the Geco; environmental analysis plug-in integrated in Grasshopper. Consequently the solar access analysis data; which has been analyzed by Autodesk Ecotect Analysis are imported in Grasshopper. Galapagos an evolutionary solver enables the generative approach. The obtained results of the solar access analysis are evaluated by Galapagos in order to search the solution from a set of available options.

The optimized geometry of the skyscaper located in Hong Kong for the cooling season and the heating season.

In order to inquire into the evolutionary algorithms approach; the research approach was applied on the instance of skyscraper with various locations. In order to evaluate the resulting geometry with the impact of the weather data; these results of the skyscaper located in Hong Kong were analyzed against the equal results with weather data of Dresden.

Optimization process of the geometry by Galapagos plug-in based on the solar access analysis data imported from Autodesk Ecotect Analysis

The optimized geometry of the skyscaper located in Dresden for the cooling season and the heating season.

Further research may refer to subsequent geometry optimization; search of sustainable solution involving the heating and the cooling season. However Galapagos evolutionary solver enables search for particular objective. The search involving the heating and the cooling season may require other evolutionary solver; which enables evaluation of incompatible objectives.

References

  1. M. Mitchell; An Introduction to Genetic Algorithms; The MIT Press; MA; US; 1990
  2. Carlos Coello Coello, Gary B. Lamont, David A. van Veldhuizen; Evolutionary Algorithms for Solving Multi-Objective Problems; Springer Science & Business Media; 2007

Diffusion-Limited Aggregation


#rhino3d #grasshopper3d #python #3dsmax #photoshop #aftereffects 

The initial intention of the research engaged in the approximating search of the string of the L-System by applying of evolutionary algorithm in order to achieve defined conditions. However the stochastic recursion of the rewriting system; thus the L-System syntax; does not enable applying of the search algorithm.

Nevertheless unlike the Lindenmayer-System the Diffusion-limited aggregation affords broader responsiveness to the requirements then the L-System syntax; thus enables applying of the evolutionary algorithm which have been used to solve optimization problems. However evolutionary algorithms do not guarantee a result; unless the defined requirements are restricted enough; the search algorithm does not achieve the result.

The initial intention of the research to apply constraints for Diffusion-limited aggregation growth process was inspired by work of Andy Lomas created for SIGGRAPH 2005.

Andy_Lomas_aggregates

“Aggregation: Complexity out of Simplicity”  by Andy Lomas for SIGGRAPH 2005

The research concerns with an application of the evolutionary algorithms and an Diffusion-limited aggregation growth process; unlike Genr8 which was written in C++ by using Autodesk Maya; the DLA script was written in Python by using GhPython; plug-in for Grasshopper algorithmic editor.

dfghjk

Diffusion-limited aggregation has been defined by physicists Thomas A. Witten and Leonard M. Sander in 1981; who published by them in 1981, titled: “Diffusion limited aggregation, a kinetic critical phenomena” in Physical Review Letters. DLA refers to iterative stochastic process of clustering (Aggregation) resulting from the Brownian motion (Diffusion); the random particle results from an evaluation of random point on sphere surface defined by the initial particle. The radius of the sphere defines the threshold distance required for the attachment of the random particle to the particle cluster (initially the seed particle).

The DLA script generates seed particle which determines the initial growth process; another random particle defined by the Brownian motion (Diffusion). The resulted growth process have a randomly branching clusters because the aggregation of the biological DLA results from randomly diffused particles. The DLA ghPython script unlike the intrinsic biological process does not consider the relevance of the Brownian motion. In order to decrease the intricacy of the ghPython script; the initial position of the particle does not result from Brownian motion.

02

In order to inquire into the Diffusion-limited aggregation process with additional constraints; the research approach was applied on the instance of chair, which was represented by the set of points. The additional constraints affect the otherwise randomly growing DLA process. The Diffusion-limited aggregation process with additional constraints does not grow quite randomly.

The resulted growth process would have a randomly branching clusters because the aggregation of the biological DLA results from randomly diffused particles; however in order to receive branching clusters with defined requirements; the DLA ghPython script requires additional constraints to decrease the randomness of the process. The attractor constraint defines grow angle; thus the sphere subsurface for the evaluation of random point. The resulted ghPython script enables DLA process with additional constraints.

Diffusion-limited aggregation with additional constraints; without any additional processing of the geometry by marching cubes algorithm

Diffusion-limited aggregation with additional constraints; additional processing of the geometry by marching cubes algorithm

Futher research may refer to self-organizing maps by using Crow plug-in for Grasshopper.

References

  1. Przemyslaw Prusinkiewicz, Aristid Lindenmayer; The Algorithmic Beauty of Plants; Springer-Verlag; 1990
  2. Przemyslaw Prusinkiewicz; Lindenmayer systems, fractals and plants; Springer-Verlag; New York; 1989
  3. Thomas A. Witten and Leonard M. Sander; Diffusion-Limited Aggregation: A Kinetic Critical Phenomenon; Physical Review Letters 47; 1981
  4. Leonard M. Sander, Z. M. Cheng, R. Richter; Diffusion-Limited Aggregation in three Dimensions; Physical Review B 28; 1983
  5. Tomoya  Sato, Masafumi  Hagiwara; IDSET: Interactive Design System using Evolutionary Techniques; Computer-Aided Design 33; 2001
  6. http://andylomas.com

Minimal Surfaces


#rhino3d #grasshopper3d #python

The research of minimal surfaces in Rrefers to Joseph-Louis Lagrange in 1762. Minimal surfaces refer to the Plateau’s problem. Minimal surfaces may be defined by Enneper-Weierstrass parameterization.

Helicoid

Helicoid was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Helicoid refers to the definition of the right conoid along with Plücker’s conoid, or Whitney umbrella. Eugène Charles Catalan proved in 1842 that the helicoid and the plane were the only ruled minimal surfaces. The helicoid may be defined by the equations

x = ρ cos(αθ)
y = ρ sin(αθ)
z = θ

where ρ and θ range from negative infinity to positive infinity, while α is a constant.

Catenoid

Catenoid was described in 1744 by the mathematician Leonhard Euler. The catenoid may be defined by the equations

x = c cosh(v/c) cos(u)
y = c cosh(v/c) sin(u)
z = v

where  u ∈ [-π,π] and v ∈ R and c is a non-zero real constant.

The Heliod-Catenoid transformation equations are

x (u,v) = cos(α) sinh(v) sin(u) + sin(a) cosh(v) cos(u)
y (u,v) = -cos(α) sinh(v) cos(u) + sin(a) cosh(v) sin(u)
z (u,v) = u cos(α) + v sin(α)

where α = 0 refers to a helicoid and α = 0.5π to a catenoid.

Further research may refer to Hoffman’s minimal surface which has been discovered in 1992

The Klein bottle

The Klein bottle, described in 1882 by the German mathematician Felix Klein, is a non-orientable closed surface; two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined.

The Klein bottle equations are for u < π and v < π

x (u,v) = 6 cos(u) (1+sin (u)) + r cos(v+π)
y (u,v) = 16 sin(u) + r sin(u) cos(v)
z (u,v) = r sin(v)

and for u > π and v > π

x (u,v) = 6 cos(u) (1+sin (u)) + r cos(v+π)
y (u,v) = 16 sin(u)
z (u,v) = r sin(v)

The Figure 8-Klein bottle

The Figure 8-Klein bottle refers to the Möbius strip which was discovered in 1858 by the German astronomer and mathematician August Ferdinand Möbius.

The Figure 8-Klein bottle equations are for 0 <= u < 2π, 0 <= v 2

x (u,v) = (r + cos(0.5u) sin(v) – sin(0.5u) sin(2v)) cos(u)
y (u,v) = (r + cos(0.5u) sin(v) – sin(0.5u) sin(2v)) sin(u)
z (u,v) = sin(0.5u) sin(v) + cos (0.5u) sin(2v)

Scherk´s second surface

Scherk’s minimal surfaces, discovered by Heinrich Scherk in 1834, were initially an attempt to solve Gergonne’s problem.

