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.
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, …
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.
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:
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
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.
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.
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.”
“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.”
“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.”
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
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)
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.
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
M. Mitchell; An Introduction to Genetic Algorithms; The MIT Press; MA; US; 1990
Carlos Coello Coello, Gary B. Lamont, David A. van Veldhuizen; Evolutionary Algorithms for Solving Multi-Objective Problems; Springer Science & Business Media; 2007
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.
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.
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.
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
Przemyslaw Prusinkiewicz, Aristid Lindenmayer; The Algorithmic Beauty of Plants; Springer-Verlag; 1990
Przemyslaw Prusinkiewicz; Lindenmayer systems, fractals and plants; Springer-Verlag; New York; 1989
Thomas A. Witten and Leonard M. Sander; Diffusion-Limited Aggregation: A Kinetic Critical Phenomenon; Physical Review Letters 47; 1981
Leonard M. Sander, Z. M. Cheng, R. Richter; Diffusion-Limited Aggregation in three Dimensions; Physical Review B 28; 1983
Tomoya Sato, Masafumi Hagiwara; IDSET: Interactive Design System using Evolutionary Techniques; Computer-Aided Design 33; 2001
The research of minimal surfaces in R3 refers 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
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)
References
Ulrich Dierkes, Albrecht Kuster, Stefan Hildebrandt, Ortwin Wohlrab; Minimal Surfaces, Vol. 1: Boundary Value Problems; New York: Springer-Verlag; 1992
Ulrich Dierkes, Albrecht Kuster, Stefan Hildebrandt, Ortwin Wohlrab; Minimal Surfaces, Vol. 2: Boundary Regularity; New York: Springer-Verlag; 1992
Scherk, H. F., Bemerkung über der kleinste Fläche innerhalb gegebener Grenzen, J. reine angew. Math. 13, 1834
Elsa Abbena, Simon Salamon, Alfred Gray; Modern Differential Geometry of Curves and Surfaces with Mathematica; 3rd edition, CRC Press, 2006
Construction of minimal surfaces; Karcher H.; Surveys in Geometry; University of Tokyo; 1989
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.
Mandelbrot Set zn+1=z2+c
Mandelbrot Set zn+1=z3+c
Mandelbrot Set zn+1=z4+c
Mandelbrot set modifications
Mandelbrot Set zn+1=-icz5 + 1 #itssofluffy
Further research may refer to Mandelbulb, which would require hypercomplex numbers.
References
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
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
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
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
Julia set for c=-0.758900-0.075300i
Julia set for c=-0.512511498387847167+0.521295573094847167i
Julia set for c=-0.08+0.8i
Julia set for c=0.28+0.008i
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 < n < 2 and values of c located close to the boundary of the Mandelbrot sets have a characteristic form and are known as Glynn fractals.
Glynn fractal for fc(z) = z1.5 – 0.2
References
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
Shishikura, Mitsuhiro; “The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets”; Annals of Mathematics; Second Series 147 (2); 1998
Cheng, J. & Tan, Jr. J.; Generalization of 3D Mandelbrot and Julia sets; Zhejiang University; 2007
Norton Alan; Julia Sets in the quaternations; Computer Graphics 13 (2); 1989
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.
The Burning Ship fractal antenna
The Burning Ship fractal zn+1 = |zn3|+ c
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 .
Julia set for c=-1.764-0.06i from the Burning Ship fractal
Julia set for c=0.87+1.52i from the Burning Ship fractal
Julia set for c=0.675-1.15i from the Burning Ship fractal
References
Michael Michelitsch and Otto E. Rössler; The “Burning Ship” and Its Quasi-Julia Sets; Computers & Graphics Vol. 16; No. 4; 1992
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
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=0.05*cos(0)* xn +0.5sin(0)* yn
yn + 1 =0.05*sin(0)* xn -0.5cos(0)*yn +1
xn + 1=0.6*cos(0.698)* xn -0.5sin(0.698)* yn
yn + 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
yn + 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
yn + 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
yn + 1 =0.55*sin(-0.6980)* xn +0.4cos(-0.6980) * yn +0.7
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.
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.
References
Barnsley, Michael Fielding; Fractals everywhere; Academic Press, Boston, 1988
Barnsley, F. and Sloan A. D.; A Better Way to Compress Images; Byte Magazine; January; 1988
Oppenheimer P. E.; Real Time Design and Animation of Fractal Plants and Trees; Computer Graphics; 1986
Demko, S., Hodges, L., and Naylor B.; Construction of Fractal Objects with Iterated Function Systems; Computer Graphics; 1985
Reghbati, H. K.; An Overview of Data Compression Techniques; Computer; 1981
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.
