# Complex systems

## Complex systems Topics

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### Hilbert curve

The Hilbert curve is a Lindenmayer system invented by Hilbert (1891) whose limit is a plane-filling function which fills a square. Traversing the polyhedron vertices of an -dimensional hypercube in Gray code order produces a generator for the -dimensional Hilbert curve. The Hilbert curve can be simply encoded with initial string "L", string rewriting rules "L" -> "+RF-LFL-FR+", "R" -> "-LF+RFR+FL-", and angle (Peitgen and Saupe 1988, p. 278). The th iteration of this Hilbert curve is implemented in the Wolfram Language as HilbertCurve[n].A related curve is the Hilbert II curve, shown above (Peitgen and Saupe 1988, p. 284). It is also a Lindenmayer system and the curve can be encoded with initial string "X", string rewriting rules "X" -> "XFYFX+F+YFXFY-F-XFYFX", "Y" -> "YFXFY-F-XFYFX+F+YFXFY",..

### Curlicue fractal

The curlicue fractal is a figure obtained by the following procedure. Let be an irrational number. Begin with a line segment of unit length, which makes an angle to the horizontal. Then define iteratively bywith . To the end of the previous line segment, draw a line segment of unit length which makes an angleto the horizontal (Pickover 1995ab). The result is a fractal, and the above figures correspond to the curlicue fractals with points for the golden ratio , , , , the Euler-Mascheroni constant , , and the Feigenbaum constant .The temperature of these curves is given in the followingtable.constanttemperaturegolden ratio 46515858Euler-Mascheroni constant 6390Feigenbaum constant 92

### Iterated function system

A finite set of contraction maps for , 2, ..., , each with a contractivity factor , which map a compact metric space onto itself. It is the basis for fractal image compression techniques.

### Minkowski sausage

A fractal curve created from the base curve and motifillustrated above (Lauwerier 1991, p. 37).As illustrated above, the number of segments after the th iteration is(1)and the length of each segment is given by(2)so the capacity dimension is(3)(4)(5)(6)(Mandelbrot 1983, p. 48).The term Minkowski sausage is also used to refer to the Minkowskicover of a curve.

### Wallis sieve

A compact set with areacreated by punching a square hole of length in the center of a square. In each of the eight squares remaining, punch out another hole of length , and so on.

### Menger sponge

The Menger sponge is a fractal which is the three-dimensionalanalog of the Sierpiński carpet. The th iteration of the Menger sponge is implemented in the Wolfram Language as MengerMesh[n, 3].Let be the number of filled boxes, the length of a side of a hole, and the fractional volume after the th iteration, then(1)(2)(3)The capacity dimension is therefore(4)(5)(6)(7)(OEIS A102447).The Menger sponge, in addition to being a fractal, is also a super-object for all compact one-dimensional objects, i.e., the topological equivalent of all one-dimensional objects can be found in a Menger sponge (Peitgen et al. 1992).The image above shows a metal print of the Menger sponge created by digital sculptorBathsheba Grossman (https://www.bathsheba.com/).

### Tetrix

The tetrix is the three-dimensional analog of the Sierpiński sieve illustrated above, also called the Sierpiński sponge or Sierpiński tetrahedron.The th iteration of the tetrix is implemented in the Wolfram Language as SierpinskiMesh[n, 3].Let be the number of tetrahedra, the length of a side, and the fractional volume of tetrahedra after the th iteration. Then(1)(2)(3)The capacity dimension is therefore(4)(5)so the tetrix has an integer capacity dimension (which is one less than the dimension of the three-dimensional tetrahedra from which it is built), despite the fact that it is a fractal.The following illustrations demonstrate how the dimension of the tetrix can be the same as that of the plane by showing three stages of the rotation of a tetrix, viewed along one of its edges. In the last frame, the tetrix "looks" like the two-dimensional plane. ..

### Mandelbrot set lemniscate

A curve on which points of a map (such as the Mandelbrot set) diverge to a given value at the same rate. A common method of obtaining lemniscates is to define an integer called the count which is the largest such that where is usually taken as . Successive counts then define a series of lemniscates, which are called equipotential curves by Peitgen and Saupe (1988).