The Scherk´s sercond surface equations for ɸ ∈ [0,2π) and r ∈ (0,1) are

x = ln((1 + r2 + 2r cos(ɸ))/(1 + r2 – 2r cos(ɸ))
y = ln((1 + r2 – 2r sin(ɸ))/(1 + r2 + 2r sin(ɸ))
z = 2 tan-1 ((2r2 sin(2 ɸ)/(r4 – 1))

Plannar Ennerper´s surface

Enneper´s surface

Enneper´s surface was introduced by Alfred Enneper in 1864. Enneper surface is a self-intersecting minimal surface.

The Enneper´s surface equations are

x (u,v) = u – u3/3 + uv2
y (u,v) = v – v3/3 + vu2
z (u,v) = u2 – v2

Boy´s surface

Boy´s surface was found by Werner Boy in 1901 on assignment from David Hilbert to prove that the projective plane could not be immersed in R3. The Boy surface is a nonorientable surface. 

x = 2/3*(cos(u)*cos(2*v) + √2*sin(u)*cos(v))*cos(u) / (√2 – sin(2*u)*sin(3*v)) 
y = 2/3*(cos(u) sin(2*v) – √2*sin(u) sin(v)) cos(u) / (√2 – sin(2*u)*sin(3*v)) 
z = √2*cos(u)*cos(u) / (√2 – sin(2*u) sin(3*v)) 

where u,v ∈ [0,2π)

fsdfgsdasdfsdfsd

 

References

  1. Ulrich DierkesAlbrecht KusterStefan HildebrandtOrtwin Wohlrab; Minimal Surfaces, Vol. 1: Boundary Value Problems; New York: Springer-Verlag; 1992
  2. Ulrich DierkesAlbrecht KusterStefan HildebrandtOrtwin Wohlrab;  Minimal Surfaces, Vol. 2: Boundary Regularity; New York: Springer-Verlag; 1992
  3. Scherk, H. F., Bemerkung über der kleinste Fläche innerhalb gegebener Grenzen, J. reine angew. Math. 13, 1834
  4. Elsa Abbena, Simon Salamon, Alfred Gray; Modern Differential Geometry of Curves and Surfaces with Mathematica; 3rd edition, CRC Press, 2006
  5. Construction of minimal surfaces; Karcher H.; Surveys in Geometry; University of Tokyo; 1989
  6. http://mathworld.wolfram.com
  7. https://www.mathcurve.com
  8. https://www.researchgate.net
  9. https://www.mathematica-journal.com

 

Mandelbrot Set


 

#rhino3d #grasshopper3d #python #chimpanzee3d

The intention of the research was to create the Mandelbrot Set by using custom ghPython component in Grasshopper for Rhino.

The Mandelbrot set, defined by Benoit B. Mandelbrot, is the set of complex numbers c obtained from the quadratic recurrence equation zn+1=zn2+c, which does not diverge when iterated. The boundary of the Mandelbrot set refers to fractal with Hausdorff dimension of 2.

casdf.png

Mandelbrot Set zn+1=z2+c

Mandelbrot Set zn+1=z3+c

Mandelbrot Set zn+1=z4+c

fsadfasdf

dasdsd.png

Mandelbrot set modifications

Mandelbrot Set zn+1=-icz5 + 1 #itssofluffy

Mandelbrot

Further research may refer to Mandelbulb, which would require hypercomplex numbers.

 

References

  1. Benoît Mandelbrot; Fractal aspects of the iteration of z→λ z(1-z) for complex λ, z; Annals of the New York Academy of Sciences 357; 2006
  2. Shishikura, Mitsuhiro; “The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets”;  Annals of Mathematics; Second Series 147 (2); 225–267; 1998
  3. Cheng Jin Tan Jian-Rong; Representation of 3-D General Mandelbrot Sets Based on Ternary Number and Its Rendering Algorithm; Chinese Journal of Computers; Vol. 27 No. 6; 2004
  4. http://www.math.harvard.edu
  5. http://www.paulbourke.net
  6. http://www.mathworld.wolfram.com
  7. http://www.bugman123.com
  8. http://www.skytopia.com
  9. http://www.juliasets.dk
  10. http://www.researchgate.net
  11. https://www.sciencedirect.com
  12. http://www.fractalforums.com
  13. http://softology.com.au
  14. https://softologyblog.wordpress.com

 

Julia Set


#rhino3d #grasshopper3d #python #chimpanzee3d

The intention of the research was to create the Julia Set by using custom ghPython component in Grasshopper for Rhino.

Julia sets were defined by french mathematician Gaston Julia who investigated the iterative properties (while the Julia set is associated with a quadratic equation of various degrees fc(z)= zn + c, Julia was interested in the iterative properties of a more general expression fc(z)= z4 + z3/(z-1) + z2/(z3 + 4 z2 + 5) + c.

Julia sets are quadratic equation of various degrees fc(z)=zn + c; where c is a complex parameter. For almost every c, this transformation generates a fractal. Depending on the value of c; the resultant Julia set may be either connected or disconnected; the disconnected Julia sets consist of individual points regardless of the detailed processing. Values of c chosen from within the Mandelbrot set are connected while those from the outside of the Mandelbrot set are disconnected.

Julia set for c=-0.74543+0.11301i

gzbhnj

Julia set for c=-0.758900-0.075300i

uiopú.jpg

Julia set for c=-0.512511498387847167+0.521295573094847167i

ztuiopúl

Julia set for c=-0.08+0.8i

Julia set for c=0.28+0.008i

retfgzbhunj.jpg

Julia set of sin(z) for c=1+0.2i

Sin(z) can be defined by sin(x + iy) = sin(x)cosh(y) + icos(x)sinh(y), the values grow exponentially to ∞ along the x-axis, but along the y-axis, the values remain bounded.

Glynn fractals

The Julia sets for 1 < < 2 and values of c located close to the boundary of the Mandelbrot sets have a characteristic form and are known as Glynn fractals.

dfsdds

Glynn fractal for fc(z) = z1.5 – 0.2

dfdgdfdghd

References

  1. Benoît Mandelbrot; Fractal aspects of the iteration of z→λ z(1-z) for complex λ, z; Annals of the New York Academy of Sciences 357, 2006
  2. Shishikura, Mitsuhiro; “The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets”; Annals of Mathematics; Second Series 147 (2); 1998
  3. Cheng, J. & Tan, Jr. J.; Generalization of 3D Mandelbrot and Julia sets; Zhejiang University; 2007
  4. Norton Alan; Julia Sets in the quaternations; Computer Graphics 13 (2); 1989
  5. http://www.mathworld.wolfram.com
  6. http://www.paulbourke.net/
  7. http://www.bugman123.com
  8. http://www.skytopia.com/
  9. http://www.juliasets.dk/
  10. http://www.evl.uic.edu
  11. http://www.researchgate.net
  12. http://www.fractalforums.com
  13. http://softology.com.au
  14. https://softologyblog.wordpress.com

Burning Ship


#rhino3d #grasshopper3d #python #chimpanzee3d

The intention of the research was to create the Burning ship fractal by using custom ghPython component in Grasshopper for Rhino.