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
The Duffing Oscillator. In: Chaos. Springer; Berlin, Heidelberg; 2008
Y. Ueda; Randomly transitional phenomena in the system governed by Duffing’s equation; J. Stat. Phys. 20; 1979
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
G. Duffing; Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz.; F. Vieweg u. Sohn; Braunschweig; 1918
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)
The Mira map for a=0.2, b=1 and x=12, y=0
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)
The Gumowski-Mira map for a=0, b=-0.31 and x=0 y=0.5
The Gumowski-Mira map for a=0, b=-0.22 and x=0.5 y=0.5
The Gumowski-Mira map for a=0, b=-0.53 and x=0 y=0.5
Mira Map modification
The modified Mira map for a=0.04, b=1, x=-1, y=8 and f(x) = ax – 3a/(a + ebx)
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
Gumowski I. Mira C.; Recurrences and Discrete Dynamic Systems; Springer; 1980
The Multifold Henon map refers to chaotic behaviour by iterating the following equations
xn+1 = 1 – a sin(xn) +byn
yn+1 = xn
References
Henon, M., A two-dimensional mapping with a strange attractor, Communications in Mathematical Physics 50, 1976
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.
The bifurcation diagram of the logistic map xn+1 = rxn (1-xn)
References
Sprott, Julien Clinton, Chaos and Time-Series Analysis, Oxford University Press, 2003
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.
References
Kensuke Ikeda; Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system; Optical Communications; 1979
Kensuke Ikeda, H. Daido & O. Akimoto; Optical turbulence: chaotic behavior of transmitted light from a ring cavity; Physical Review Letters; 1980
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.
The Hopalong attractor refers to chaotic behaviour by iterating the following equations
xn+1 = axn – (xn)3 – yn
yn+1 = xn
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
MacMillan map for a=1.6 und b=0.2
MacMillan map for a=1.6 und b=0.4
References
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
MacMillan, W.; An integrable case in the restricted problem of three bodies; 1911
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
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
Projection of the Lorenz system on the x-z plane
Projection of the Lorenz system on the y-z plane
References
E. N. Lorenz; Deterministic nonperiodic flow; J. Atmos. Sci., Vol. 20; 1963
K. Mischaikow, M. Mrozek; Chaos in Lorenz equations: a computer assisted proof; Bull. Amer. Math. Soc.; 1995
K. Mischaikow, M. Mrozek; Chaos in the Lorenz equations: a computer assisted proof. Part II: details; Mathematics of Computation; 67; 1998
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
Morris W. Hirsch,Stephen Smale,Robert L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2013
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
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
J. Lu, G. Chen, Multi-scroll chaos generation: Theories, methods and applications, International Journal Bifurcation and Chaos, Volume. 16, No. 4, 2006
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.
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
J. Lü, G. Chen; Generating multiscroll chaotic attractors: Theories, Methods and Applications; Int. J. Bifur Chaos, 16 (4); 2006
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
Pecora L. M. and Carroll T. L.; Synchronization of Chaotic Systems. Physical Review Letters 64; 1990
Khan A. and Singh P.; Non-Linear Dynamical System and Chaos Synchronization; International Journal of Bifurcation and Chaos 18; 2008
Joos Vandewalle, Müştak E. Yalçin, Cellular Neural Networks, Multi-scroll Chaos and Synchronization,World Scientific Series on Nonlinear Science Series A: Volume 50, 2005
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.
Projection of the Sprott C attractor on the x-z plane
Projection of the Sprott C system on the y-z plane
2-scroll Sprott C system
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
Julien Clinton Sprott; Strange Attractors: Creating Patterns in Chaos; M&T Press; 1993
Joos Vandewalle, Müştak E. Yalçin, Cellular Neural Networks, Multi-scroll Chaos and Synchronization,World Scientific Series on Nonlinear Science Series A: Volume 50, 2005
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.
Projection of the Nosé-Hoover system on the x-z plane
Projection of the Nosé-Hoover system on the y-z plane
References
S. Nosé; A molecular dynamics method for simulations in the canonical ensemble; Molecular Physics; 1984
W. G. Hoover; Nonlinear conductivity and entropy in the two-body Boltzmann gas; Journal of Statistical Physics; 1986
H. A. Posch, W.oover & F. J. Vesely; Canonical dynamics of the Nosé oscillator: Stability, order, and chaos; Physical Review A; 1986
Sprott, J. C.; Some simple chaotic flows; Phys. Rev.; 1994
Mahdi Adam; Valls Clàudia; The integrability of the Nose-Hoover equation; Journal of Geometry and Physics, Vol. 62; 2011
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)
Rössler studied the chaotic system with a=0.2, b=0.2, c=5.7
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
O. E. Rössler; An Equation for COntinous Chaos; Physics Letters; 1976
O. E. Rössler; An Equation for Hyperchaos; Physics Letters; 1979
O. E. Rössler; Different types of chaos in two simple differential equations; Zeitschrift für Naturforsch A; 1976
Morris W. Hirsch,Stephen Smale,Robert L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2013
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
Projection of the Dadras-Momeni system on the x-z plane
Projection of the Dadras-Momeni system on the y-z plane
Projection of the Dadras-Momeni system on the x-y plane
References
Dadras S., Momeni H. R.: A novel three-dimensional autonomos chaotic system generating two, three and four-scroll attractors; Phys. Lett. A 373; 2009
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
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
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
D. Li, A three-scroll chaotic attractor; Physics Letters A; vol. 372; 2008
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
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.