### Mandelbrot set

The term Mandelbrot set is used to refer both to a general class of fractal sets and to a particular instance of such a set. In general, a Mandelbrot set marks the set of points in the complex plane such that the corresponding Julia set is connected and not computable."The" Mandelbrot set is the set obtained from the quadraticrecurrence equation(1)with , where points in the complex plane for which the orbit of does not tend to infinity are in the set. Setting equal to any point in the set that is not a periodic point gives the same result. The Mandelbrot set was originally called a molecule by Mandelbrot. J. Hubbard and A. Douady proved that the Mandelbrot set is connected.A plot of the Mandelbrot set is shown above in which values of in the complex plane are colored according to the number of steps required to reach . The kidney bean-shaped portion of the Mandelbrot set turns out to be bordered by a cardioid with equations(2)(3)The..

### Feigenbaum constant

The Feigenbaum constant is a universal constant for functions approaching chaos via period doubling. It was discovered by Feigenbaum in 1975 (Feigenbaum 1979) while studying the fixed points of the iterated function(1)and characterizes the geometric approach of the bifurcation parameter to its limiting value as the parameter is increased for fixed . The plot above is made by iterating equation (1) with several hundred times for a series of discrete but closely spaced values of , discarding the first hundred or so points before the iteration has settled down to its fixed points, and then plotting the points remaining.A similar plot that more directly shows the cycle may be constructed by plotting as a function of . The plot above (Trott, pers. comm.) shows the resulting curves for , 2, and 4.Let be the point at which a period -cycle appears, and denote the converged value by . Assuming geometric convergence, the difference between this value and..

### H&eacute;non map

There are at least two maps known as the Hénon map.The first is the two-dimensional dissipative quadraticmap given by the coupled equations(1)(2)(Hénon 1976).The strange attractor illustrated above is obtained for and .The illustration above shows two regions of space for the map with and colored according to the number of iterations required to escape (Michelitsch and Rössler 1989).The plots above show evolution of the point for parameters (left) and (right).The Hénon map has correlation exponent (Grassberger and Procaccia 1983) and capacity dimension (Russell et al. 1980). Hitzl and Zele (1985) give conditions for the existence of periods 1 to 6.A second Hénon map is the quadratic area-preserving map(3)(4)(Hénon 1969), which is one of the simplest two-dimensional invertible maps...

### Sierpi&324;ski sieve

The Sierpiński sieve is a fractal described by Sierpiński in 1915 and appearing in Italian art from the 13th century (Wolfram 2002, p. 43). It is also called the Sierpiński gasket or Sierpiński triangle. The curve can be written as a Lindenmayer system with initial string "FXF--FF--FF", string rewriting rules "F" -> "FF", "X" -> "--FXF++FXF++FXF--", and angle .The th iteration of the Sierpiński sieve is implemented in the Wolfram Language as SierpinskiMesh[n].Let be the number of black triangles after iteration , the length of a side of a triangle, and the fractional area which is black after the th iteration. Then(1)(2)(3)The capacity dimension is therefore(4)(5)(6)(7)(OEIS A020857; Wolfram 1984; Borwein and Bailey2003, p. 46).The Sierpiński sieve is produced by the beautiful recurrenceequation(8)where denote bitwise..

### Correlation exponent

A measure of a strange attractor which allows the presence of chaos to be distinguished from random noise. It is related to the capacity dimension and information dimension , satisfying(1)It satisfies(2)where is the Kaplan-Yorke dimension. As the cell size goes to zero,(3)where is the correlation dimension.

### Sierpi&324;ski curve

There are several fractal curves associated with Sierpiński.The area for the first Sierpiński curve illustratedabove (Sierpiński curve 1912) isThe curve is called the Sierpiński curve by Cundy and Rollett (1989, pp. 67-68), the Sierpiński's square snowflake by Wells (1991, p. 229), and is pictured but not named by Steinhaus (1999, pp. 102-103). The th iteration of the first Sierpiński curve is implemented in the Wolfram Language as SierpinskiCurve[n].The limit of the second Sierpiński's curve illustrated above has areaThe Sierpiński arrowhead curveis another Sierpiński curve.

### Koch snowflake

The Koch snowflake is a fractal curve, also known as the Koch island, which was first described by Helge von Koch in 1904. It is built by starting with an equilateral triangle, removing the inner third of each side, building another equilateral triangle at the location where the side was removed, and then repeating the process indefinitely. The Koch snowflake can be simply encoded as a Lindenmayer system with initial string "F--F--F", string rewriting rule "F" -> "F+F--F+F", and angle . The zeroth through third iterations of the construction are shown above.Each fractalized side of the triangle is sometimes known as a Koch curve.The fractal can also be constructed using a base curve and motif, illustrated above.The th iterations of the Koch snowflake is implemented in the Wolfram Language as KochCurve[n].Let be the number of sides, be the length of a single side, be the length of the perimeter, and the snowflake's..