The Burning Ship fractal, described by Michael Michelitsch and Otto E. Rössler in 1992, is defined by iterating the function zn+1 = |zn2|+ c. The difference between the formula of the Bruning Ship fractal and the Mandelbrot set is that the real and imaginary components are set to the absolute values before squaring.

safkjl.jpg

The Burning Ship fractal antenna

burn3

The Burning Ship fractal zn+1 = |zn3|+ c

burning4

The Burning Ship fractal zn+1 = |zn4|+ c

Julia Sets from the Burning Ship

The Burning Ship and the Mandelbrot do not share Julias for any other poins where y!=0 because the orbits of all points will eventually take a point to somewhere in the second or fourth quadrants (-x,y) or (x,-y). The y value of the next iteration from within one of these quadrants will be i*2*|-x|*|y| or i*2*|x|*|-y|. Neither of these are the same as the y-value for a point iterated through the Mandelbrot, i*2*x*y, because of the absolute value operation .

dfgdfgdfg

Julia set for c=-1.764-0.06i from the Burning Ship fractal

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Julia set for c=0.87+1.52i from the Burning Ship fractal

tztzgzuh

Julia set for c=0.675-1.15i from the Burning Ship fractal

References

  1. Michael Michelitsch and Otto E. Rössler; The “Burning Ship” and Its Quasi-Julia Sets; Computers & Graphics Vol. 16; No. 4; 1992
  2. http://www.mathworld.wolfram.com
  3. http://www.paulbourke.net
  4. https://www.sciencedirect.com
  5. https://softologyblog.wordpress.com
  6. http://www.fractalforums.com

Iterated Function System


 

#rhino3d #grasshopper3d #python #chimpanzee3d

The intention of the research was to create the Iterated function system (IFS) by using custom ghPython component in Grasshopper for Rhino.

An iterated function system (IFS) introduced in 1981 is a set of affine transformations. The application of iterated function systems refers to fractals.

IFS Maple Leaf

uiiéo

Barnsley Fern

The Barnsley Fern, described by the British mathematician Michael Barnsley, refers to an iterated function system (IFS). Barnsley fern uses recursive affine transformations represented by matrix

xn + 1 = 0
yn + 1 = 0.16 yn
xn + 1 = 0.85 xn + 0.04 yn
yn + 1 = −0.04 xn + 0.85 yn + 1.6
xn + 1 = 0.2 xn − 0.26 yn
yn + 1 = 0.23 xn + 0.22 yn + 1.6
xn + 1 = −0.15 xn + 0.28 yn
yn + 1 = 0.26 xn + 0.24 yn + 0.44 

IFS Dragon

IFS Dragon uses recursive affine transformations represented by matrix

xn + 1 = 0.824074 x+ 0.281428 y– 1.882290

yn + 1 = -0.212346 x+ 0.864198 y– 0.110607

xn + 1 = 0.088272 x+ 0.520988 y+ 0.785360

yn + 1 = – 0.463889 x-0.377778 yn + 8.095795

 

dragon

IFS Tree

xn + 1 =0.05*cos(0)* xn -0.6 sin(0)*yn
y+ 1 =0.05*sin(0)* xn +0.6cos(0) *y2

xn + 1 =0.05*cos(0)* xn +0.5sin(0)* yn
y+ 1 =0.05*sin(0)* xn -0.5cos(0)*yn +1

xn + 1 =0.6*cos(0.698)* xn -0.5sin(0.698)* yn
y+ 1 =0.6*sin(0.698)* xn +0.5cos(0.698)* yn +0.6

xn + 1 =0.5*cos(0.3490)* xn -0.45sin(0.3492)* yn
y+ 1 =0.5*sin(0.3490)* xn +0.45cos(0.3492)* yn +1.1

xn + 1 =0.5*cos(-0.5240)* xn -0.55sin(-0.5240) * yn
y+ 1 =0.5*sin(-0.5240)* xn +0.55cos(-0.5240) * yn +1

xn + 1 =0.55*cos(-0.6980)* xn -0.4sin(-0.6980)*yn
y+ 1 =0.55*sin(-0.6980)* xn +0.4cos(-0.6980) * yn +0.7

sdf

dsfdsfsdf.png

Sierpinski Carpet

Sierpinski carpet was described by Wacław Sierpiński in 1916. Sierpinski carpet refers to concept common with Cantor set and Menger spoonge, described by Karl Menger in 1926.

sdfsdfsd

Sierpinski Triangle

The Sierpiński sieve is a fractal described by Wacław Sierpiński in 1915 and appearing in Italian art from the 13th century; however these patterns appear already in the 13th-century n the cathedral of Anagni, Italy.

fdgdfgdf

 

References

  1. Barnsley, Michael Fielding; Fractals everywhere; Academic Press, Boston, 1988
  2. Barnsley, F. and Sloan A. D.; A Better Way to Compress Images; Byte Magazine; January; 1988
  3. Oppenheimer P. E.; Real Time Design and Animation of Fractal Plants and Trees; Computer Graphics; 1986
  4. Demko, S., Hodges, L., and Naylor B.; Construction of Fractal Objects with Iterated Function Systems; Computer Graphics; 1985
  5. Reghbati, H. K.; An Overview of Data Compression Techniques; Computer; 1981
  6. http://www.cs.lmu.edu
  7. http://www.paulbourke.net
  8. http://www.bugman123.com
  9. https://www.sciencedirect.com
  10. http://www.mathworld.wolfram.com
  11. http://softology.com.au

 

Maps


#rhino3d #grasshopper3d #python #chimpanzee3d

A dynamical system may be mathematically expressed either by continuous set of equations, or by discrete system, called map.

Duffing Oscillator

The Duffing oscillator named after German engineer Georg Wilhelm Christian Caspar Duffing.  The Duffing map refers to a non-linear second-order differential equation (The Duffing oscillator) that exhibits chaotic behavior.  In 1961, Youshisuke Ueda discovered that this system indicates chaotic behavior.

fgfdg.png

The Duffing map is a discrete-time dynamical system which constrains are usually set to a = 2.75 and b = 0.2 to produce chaotic behaviour by iterating the following equations

xn+1  = yn
yn+1  = -bxn + ayn -yn3

The Duffing equation is defined by

d2x /dt2 = – δ dx/dt + x – x3 + γ cos(ω t)

A Poincaré section of the forced Duffing oscillator

Poincaré map constitutes a procedure employed to eliminate a dimension of the system and, therefore, a continuous system is transformed into a discrete one (Thompsom & Stewart, 1986).

References

  1. The Duffing Oscillator. In: Chaos. Springer; Berlin, Heidelberg; 2008
  2. Y. Ueda; Randomly transitional phenomena in the system governed by Duffing’s equation; J. Stat. Phys. 20; 1979
  3. Y. Ueda; Steady Motions Exhibited by Duffing’s Equation: A Picture Book of Regular and Chaotic Motions; in Hao Bai-Lin, D. H. Feng, and J.-M. Yuan, New Approaches to Nonlinear Problems; SIAM; Philadelphia; 1980
  4. G. Duffing; Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz.; F. Vieweg u. Sohn; Braunschweig; 1918
  5. Morris W. Hirsch, Stephen Smale, Robert L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2013

Mira Map

Mira maps result  from equations

xn+1 = ayn + f(xn)
yn+1 = −xn + f(xn+1)

f(x) = bx + 2x2(1 – b)/(1 + x2)

fdsfsd

The Mira map for a=0.2, b=1 and x=12, y=0

gfd

The Mira map for a=0.3, b=1 and x=5, y=0

Gumowski-Mira Map

The Gumowski-Mira equation was developed in 1980 at CERN by I. Gumowski and C. Mira to calculate the trajectories of sub-atomic particles. Gumowski-Mira maps are chaotic strange attractors that result  from equations

xn+1 = yn + a (1 – 0.005 yn2) + f(xn)
yn+1 = −xn + f(xn+1)

f(x) = bx + 2x2(1 – b)/(1 + x2)

dsfsddf

The Gumowski-Mira map for a=0, b=-0.31 and x=0 y=0.5

sdasd

The Gumowski-Mira map for a=0, b=-0.22 and x=0.5 y=0.5

dfsdf

The Gumowski-Mira map for a=0, b=-0.53 and x=0 y=0.5

qweqw.png

Mira Map modification

fsdfe.png

The modified Mira map for a=0.04, b=1, x=-1, y=8 and f(x) = ax – 3a/(a + ebx)

fdsdf

The modified Mira map for a=-0.48, b=.9924, x=1, y=1 and f(x) = ax – 2(1-a)x2/(1 + x2)