References
Leon O. Chua, Memristor, The missing circuit element, IEEE Transaction Circuit Theory, No. 18, 1971
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
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
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 + 3de2
References
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
S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, C. S. Zhou, The synchronization of chaotic systems, Physics Reports, 2002
Pecora and Carroll, L. M. Pecora, T. L. Carroll, Synchronization in chaotic systems, Physical review letters 64, 1990
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
S. Jafari, J. C. Sprott, Simple chaotic flows with a line equilibrium. Chaos, Solitons & Fractals, No. 57, 2013
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
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
B. C. Bao, Z. Liu, J. P. Xu, Dynamical analysis of memristor chaotic oscillator, Acta Physica Sinica, 2010
X. Wang, G. Chen, Constructing a chaotic system with any number of equilibria, Nonlinear Dynamics, No. 71, 2013
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
Thomas René; Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, ‘labyrinth chaos’; Int. J. Bifurcation and Chaos 9 (10); 1999
H. Zhang, D. Liu and Z. Wang; Controlling Chaos Suppression, Synchronization and Chaotification; Springer; 2009
Kaneko T., Tsuda I.; Complex Systems: Chaos and Beyond. A Constructive Approach with Applications in Life Sciences; New York: Springer; 2001
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
G. Chen, X. Dong; On feedback control of chaotic continuous time systems; IEEE Trans. Circuit Systems; 1993;
M. T. Yassen; Chaos control of Chen chaotic dynamical system. Chaos; Solitons & Fractals; 2003
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
Safieddine Bouali; A Novel Strange attractor with a Stretched Loop; Nonlinear Dynamics 70; 2012; DOI 10.1007/s11071-012-0625-6
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
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”.
Safieddine Bouali; A Novel Strange attractor with a Stretched Loop; Nonlinear Dynamics 70; 2012; DOI: 10.1007/s11071-012-0625-6
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
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;
Kapitaniak T., Ponce E. and Wojewoda J., Route to chaos via strange non-chaotic attractors, Journal of Physics A: Mathematical and General, Volume 23, No. 8, 1990;
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.
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
Projection of the hyperchaotic 4D Bouali system on the w-y plane
Projection of the hyperchaotic 4D Bouali system on the x-y plane
Projection of the hyperchaotic 4D Bouali system on the y-z plane
Projection of the hyperchaotic 4D Bouali system on the w-z plane
Projection of the hyperchaotic 4D Bouali system on the w-y plane
Safieddine Bouali; A Novel Strange attractor with a Stretched Loop; Nonlinear Dynamics 70; 2012; DOI: 10.1007/s11071-012-0625-6
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
Rössler O. E.; An Equation for Hyperchaos; Physics Letters; 1979
The quantity of the equilibrium points depends on the values of the parameter d2
Projection of the modified 6-scrollChen system on the x-z plane
Projection of the modified 6-scroll Chen system on the y-z plane
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
Liu, Shen, Zhang; Multi-scroll chaotic and hyperchaotic attractors generated from Chen system; International Journal of Bifurcation and Chaos; Vol. 22; No. 2; 2012
Wang L. & Yang X.; Generation of multi-scroll delayed chaotic oscillator; 2006
Rossler, O. E.; An equation for hyperchaos; 1979
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.
Projection of the Chen-Lee system on the x-y plane
Projection of the Chen-Lee system on the x-z plane
References
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
Li C., Chen G.; Chaos in the fractional order Chen system and its control; Chaos Solitons Fractals 22; 2004
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 = zn – 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
The Nova fractal for zn+1 = zn – R (zn3 – 1) / (3 zn2) + c
References:
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
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
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
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.
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.
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
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.
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
Susan Tufts Fiske; Social beings; a core motives approach to social psychology; Princeton University; 2004
Nonaka Ikujiro; Takeuchi Hirotaka; The knowledge creating company; Oxford; 1995
James Jerome Gibson; The Perception of the visual world; Boston; Houghton Mifflin; 1950
James Jerome Gibson, Eleanor Jack Gibson; Perceptual learning; Differentiation or enrichment; 1955
John A. Bargh; Social Psychology and the Unconscious; The Automaticity of Higher Mental Processes; 2007