### Sierpi&#324;ski carpet

The Sierpiński carpet is the fractal illustrated above which may be constructed analogously to the Sierpiński sieve, but using squares instead of triangles. It can be constructed using string rewriting beginning with a cell [1] and iterating the rules(1)The th iteration of the Sierpiński carpet is implemented in the Wolfram Language as MengerMesh[n].Let be the number of black boxes, the length of a side of a white box, and the fractional area of black boxes after the th iteration. Then(2)(3)(4)(5)The numbers of black cells after , 1, 2, ... iterations are therefore 1, 8, 64, 512, 4096, 32768, 262144, ... (OEIS A001018). The capacity dimension is therefore(6)(7)(8)(9)(OEIS A113210).

### Koch antisnowflake

A fractal derived from the Kochsnowflake. The base curve and motif for the fractal are illustrated below.The area enclosed by pieces of the curve after the th iteration iswhere is the area of the original equilateral triangle, so from the derivation for the Koch snowflake, the total area enclosed is

### Ces&agrave;ro fractal

The Cesàro fractal is a fractal also known as the torn square fractal. The base curves and motifs for the two fractals illustrated above are shown below.Starting with a unit square, the area of the interior for bend angle is given by(K. Sinha, pers. comm., Jul. 20, 2005).

### Julia set

Let be a rational function(1)where , is the Riemann sphere , and and are polynomials without common divisors. The "filled-in" Julia set is the set of points which do not approach infinity after is repeatedly applied (corresponding to a strange attractor). The true Julia set is the boundary of the filled-in set (the set of "exceptional points"). There are two types of Julia sets: connected sets (Fatou set) and Cantor sets (Fatou dust).Quadratic Julia sets are generated by the quadratic mapping(2)for fixed . For almost every , this transformation generates a fractal. Examples are shown above for various values of . The resulting object is not a fractal for (Dufner et al. 1998, pp. 224-226) and (Dufner et al. 1998, pp. 125-126), although it does not seem to be known if these two are the only such exceptional values.The special case of on the boundary of the Mandelbrot set is called a dendrite fractal (top left figure),..

### Capacity dimension

A dimension also called the fractal dimension, Hausdorff dimension, and Hausdorff-Besicovitch dimension in which nonintegral values are permitted. Objects whose capacity dimension is different from their Lebesgue covering dimension are called fractals. The capacity dimension of a compact metric space is a real number such that if denotes the minimum number of open sets of diameter less than or equal to , then is proportional to as . Explicitly,(if the limit exists), where is the number of elements forming a finite cover of the relevant metric space and is a bound on the diameter of the sets involved (informally, is the size of each element used to cover the set, which is taken to approach 0). If each element of a fractal is equally likely to be visited, then , where is the information dimension.The capacity dimension satisfieswhere is the correlation dimension (correcting the typo in Baker and Gollub 1996)...

### Information dimension

Define the "information function" to be(1)where is the natural measure, or probability that element is populated, normalized such that(2)The information dimension is then defined by(3)(4)If every element is equally likely to be visited, then is independent of , and(5)so(6)and(7)(8)(9)(10)where is the capacity dimension.It satisfies(11)where is the capacity dimension and is the correlation dimension (correcting the typo in Baker and Gollub 1996).

### Cantor square fractal

A fractal which can be constructed using stringrewriting beginning with a cell [1] and iterating the rules(1)The size of the unit element after the th iteration is(2)and the number of elements is given by the recurrencerelation(3)where , and the first few numbers of elements are 5, 65, 665, 6305, ... (OEIS A118004). Expanding out gives(4)The capacity dimension is therefore(5)(6)Since the dimension of the filled part is 2 (i.e., the square is completely filled), Cantor's square fractal is not a true fractal.

### Ice fractal

The term "ice fractal" refers to a fractal (square, triangle, etc.) that is based on a simple generating motif. The above plots show the ice triangle, antitriangle, square, and antisquare. The base curves and motifs for the fractals illustrated above are shown below.