References

  1. Gumowski I. Mira C.; Recurrences and Discrete Dynamic Systems; Springer; 1980
  2. Lauwerier, H.; Fractals: Endlessly Repeated Geometric Figures; Princeton, NJ: Princeton University Press; 1991
  3. Albert D. Morozov; Invariant Sets for Windows: Resonance Structures, Attractors, Fractals and Patterns; World Scientific; 1999
  4. Morris W. Hirsch, Stephen Smale, Robert L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2013
  5. http://mathworld.wolfram.com
  6. http://www.3d-meier.de
  7. https://softologyblog.wordpress.com

Multifold Henon Map

The Multifold Henon map refers to chaotic behaviour by iterating the following equations

xn+1 = 1 – a sin(xn) +byn
yn+1 = xn

fghfg

sdffsd.png

References

  1. Henon, M., A two-dimensional mapping with a strange attractor, Communications in Mathematical Physics 50, 1976
  2. Morris W. Hirsch, Stephen Smale, Robert L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2013

Logistic map

The logistic map introduced in a 1976 paper by the biologist Robert May in part as a discrete-time demographic model . The logistic map is a polynomial mapping with quadratic recurrence equation

xn+1 = rxn (1-xn)

where r is a control parameter and the map iterations are mapped on the limited range 0<x<1 while r ≤ 4.

adasd.png

The bifurcation diagram of the logistic map xn+1 = rxn (1-xn)

References

  1. Sprott, Julien Clinton, Chaos and Time-Series Analysis, Oxford University Press, 2003
  2. R. Egydio de Carvalho, Edson D. Leonel, Squared sine logistic map, Physica A: Statistical Mechanics and its Applications, Volume 463, 2016

Ikeda map

The Ikeda map defined by Kensuke Ikeda in 1980 is a discrete-time dynamical system defined by equations

xn+1 = a + b(xn cos(t) – yn sin(t))
yn+1 = b(xn sin(t) – yn cos(t))

t = c – d/(xn2 + yn2+ 1) 

The system is insensitive to the initial values.

dfssd.png

References

  1. Kensuke Ikeda; Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system; Optical Communications; 1979
  2. Kensuke Ikeda, H. Daido & O. Akimoto; Optical turbulence: chaotic behavior of transmitted light from a ring cavity; Physical Review Letters; 1980
  3. Morris W. Hirsch, Stephen Smale, Robert L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2013

Hyperchaotic Folded-Towel Map

The Folded-Towel map was introduced by O. E. Rössler in 1979 as an invertible map with two positive and one negative Lyapunov exponents which refers to hyperchaotic behavior.

xn+1 = ax(1 – xn) − 0.05 (y+ 0.35) (1 – 2zn)
yn+1 = 0.1 ((y+ 0.35) (1 + 2zn) – 1) (1 – 1.9xn)
zn+1 = 3.78z(1 – zn) + byn

dfsf

Projection of the hyperchaotic Folded-Towel map on the x-y plane for a=3, b=0.2

Refeences

  1. Charalampos Haris Skokos, Georg A. Gottwald, Jacques Laskar, Chaos Detection and Predictability, Springer, 2006
  2. Albert C. J. Luo, Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, Springer Science & Business Media, 2010
  3. http://www.scholarpedia.org

Cubic Henon Map

The Hopalong attractor refers to chaotic behaviour by iterating the following equations

xn+1 = axn – (xn)3 – yn
yn+1 = xn

dfs

MacMillan Map

The MacMillan map refers to the Sitnikov problem; the Sitnikov problem is named after Russian mathematician Kirill Alexandrovitch Sitnikov that attempts to describe the movement of three objects due to their mutual gravitational attraction.

A peculiar case of the Sitnikov problem was first discovered by the American scientist William Duncan MacMillan in 1911, but the problem wasn’t discovered until 1961 by Sitnikov.

xn+1 = yn
yn+1 = −xn + 2ayn/(1+yn2)+ byn

sdfsd

MacMillan map for a=1.6 und b=0.2

sdfsgf

MacMillan map for a=1.6 und b=0.4

References

  1. Ruiz Lopez G., Bountis, T. and Tsallis, C.; Time-Evolving Ststistics of Chaotic
    Orbits in Conservative Maps in the Spirit of the Central Limit Theorem; 2010
  2. MacMillan, W.; An integrable case in the restricted problem of three bodies; 1911
  3. K. A. Sitnikov; The existence of oscillatory motions in the three-body problems; In: Doklady Akademii Nauk SSSR; 1960

Hopalong Attractor

Hopalong attractor, also known as Martin Attractor, was introduced in 1986 by Barry Martin of Aston University in Birmingham. The Hopalong attractors were presented by A. K. Dewdney in the “Scientific American” magazine in 1986.

The Hopalong attractor refers to chaotic behaviour by iterating the following equations

xn+1 = yn + f(xn)
yn+1 = a – xn

f(xn) = – SGN(xn)√|bxn – c)|

ljk.png

Hopalong attractor for a=0,6 und b=0,5 und c=0

fdh

Hopalong attractor for a=0.1, b=0.5 and c=0

ctf

Hopalong attractor for a=0.5, b=-0.3 and c=0.7

dfgf.png

Hopalong attractor for a=0.1, b=0.5 and c=0.5

References

  1. http://www.3d-meier.de
  2. http://www.math.harvard.edu/archive
  3. http://www.paulbourke.net
  4. https://softologyblog.wordpress.com

Peter de Jong Attractor

Peter de Jong attractor results  from equations

xn+1 = sin(ayn) – cos(bxn)
yn+1 = sin(cxn) – cos(dyn)

fgdffd

Peter de Jong attractor for a=2.01, b=-2.53, c=1.61, d=-0.33

dfsdfsew.png

Peter de Jong attractor for a=1.4, b=-2.3, c=2.4, d=-2.1

grt

Peter de Jong attractor for a=-2.7, b=-0.09, c=-0.86, d=-2.2

afdsfd.png

Peter de Jong attractor for a=0.970, b=-1.899, c=1.381, d=-1.506

sdfsd

Peter de Jong attractor for a=-0.709, b=1.638, c=0.452, d=1.740

Svennson Attractor

Svennson Attractor is a modification of the Peter de Jong attractor.

xn+1 = d*sin(ayn) – cos(bxn)
yn+1 = c*sin(axn) + cos(byn)

sada

Svennson Attractor  for a=1,4, b=1,56, c=1,4 and d=-6,56

qwe

Svennson Attractor  for a=1.3, b=1.3, c=1.3 and d=1.3

References

  1. https://softologyblog.wordpress.com

Strange Attractors


#rhino3d #grasshopper3d #python #chimpanzee3d

The intention of the research was to create strange attractors based on mathematic equations by using custom ghPython component in Grasshopper for Rhino.

Chaotic systems are dynamical systems that are highly sensitive to initial conditions. This sensitivity is known as the butterfly effect. A dynamical system may be mathematically expressed either by continuous set of equations, or by discrete system, called map. The term strange attractor was coined by David Ruelle and Floris Takens.

Lorenz System

The Lorenz system is defined by three non-linear differential equations (Lorenz equations), which were defined by Edward N. Lorenz in 1963. The Lorenz system  is strange attractor with nonlinear and deterministic characteristics, thus the behavior is determined by the initial conditions, without any randomness involved.