### Reverend back's abbey floor

Consider the sequence defined by and , where denotes the reverse of a sequence . The first few terms are then 01, 010110, 010110010110011010, .... All words are cubefree (Allouche and Shallit 2003, p. 28, Ex. 1.49). Iterating gives the sequence 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, ... (OEIS A118006)Plotting (mod 2), where denotes the th digit of the infinitely iterated sequence, gives the beautiful pattern shown above, known as Reverend Back's abbey floor (Wegner 1982; Siromoney and Subramanian 1983; Allouche and Shallit 2003, pp. 410-411). Note that this plot is identical to the recurrence plot (mod 2).

### Cantor function

The Cantor function is the continuous but not absolutely continuous function on which may be defined as follows. First, express in ternary. If the resulting ternary digit string contains the digit 1, replace every ternary digit following the 1 by a 0. Next, replace all 2's with 1's. Finally, interpret the result as a binary number which then gives .The Cantor function is a particular case of a devil's staircase (Devaney 1987, p. 110), and can be extended to a function for , with corresponding to the usual Cantor function (Gorin and Kukushkin 2004).Chalice (1991) showed that any real-valued function on which is monotone increasing and satisfies 1. , 2. , 3. is the Cantor function (Chalice 1991; Wagon 2000, p. 132).Gorin and Kukushkin (2004) give the remarkable identityfor integer . For and , 2, ..., this gives the first few values as 1/2, 3/10, 1/5, 33/230, 5/46, 75/874, ... (OEIS A095844 and A095845).M. Trott (pers. comm., June..

### Cantor dust

Cantor dust is a fractal that can be constructed using string rewriting beginning with a cell [0] and iterating the rules(1)The th iteration of Cantor dust is implemented in the Wolfram Language as CantorMesh[n, 2].Let be the number of black boxes, the length of a side of a box, and the fractional area of black boxes after the th iteration, then(2)(3)(4)(5)The number of black squares after , 1, 2, ... iterations is therefore 1, 4, 16, 64, 256, 1024, 4096, 16384, ... (OEIS A000302). The capacity dimension is therefore(6)(7)(8)(9)

### Randelbrot set

The fractal-like figure obtained by performing the same iteration as for the Mandelbrot set, but adding a random component ,In the above plot, , where .

### Hausdorff dimension

Informally, self-similar objects with parameters and are described by a power law such aswhereis the "dimension" of the scaling law, knownas the Hausdorff dimension.Formally, let be a subset of a metric space . Then the Hausdorff dimension of is the infimum of such that the -dimensional Hausdorff measure of is 0 (which need not be an integer).In many cases, the Hausdorff dimension correctly describes the correction term for a resonator with fractal perimeter in Lorentz's conjecture. However, in general, the proper dimension to use turns out to be the Minkowski-Bouligand dimension (Schroeder 1991).

### Brown function

For a fractal process with values and , the correlation between these two values is given by the Brown functionalso known as the Bachelier function, Lévy function, or Wiener function.

### Haferman carpet

The Haferman carpet is the beautiful fractal constructed using string rewriting beginning with a cell [1] and iterating the rules(1)(Allouche and Shallit 2003, p. 407).Taking five iterations gives the beautiful pattern illustrated above.This fractal also appears on the cover of Allouche and Shallit (2003).Let be the number of black boxes, the length of a side of a white box, and the fractional area of black boxes after the th iteration. Then(2)(3)The numbers of black cells after , 1, 2, ... iterations are therefore 1, 4, 61, 424, 4441, 36844, ... (OEIS A118005). The capacity dimension is therefore(4)(5)

### Box fractal

The box fractal is a fractal also called the anticross-stitch curve which can be constructed using string rewriting beginning with a cell [1] and iterating the rules(1)An outline of the box fractal can encoded as a Lindenmayer system with initial string "F-F-F-F", string rewriting rule "F" -> "F-F+F+F-F", and angle (J. Updike, pers. comm., Oct. 26, 2004).Let be the number of black boxes, the length of a side of a white box, and the fractional area of black boxes after the th iteration.(2)(3)(4)(5)The sequence is then 1, 5, 25, 125, 625, 3125, 15625, ... (OEIS A000351). The capacity dimension is therefore(6)(7)(8)(9)(OEIS A113209).

### Gosper island

The Gosper island (Mandelbrot 1977), also known as a flowsnake (Gardner 1989, p. 41), is a fractal that is a modification of the Koch snowflake. The term "Gosper island" was used by Mandelbrot (1977) because this curve bounds the space filled by the Peano-Gosper curve.It has fractal dimension(OEIS A113211).Gosper islands can tile the plane(Gardner 1989, p. 41).

### Barnsley's tree

A Julia set fractal obtainedby iterating the functionwhere is the sign function and is the real part of . The plot above sets and uses a maximum of 50 iterations with escape radius 2.

### Pentaflake

The pentaflake is a fractal with 5-fold symmetry. As illustrated above, five pentagons can be arranged around an identical pentagon to form the first iteration of the pentaflake. This cluster of six pentagons has the shape of a pentagon with five triangular wedges removed. This construction was first noticed by Albrecht Dürer (Dixon 1991).For a pentagon of side length 1, the first ring of pentagons has centers at radius(1)where is the golden ratio. The inradius and circumradius are related by(2)and these are related to the side length by(3)The height is(4)giving a radius of the second ring as(5)Continuing, the th pentagon ring is located at(6)Now, the length of the side of the first pentagon compound is given by(7)so the ratio of side lengths of the original pentagon to that of the compound is(8)We can now calculate the dimension of the pentaflake fractal. Let be the number of black pentagons and the length of side of a pentagon after the iteration,(9)(10)The..

### Fractal process

A one-dimensional map whose increments are distributed according to a normal distribution. Let and be values, then their correlation is given by the Brown functionWhen , and the fractal process corresponds to one-dimensional Brownian motion. If , then and the process is called a persistent process. If , then and the process is called an antipersistent process.

### Barnsley's fern

The attractor of the iteratedfunction system given by the set of "fern functions"(1)(2)(3)(4)(Barnsley 1993, p. 86; Wagon 1991). These affine transformations are contractions. The tip of the fern (which resembles the black spleenwort variety of fern) is the fixed point of , and the tips of the lowest two branches are the images of the main tip under and (Wagon 1991).

### Fractal dimension

The term "fractal dimension" is sometimes used to refer to what is more commonly called the capacity dimension of a fractal (which is, roughly speaking, the exponent in the expression , where is the minimum number of open sets of diameter needed to cover the set). However, it can more generally refer to any of the dimensions commonly used to characterize fractals (e.g., capacity dimension, correlation dimension, information dimension, Lyapunov dimension, Minkowski-Bouligand dimension).

### Fractal

A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales. The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales. A plot of the quantity on a log-log graph versus scale then gives a straight line, whose slope is said to be the fractal dimension. The prototypical example for a fractal is the length of a coastline measured with different length rulers. The shorter the ruler, the longer the length measured, a paradox known as the coastline paradox.Illustrated above are the fractals known as the Gosper island, Koch snowflake, box fractal, Sierpiński sieve, Barnsley's fern, and Mandelbrot set.

### Fatou set

A Julia set consisting of a set of isolated points which is formed by taking a point outside an underlying set (e.g., the Mandelbrot set). If the point is outside but near the boundary of , the Fatou set resembles the Julia set for nearby points within . As the point moves further away, however, the set becomes thinner and is called Fatou dust.

### Mira fractal

A fractal based on iterating the map(1)according to(2)(3)The plots above show iterations of this map for various starting values and parameters .

A two-dimensional piecewise linear map defined by(1)(2)The map is chaotic in the filled region above and stable in the six hexagonal regions. Each point in the interior hexagon defined by the vertices (0, 0), (1, 0), (2, 1), (2, 2), (1, 2), and (0, 1) has an orbit with period six (except the point (1, 1), which has period 1). Orbits in the other five hexagonal regions circulate from one to the other. There is a unique orbit of period five, with all others having period 30. The points having orbits of period five are (, 3), (, ), (3, ), (5, 3), and (3, 5), indicated in the above figure by the black line. However, there are infinitely many distinct periodic orbits which have an arbitrarily long period.

### Zaslavskii map

The two-dimensional map(1)(2)where(3)(Zaslavskii 1978). It has correlation exponent (Grassberger and Procaccia 1983) and capacity dimension 1.39 (Russell et al. 1980).