The Lorenz system refers to chaotic behavior for certain values and initial conditions (Rayleigh ρ=28; Prandlt σ=10; and β=8/3). The Lorenz equations are commonly expressed as coupled non-linear differential equations.

dx / dt = σ (y – x)
dy / dt = x (ρ – z) – y
dz / dt = xy – βz

xz

Projection of the Lorenz system on the x-z plane

Projection of the Lorenz system on the y-z plane

References

  1. E. N. Lorenz; Deterministic nonperiodic flow; J. Atmos. Sci., Vol. 20; 1963
  2. K. Mischaikow, M. Mrozek; Chaos in Lorenz equations: a computer assisted proof; Bull. Amer. Math. Soc.; 1995
  3. K. Mischaikow, M. Mrozek; Chaos in the Lorenz equations: a computer assisted proof. Part II: details; Mathematics of Computation; 67; 1998
  4. K. Mischaikow, M. Mrozek, A. Szymczak; Chaos in the Lorenz equations: a computer assisted proof. Part III: the classical parameter values; J. Diff. Equ. 169; 2001
  5. Morris W. Hirsch, Stephen Smale, Robert L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2013
  6. http://www.3d-meier.de
  7. http://www.mathworld.wolfram.com

Multi-scroll modified Lorenz System

Based on the sawtooth wave function theory, the generalized first type of modified Lorenz system is described by

dx/dt = a (y − x)
dy/dt = (c − z) sgn(x)
dz/dt = g(x) − bz

where a = 0.5, b = 0.1, c = 2, sgn(x) is a sign function, and g(x) is a sawtooth wave function with A = 0.43, k = 2.5, A= 2.06

g0(x) = k|x|
g(x)= g0(x) +∑N(n=1) gn(x)
gn(x) = −Gn[2 + sgn(x − En) − sgn(x + En)]
Gn = A/An
En = nA/k

sadas.png

10-scroll modified Lorenz system

dsfvg

References

  1. A. S. Elwakil, S. Ozoguz, and M. P. Kennedy, Creation of a complex
    butterfly attractor using a novel Lorenz-type system, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Volume. 49, 2002
  2. S. Ozoguz, A. S. Elwakil, and M. P. Kennedy, Experimental verification
    of the butterfly attractor in a modified Lorenz system, International Journal Bifurcation and Chaos, Volume 12, No. 7, 2002
  3. J. Lu, G. Chen, Multi-scroll chaos generation: Theories, methods
    and applications, International Journal Bifurcation and Chaos, Volume. 16, No. 4, 2006
  4. https://www.sciencedirect.com
  5. https://www.researchgate.net
  6. https://www.semanticscholar.org

Chaos synchronization problem was first described by Fujisaka and Yemada in 1983, which did not receive any attention until Pecora and Carroll published their results on chaos synchronization. Synchronization of chaotic systems is a phenomenon that may occur when more chaotic oscillators are coupled. Because of the butterfly effect, which
causes the exponential divergence of the trajectories of the identical chaotic systems defined with the almost same initial conditions, synchronizing more chaotic systems may certainly be an issue.

Coupled Lorenz System

Coupled Lorenz system refers to the synchronization of chaotic systems which may not be supposted because of the sensitivity to the permutations on the initial conditions. Chaotic behaviour occurs in the response system regardless of synchronization. However certain subsystems of chaotic systems can be synchronized by linking them. Subsequently was discovered that synchronization of non-identical chaotic systems may also be possible.

cl02b

The initial parameters are (Rayleigh ρ=28; Prandlt σ=10; β=8/3 and r1=35 and r2=1,15). The equations of the Coupled Lorenz System are commonly expressed

dx1 / dt = σ (y1-x1)
dy1 / dt = r1x1 – y1 – x1z1
dz1 / dt = -bz1 + x1y1

dx2 / dt = σ (y2 – x2) + ε (x1 – x2)
dy2 / dt = r2x2 – y2 – x2z2
dz2 / dt = -bz2 + x2y2

3-scroll coupled Lorenz system for  ε=2.85

6-scroll coupled Lorenz system for  ε=0.95

References

  1. J. Lü, G. Chen; Generating multiscroll chaotic attractors: Theories, Methods and Applications; Int. J. Bifur Chaos, 16 (4); 2006
  2. S. Yu, J. Lü, W. K. S. Tang, G. Chen; A general multiscroll Lorenz system family and its realization via digital signal processors; Chaos, 16; 2006
  3. Pecora L. M. and Carroll T. L.; Synchronization of Chaotic Systems. Physical Review Letters 64; 1990
  4. Khan A. and Singh P.; Non-Linear Dynamical System and Chaos Synchronization; International Journal of Bifurcation and Chaos 18; 2008
  5. Joos Vandewalle, Müştak E. Yalçin, Cellular Neural Networks, Multi-scroll Chaos and Synchronization,World Scientific Series on Nonlinear Science Series A: Volume 50, 2005
  6. http://www.3d-meier.de
  7. http://www.mathworld.wolfram.com
  8. https://chaoticatmospheres.com
  9. https://www.sciencedirect.com
  10. https://www.researchgate.net
  11. https://www.semanticscholar.org

Sprott C System

In 1994, Julien Clinton Sprott reported 19 cases of three-dimensional autonomous differential systems with either five terms and two quadratic nonlinearities or six
terms and one quadratic nonlinearity, whose solutions are chaotic. These systems were named as Sprott A-S systems, where the Sprott C system equations were defined as
follows

dx/dt = a(y − x)
dy/dt = xz
dz/dt = b – y2

The Sprott C system refers a close resemblance to the Lorenz system in that they exist as a butterfly strange attractor and two symmetrical equilibria.

fdgdfdf

Projection of the Sprott C attractor on the x-z plane

fgdgh

Projection of the Sprott C system on the y-z plane

ghjmg.png

2-scroll Sprott C system

fdfg.png

3-scroll Sprott C system for f(z) = 5 [ sgn (z-5) + sgn (z+5)]

The 4D multi scroll Sprott C system is distinguished by the equations

dx/dt = a (y − x)
dy/dt = xz – xw
dz/dt = b – y2
dw/dt= f(z) – w

where

f(z) = 5 sgn(z) for the 2-scroll Sprott C system
f(z) = 5 [ sgn (z-5) + sgn (z+5)] for the 3-scroll Sprott C system
f(z) = 5 [ sgn (z10) + sgn(z) + sgn (z+10)] for the 4-scroll Sprott C system
f(z) = 5 [ sgn (z15) + sgn (z-5) sgn (z+5)] + sgn (z+15)] for the 5-scroll Sprott C system

The 5-scroll Sprott C system f(z) = 5 [ sgn (z15) + sgn (z-5) sgn (z+5)] + sgn (z+15)]

References

    1. Julien Clinton Sprott; Strange Attractors: Creating Patterns in Chaos; M&T Press; 1993
    2. Joos Vandewalle, Müştak E. Yalçin, Cellular Neural Networks, Multi-scroll Chaos and Synchronization,World Scientific Series on Nonlinear Science Series A: Volume 50, 2005
    3. http://sprott.physics.wisc.edu
    4. http://www.3d-meier.de
    5. http://cpb.iphy.ac.cn
    6. https://www.sciencedirect.com/
    7. https://www.researchgate.net/
  1.  

Nosé-Hoover System

Sprott developed a search method which appliedin order to get sets of coupled chaotic ordinary differential equations. Each of the sets contained three coupled quadratic equations. Of these, Sprott’s A system, is the set of equations that Posch, William Hoover, and Vesely used to describe a one-dimensional Nosé-Hoover oscillator. The Sprott A system is a particular case of the Nose-Hoover oscillator.

fsdf

Projection of the Nosé-Hoover system on the x-z plane

dfdfsfd

Projection of the Nosé-Hoover system on the y-z plane

References

  1. S. Nosé; A molecular dynamics method for simulations in the canonical ensemble; Molecular Physics; 1984
  2. W. G. Hoover; Nonlinear conductivity and entropy in the two-body Boltzmann gas; Journal of Statistical Physics; 1986
  3. H. A. Posch, W.oover & F. J. Vesely; Canonical dynamics of the Nosé oscillator: Stability, order, and chaos; Physical Review A; 1986
  4. Sprott, J. C.; Some simple chaotic flows; Phys. Rev.; 1994
  5. Mahdi Adam; Valls Clàudia; The integrability of the Nose-Hoover equation; Journal of Geometry and Physics, Vol. 62; 2011
  6. http://www.gsd.uab.cat

Rössler System

Rössler system is a system of three non-linear differential equations defined by Otto Rössler German biochemist in 1976.

dx/dt = – y – z
dy/dt = x + a y
dz/dt = b + z (x – c)

fgfd

Rössler studied the chaotic system with  a=0.2, b=0.2, c=5.7

fsdf

Hyperchaos was first described in 1979 by O. E. Rössler in a four-dimensional system with two unstable equilibrium points. The degree of hyperchaos can be quantified by the value of the Lyapunov exponent.

References

  1. O. E. Rössler; An Equation for COntinous Chaos; Physics Letters; 1976
  2. O. E. Rössler; An Equation for Hyperchaos; Physics Letters; 1979
  3. O. E. Rössler; Different types of chaos in two simple differential equations; Zeitschrift für Naturforsch A; 1976
  4. Morris W. Hirsch, Stephen Smale, Robert L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2013
  5. https://www.library.cornell.edu

Dadras-Momeni System

The Dadras-Momeni system refers to an autonomous multi-scroll chaotic system distinguished by five equilibrium points. By varying only the system parameter c has am substantial  impact on the quantity of scrolls.

The Dadras-Momeni system is described by the following nonlinear differential equations,

dx/dt = y − ax + byz
dy/dt = cy − xz + z
dz/dt = dxy − hz

dsfgh

Projection of the Dadras-Momeni system on the x-z plane

fasdfdasf.png

Projection of the Dadras-Momeni system on the y-z plane

sffs

Projection of the Dadras-Momeni system on the x-y plane

References

  1. Dadras S., Momeni H. R.: A novel three-dimensional autonomos chaotic system generating two, three and four-scroll attractors; Phys. Lett. A 373; 2009
  2. Dadras S., Momeni, H.; Four-scroll hyperchaos and four-scroll chaos evolved from a novel 4D nonlinear smooth autonomous system; Phys. Lett. A 374; 2010
  3. Dadras S., Momeni H. R., Qi, G.; Analysis of a new 3D smooth autonomous system with different wing chaotic attractors and transient chaos.; Nonlinear Dyn. 62; 2010
  4. http://www.3d-meier.de

Three-Scroll Unified Chaotic System (TSUCS)

The Three-Scroll Unified Chaotic System (TSUCS) was introduced by Lin Pan, Wuneng Zhou, Jian’an Fang and Dequan Li in 2010. The Three-Scroll Unified Chaotic System (TSUCS), distinguished by three equilibrium points, represents continued transition from the Lorenz subsystem to the Chen subsystem.

The Three-Scroll Unified Chaotic System is chaotic independent on the input variables (a=40, b=55, c=11/6, d=0.16, e=0.65, f=20). The equations of the Three-Scroll Unified Chaotic System (TSUCS) are commonly expressed

dx / dt = a (y – x) + dxz
dy / dt = bx – xz +fy
dz / dt = -ex2 + xy + cz

Projection of the Three-Scroll Unified Chaotic System (TSUCS) on the x-z plane

sadfg.jpg

Projection of the Three-Scroll Unified Chaotic System (TSUCS) on the y-z plane

References

  1. L. Pan, W. Zhou, J. Fang, and D. Li; A new three-scroll unified chaotic system coined; International Journal of Nonlinear Science; vol. 10; 2010
  2. D. Li, A three-scroll chaotic attractor; Physics Letters A; vol. 372; 2008
  3. S. Dadras and H.R. Momeni; Generating one-, two-, three- and four-scroll attractors from a novel four-dimensional smooth autonomous chaotic system; Chinese Physics B, vol. 19; 2009
  4. http://www.3d-meier.de
  5. http://www.mathworld.wolfram.com
  6. https://chaoticatmospheres.com
  7. https://www.library.cornell.edu/
  8. http://cpb.iphy.ac.cn

Chua’s circuit

Chua´s circuit was proposed in 1984 by Leon O. Chua. The Chua´s circuit is determined by three equlibrium points.

3-double-scroll Chua´s circuit

7-scroll Chua´s circuit

Further research may refer to multi-dimensional multi-scroll systems.

References

  1. Zou Y., Luo X., Jiang P., Wang B., Chen G., Fang J. & Quan H.;Controlling the chaotic n-scroll Chua’s circuit; Int. J. Bifurcation and Chaos 13; 2003
  2. Suykens J. A. K., Huang A. & Chua L. O.; A family of n-scroll attractors from a generalized Chua’s circuit; Int. J. Electron. Commun. 51; 1997
  3. Kennedy M. P.; Three steps to chaos, Part II: A Chua’s circuit primer; IEEE Trans. Circuits Syst.-I 40; 1993
  4. Cafagna D. & Grassi G.; New 3D-scroll attractors in hyperchaotic Chua’s circuit forming a ring; Int. J. Bifurcation and Chaos 13; 2003
  5. Matsumuto T.; A chaotic attractor from Chua’s circuit; IEEE Trans. Circuits Syst. 31(12); 1984
  6. http://www.3d-meier.de
  7. http://cpb.iphy.ac.cn
  8. http://www.mathworld.wolfram.com
  9. https://www.library.cornell.edu/
  10. https://www.actaphys.uj.edu.pl
  11. https://www.sciencedirect.com/
  12. https://www.researchgate.net/

Matsumoto–Chua–Kobayashi System

Matsumoto–Chua–Kobayashi (MCK) circuit refers to a hyperchaotic variation of Chua´s circuit with equations 

dx/dt = α (g(y − x)−z)
dy/dt = β(−g(y −x)− w)
dz/dt = γ(x + z)
dw/dt = δy

where g = ε(y − x) + 0.5(θ − ε)(|y − x + μ| − |y − x − μ|)

By obtaining generalized g function results in n-scroll hyperchaotic systems.

dsfsd

Double-scroll Matsumoto–Chua–Kobayashi (MCK) circuit

dsfg

wyz

References

  1. T. Matsumoto, Leon O. Chua, T. Kobayashi, IEEE Trans. Circuits Systems, 1986
  2. Gianluca Setti, Gianluca Mazzin, Riccardo Rovatti, Nonlinear Dynamics of Electronic Systems, Proceedings of the IEEE Workshop, 2000, https://doi.org/10.1142/4453
  3. Simin Yu, Jinhu Lü, Guanrong Chen, A family of n-scroll hyperchaotic attractors and their realization, Physics Letters, 2007, http://dx.doi.org/10.1016/j.physleta.2006.12.029

Memristive Chua’s circuit

After the invention by Leon O. Chua in 1971, a memristor is known as the fourth basic circuit element after the the resistor, capacitor, and inductor. A physical memristor was successfully fabricated during 2008. Memristor refers to the correlation between charge and flux which are the fundamental circuit variables. Memristor may be either flux controlled or charge controlled.

dasd

dfsd

References

  1. Leon O. Chua, Memristor, The missing circuit element, IEEE Transaction Circuit Theory, No. 18, 1971
  2. A. L. Fitch, D. Yu, H. H. Iu, V. Sreeram, Hyperchaos in a memristor-based modified canonical Chua’s circuit, International Journal Bifurcation Chaos, No. 22, 2012
  3. B. C. Bao, Z. H. Ma, J. P. Xu, Z. Liu, Q. Xu, A simple memristor chaotic circuit with complex dynamics, International Journal Bifurcation Chaos, No. 21, 2011
  4. B. C. Bao, Z. Liu, J. P. Xu, Dynamical analysis of memristor chaotic oscillator, Acta Physica Sinica, 2010

Multi-Scroll Memristive Chaotic System

The multi-scroll memristive chaotic system has no equilibriums which refers to the characteristics of a hidden attractor system. The multi-scroll memristive chaotic system is defined by

dx/dt = y
dy/dt = -x + axz + by sin(z)
dz/dt = 1 – y2 – W(x)y

where the nonlinearity function for the flux dependency

W(x) = c + 3de

mem.png

fsdf.png

References

  1. Nalini Prasad Mohanty, Rajeeb Dey, Binoy Krishna Roy, A New 3-D Memristive Time-delay Chaotic System with Multi-scroll and Hidden Attractors, IFAC-PapersOnLine, Volume 51, No. 1, 2018, https://doi.org/10.1016/j.ifacol.2018.05.097
  2. S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, C. S. Zhou, The synchronization of chaotic systems, Physics Reports, 2002
  3. Pecora and Carroll, L. M. Pecora, T. L. Carroll, Synchronization in chaotic systems, Physical review letters 64, 1990
  4. S. Jafari, V. T. Pham, T. Kapitaniak, Multiscroll chaotic sea obtained from a simple 3D system without equilibrium. International Journal of Bifurcation and Chaos, No. 26, 2016
  5. S. Jafari, J. C. Sprott, Simple chaotic flows with a line equilibrium. Chaos, Solitons & Fractals, No. 57, 2013
  6. A. L. Fitch, D. Yu, H. H. Iu, V. Sreeram, Hyperchaos in a memristor-based modified canonical Chua’s circuit. International Journal of Bifurcation and Chaos, No. 22, 2012
  7. B. C. Bao, Z. H. Ma, J. P. Xu, Z. Liu, Q. Xu, A simple memristor chaotic circuit with complex dynamics. International Journal Bifurcation Chaos, No. 21, 2011
  8. B. C. Bao, Z. Liu, J. P. Xu, Dynamical analysis of memristor chaotic oscillator, Acta Physica Sinica, 2010
  9. X. Wang, G. Chen, Constructing a chaotic system with any number of equilibria, Nonlinear Dynamics, No. 71, 2013
  10. L. Wang, S. Duan, A chaotic attractor in delayed memristive system, Abstract and Applied Analysis, 2012

Thomas’ Cyclically Symmetric Attractor

Thomas’ Cyclically Symmetric attractor, proposed by René Thomas, is cyclically symmetric in the x,y, and z variables.

References

  1. Thomas René; Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, ‘labyrinth chaos’; Int. J. Bifurcation and Chaos 9 (10); 1999
  2. H. Zhang, D. Liu and Z. Wang; Controlling Chaos Suppression, Synchronization and Chaotification; Springer; 2009
  3. Kaneko T., Tsuda I.; Complex Systems: Chaos and Beyond. A Constructive Approach with Applications in Life Sciences; New York: Springer; 2001
  4. http://www.3d-meier.de
  5. http://www.mathworld.wolfram.com
  6. https://www.library.cornell.edu/

Newton-Leipnik System

The Newton–Leipnik system is a chaotic system with two strange attractors. The Newton–Leipnik system is described by differencial equations

dx / dt = – ax + y + 10 yz
dy / dt = −x – 4y + 5 xz
dz / dt = bz – 5 xy

In 1981, Newton and Leipnik constructed a set of differential equations from Euler’s rigid body equations which were modified with a linear feedback. In 2002, B. Marlin established the existence of closed orbits which were not asymptotically stable for this system. In 2002, Chen studied chaos control and synchronization of the Newton–Leipnik system by using a stable-manifoldbased method.  More recent studies by Wang and Tian indicated that this chaotic system can be controlled to unstable period orbits and torus with a suited linear controller.

References

  1. G. Chen, X. Dong; On feedback control of chaotic continuous time systems; IEEE Trans. Circuit Systems; 1993;
  2. M. T. Yassen; Chaos control of Chen chaotic dynamical system. Chaos; Solitons & Fractals; 2003
  3.  Xuedi Wang, Chao Ge; Adaptive control and synchronization of the Newton-Leipnik systems, Journal of Information and Computing Science; Vol. 3; No. 4; 2008

Bouali System 2

The Bouali system, introduced by Safieddine Bouali in research paper in Nonlinear Dynamics in 2012, derives from the established Lotka-Volterre oscillator.

The initial variables of the Bouali equations are (a=3, α=2.2, b=1, β=1, c=0.002 and s=1). The equations of the Bouali systemis expressed by three-nonlinear differential equations.

dx / dt = x (a − y) + αz
dy / dt = −y (b − x2)
dz / dt = −x (c − sz) − βz

References

  1. Safieddine Bouali; A Novel Strange attractor with a Stretched Loop; Nonlinear Dynamics 70; 2012; DOI 10.1007/s11071-012-0625-6
  2. Safieddine Bouali; Feedback Loop in Extended van der Pol’s Equation Applied to an Economic Model of Cycles; International Journal of Bifurcation and Chaos; Vol. 9; 1999

  3. http://chaos-3d.e-monsite.com

Bouali System 3

The Bouali system refers to the sensitive dependency on initial conditions (SDIC). The equations of the Bouali system are expressed by three-nonlinear differential equations.

dx/dt = α x (1 – y) – β z
dy/dt = – γ y (1 – x2)
dz/dt = μ x

The Bouali system for α=3, β=2.2, γ=1 μ=0.001

The bifurcation of the dynamical system seems analog to the “Ruelle-Takens-Newhouse route to chaos”; the modification of μ value validates the relevance of the “Ruelle-Takens-Newhouse route to chaos”.

References

    1. Bouali, Safieddine; A 3D Strange Attractor with a Distinctive Silhouette. The Butterfly Effect Revisited; 2013; 
    2. Safieddine Bouali; A Novel Strange attractor with a Stretched Loop; Nonlinear Dynamics 70; 2012; DOI: 10.1007/s11071-012-0625-6
    3. Safieddine Bouali; Feedback Loop in Extended van der Pol’s Equation Applied to an Economic Model of Cycles; International Journal of Bifurcation and Chaos; Vol. 9; 1999; DOI: 10.1142/S0218127499000535
    4. Newhouse, S., Ruelle, D., Takens, F., Occurrence of strange Axiom A attractors near quasiperiodic flows on Tm, m≥3, Communications in Mathematical Physics 64, No. 1, 1978;
    5. Kapitaniak T.Ponce E. and Wojewoda J., Route to chaos via strange non-chaotic attractors, Journal of Physics A: Mathematical and GeneralVolume 23No. 8, 1990;
    6. http://chaos-3d.e-monsite.com
    7. https://www.researchgate.net
    8. https://arxiv.org

Hyperchaotic 4D Bouali System A

A hyperchaotic systems are defined as chaotic behavior with at least two positive Lyapunov exponents. The initial 4D strange system was described in 1979 by Otto Rössler. Some of hyperchaotic systems are thereafter defined mainly by the extension of another dimension; hyperchaotification techniques. The hyperchaotic 4D Bouali system A does not refer to the application of the hyperchaotification techniques. However, the present system retains a modified 2D Lotka-Volterra oscillator.

xyw

The hyperchaotic 4D Bouali system described by four nonlinear differential equations

dx/dt = x (1 – y) + α z
dy/dt = β (x² – 1) y
dz/dt = γ (1 – y) w
dw/dt = η z

where x, y, z, w are the variables, and α, β, γ, and η real parameters. The core of the hyperchaotic 4D Bouali system A refers to modification of the Lotka-Volterra oscillator.

dx/dt = x (1 – y)
dy/dt = ( x² – 1) y

xz, wyz

Projection of the hyperchaotic 4D Bouali system on the w-y plane

xz, xyz

Projection of the hyperchaotic 4D Bouali system on the x-y plane

yz, xyz

Projection of the hyperchaotic 4D Bouali system on the y-z plane

xz, xwz

Projection of the hyperchaotic 4D Bouali system on the w-z plane

xy, wyz2

Projection of the hyperchaotic 4D Bouali system on the w-y plane

References

  1. Safieddine Bouali; A New Hyperchaotic Attractor with Complex Patterns; 2015;
  2. Safieddine Bouali; A Novel Strange attractor with a Stretched Loop; Nonlinear Dynamics 70; 2012; DOI: 10.1007/s11071-012-0625-6
  3. Safieddine Bouali; Feedback Loop in Extended van der Pol’s Equation Applied to an Economic Model of Cycles; International Journal of Bifurcation and Chaos; Vol. 9; 1999; DOI: 10.1142/S0218127499000535

  4. Rössler O. E.; An Equation for Hyperchaos; Physics Letters; 1979

  5. http://www.mathworld.wolfram.com
  6. http://chaos-3d.e-monsite.com
  7. http://cpb.iphy.ac.cn
  8. http://www.3d-meier.de
  9. https://www.library.cornell.edu

Modified multi-scroll Chen System

Chen system was introduced in 1999 while studying how to control the Lorenz system.

dx/dt = a(y – x)
dy/dt = (c – a)x – xz + cy
dz/dt = xy – bz

The modified multi-scroll Chen system is defined by equations

dx/dt = a(y – x)
dy/dt = (c – a)x – xu + cy
dz/dt = xy – bz

where u = d1z – d2 sin(z)

The quantity of the equilibrium points depends on the values of the parameter d2

asdf

Projection of the modified 6-scrollChen system on the x-z plane

sdfads

Projection of the modified 6-scroll Chen system on the y-z plane

dsfgd

Projection of the modified 6-scroll Chen system on the x-y plane

The delay differential equation may generate hyperchaotic dynamic behavior. The modified multi-scroll Chen system equations

u(t) = d0z(t) + d1z(t −τ ) − d2 sin(z(t − τ ))

where d0, d1, d2 are constants and τ is the time delay.

References

  1. Liu, Shen, Zhang; Multi-scroll chaotic and hyperchaotic attractors generated from Chen system; International Journal of Bifurcation and Chaos; Vol. 22; No. 2; 2012
  2. Wang L. & Yang X.; Generation of multi-scroll delayed chaotic oscillator; 2006
  3. Rossler, O. E.; An equation for hyperchaos; 1979
  4. Lu J., Han F., Yu X. & Chen G.; Generating 3-D multi-scroll chaotic attractors: A hysteresis series switching method; 2004

Xing-Yun System

The Xing-Yun system is a four-wing autonomous chaotic system which each equation contains a cubic product term. The Xing-Yun four-wing autonomous chaotic system indicates a broad magnitude bandwidth compared to the Lorenz system and other four-wing chaotic systems.

dx / dt = a(y − x) + yzz
dy / dt = b(x + y) − xzz
dz / dt = -cz + dy + xyz

dsf

Projection of the Xing-Yun system on the x-z plane for a=50, b=13, c=13, d=6

fdgdf

Projection of the Xing-Yun system on the y-z plane for a=50, b=13, c=13, d=6

References

  1. Liu Xing-Yun; A New 3D Four-Wing Chaotic System with Cubic Nonlinearity and Its Circuit Implementation; Chinese physics letters Vol. 26; No. 9; 2009
  2.  Vaněček and Čelikovský; Control Systems: From Linear Analysis to Synthesis of Chaos; 1996
  3. http://cpb.iphy.ac.cn

Chen-Lee System

Projection of the Chen-Lee system on the x-y plane

Projection of the Chen-Lee system on the x-z plane

References

  1. Tam L., Chen J., Chen H., & Tou W.; Generation of hyperchaos from the Chen–Lee system via sinusoidal perturbation; Chaos, Solitons and Fractals; Vol. 38; 2008
  2. Li C., Chen G.; Chaos in the fractional order Chen system and its control; Chaos Solitons Fractals 22; 2004

  3. https://www.library.cornell.edu/
  4. http://cpb.iphy.ac.cn
  5.  www.3d-meier.de

 

Nova Fractal


 

#rhino3d #grasshopper3d #python

The Nova fractal was created by Paul Derbyshire while estimating the Newton fractals. Paul Derbyshire modified the Newton’s method in order to improve the the rate of convergence. The Nova fractal is derived from this equation zn+1 = z– R * f(zn) / f ‘(zn) while the initial function was f(z) = zp-1, and therefore the equation that is iterated  zn+1 = zn – R (znp – 1) / (p znp-1) + c

hgfjhgjgh

The Nova fractal for zn+1 = zn – R (zn3 – 1) / (3 zn2) + c

 

References:

  1. Benoît Mandelbrot; Fractal aspects of the iteration of z→λ z(1-z) for complex λ, z; Annals of the New York Academy of Sciences 357; 2006
  2. http://www.researchgate.net
  3. http://www.mathworld.wolfram.com
  4. http://www.paulbourke.net
  5. http://www.hpdz.net
  6. http://www.fractalforums.com

other fractals


#rhino3d #grasshopper3d #python

Curlicue fractal

The curlicue fractal refers to the recurrence of the connected Fresnel spirals. The characteristics of the curlicue fractal are determined by the parameter, which affects the scale, etc. The curlicue fractal equations are

θ(n+1) = (θ(n) + 2π*s) mod(2π)
ϕ(n+1) = (ϕ (n) + θ(n)) mod(2π)

where

x(n) = cos(ϕ (n))
y(n) = sin(ϕ (n))

odm2

The curlicue fractal for s=√2

odmi3

The curlicue fractal for s=√3

e5

The curlicue fractal for s=e5

References

  1. Berry M.; Goldberg J.; Renormalization of Curlicues; Nonlinearity; 1988
  2. Moore R.; van der Poorten A.; On the Thermodynamics of Curves and Other Curlicues; McQuarie University; 1989
  3. Pickover C. A.; Is the Fractal Golden Curlicue Cold?; Visual Computer 11; 1995
  4. Sedgewick R.; Algorithms in C, 3rd edition; Reading; MA: Addison-Wesley; 1998
  5. http://mathworld.wolfram.com

Simurgh fractal

Simurgh fractal z(n+1) = z(n) a + c*z(n)b + p*z(n-1) + q*z(n-2); subset of the the Phoenix fractals, discovered by Shigehiro Ushiki in 1988, are the Simurgh fractals

sim2

Phoenix set

Phoenix fractal was discovered by Shigehiro Ushiki in 1998 in paper published in the ‘IEEE Transactions on Circuits and Systems’ journal. The Phoenix fractal is a variant of the Mandelbrot and Julia sets; however the properties of the Phoenix fractals are different from the Mandelbrot and Julia sets. 

gzbhrtffztfz

The initial formula of the Phoenix set discovered by Shigehiro Ushiki was z(n+1) = z(n)**p + Re(c) + Im(c)*z(n-1); where p ≥ 2 and n and c are constants.

References

  1. https://www.formulas.ultrafractal.com
  2. https://www.researchgate.net
  3. Complex dynamics of superior Phoenix set; Sunil Shukla, Ashish Negi; International journal of computer engineering & Technology (IJCET); Volume 4, Issue 1, pp. 263-274; 2013

Social Knowledge Processes


#rhino3d #grasshopper3d

The thesis concerns with the core social knowledge processes including communication, interaction, attention, perception, apperception and selfperception, which provide the required integration of individuals into the social environment.00

The thesis aims to approximate the progressive, responsible integration of individuals into the social environment; and the understanding of the social environment, often underestimated, or even not considered by the individuals. The thesis approaches to the social environment from the individual’s view; which contributes to understanding of the social influence.

The thesis concerns with the question of the social influence of the core social knowledge processes to the individuals. The thesis aims to approximate the process progress incipient from the perception, thus process without conscious awareness, approaching to the intentionally process of communication. The process progress consists of different processes including perception, attention, apperception, selfperception, interaction; which are influenced by the motivation. The thesis considers the requirements of the core social knowledge processes including motivation, intention, as reaction to internal or external initiative, resulting from the premise of the motivation of individuals to integration into the social environment.

References

  1. Susan Tufts Fiske; Social beings; a core motives approach to social psychology; Princeton University; 2004
  2. Nonaka Ikujiro; Takeuchi Hirotaka; The knowledge creating company; Oxford; 1995
  3. James Jerome Gibson; The Perception of the visual world; Boston; Houghton Mifflin; 1950
  4. James Jerome Gibson, Eleanor Jack Gibson; Perceptual learning; Differentiation or enrichment; 1955
  5. John A. Bargh; Social Psychology and the Unconscious; The Automaticity of Higher Mental Processes; 2007

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