### Feigenbaum function

Consider an arbitrary one-dimensional map(1)(with implicit parameter ) at the onset of chaos. After a suitable rescaling, the Feigenbaum function(2)is obtained. This function satisfies(3)with .Proofs for the existence of an even analytic solution to this equation, sometimes called the Feigenbaum-Cvitanović functional equation, have been given by Campanino and Epstein (1981), Campanino et al. (1982), and Lanford (1982, 1984).The picture above illustrate the Feigenbaum function for the logistic map with ,(4)along the real axis (M. Trott, pers. comm., Sept. 9, 2003).The images above show two views of a sculpture presented by Stephen Wolfram to Mitchell Feigenbaum on the occasion of his 60th birthday that depicts the Feigenbaum function in the complex plane. The sculpture (photos courtesy of A. Young) was designed by M. Trott and laser-etched into a block of glass by Bathsheba Grossman (https://www.bathsheba.com/)...

### Map winding number

The winding number of a map with initial value is defined bywhich represents the average increase in the angle per unit time (average frequency). A system with a rational winding number is mode-locked, whereas a system with an irrational winding number is quasiperiodic. Note that since the rationals are a set of zero measure on any finite interval, almost all winding numbers will be irrational, so almost all maps will be quasiperiodic.

### Correlation integral

Consider a set of points on an attractor, then the correlation integral iswhere is the number of pairs whose distance . For small ,where is the correlation exponent.

### Strange attractor

An attracting set that has zero measure in the embedding phase space and has fractal dimension. Trajectories within a strange attractor appear to skip around randomly.A selection of strange attractors for a general quadraticmap(1)(2)are illustrated above, where the letters to stand for coefficients of the quadratic from to 1.2 in steps of 0.1 (Sprott 1993c). These represent a small selection of the approximately 1.6% of all possible such maps that are chaotic (Sprott 1993bc).

### Standard map

A two-dimensional map also called the Taylor-Greene-Chirikovmap in some of the older literature and defined by(1)(2)(3)where and are computed mod and is a positive constant. Surfaces of section for various values of the constant are illustrated above.An analytic estimate of the width of the chaotic zone (Chirikov1979) finds(4)Numerical experiments give and . The value of at which global chaos occurs has been bounded by various authors. Greene's Method is the most accurate method so far devised.authorboundexactapprox.Hermann0.029411764Celletti and Chierchia (1995)0.838Greene-0.971635406MacKay and Percival (1985)0.984375000Mather1.333333333Fixed points are found by requiring that(5)(6)The first gives , so and(7)The second requirement gives(8)The fixed points are therefore and . In order to perform a linear stability analysis, take differentials of the variables(9)(10)In matrix form,(11)The eigenvalues are found..

### Chaos game

An algorithm originally described by Barnsley in 1988. Pick a point at random inside a regular -gon. Then draw the next point a fraction of the distance between it and a polygon vertex picked at random. Continue the process (after throwing out the first few points). The result of this "chaos game" is sometimes, but not always, a fractal. The results of the chaos game are shown above for several values of .The above plots show the chaos game for points in the regular 3-, 4-, 5-, and 6-gons with . The case gives the interior of a square with all points visited with equal probability.The above plots show the chaos game for points in the square with , 0.4, 0.5, 0.6, 0.75, and 0.9.

### Chaos

"Chaos" is a tricky thing to define. In fact, it is much easier to list properties that a system described as "chaotic" has rather than to give a precise definition of chaos.Gleick (1988, p. 306) notes that "No one [of the chaos scientists he interviewed] could quite agree on [a definition of] the word itself," and so instead gives descriptions from a number of practitioners in the field. For example, he quotes Philip Holmes (apparently defining "chaotic") as, "The complicated aperiodic attracting orbits of certain, usually low-dimensional dynamical systems." Similarly, he quotes Bai-Lin Hao describing chaos (roughly) as "a kind of order without periodicity."It turns out that even textbooks devoted to chaos do not really define the term. For example, Wiggins (1990, p. 437) says, "A dynamical system displaying sensitive dependence on initial conditions..

### Bogdanov map

A two-dimensional map which is conjugate to the Hénonmap in its nondissipative limit. It is given by(1)(2)

### Resonance overlap

Isolated resonances in a dynamical system can cause considerable distortion of preserved tori in their neighborhood, but they do not introduce any chaos into a system. However, when two or more resonances are simultaneously present, they will render a system nonintegrable. Furthermore, if they are sufficiently "close" to each other, they will result in the appearance of widespread (large-scale) chaos.To investigate this problem, Walker and Ford (1969) took the integrable Hamiltonianand investigated the effect of adding a 2:2 resonance and a 3:2 resonanceAt low energies, the resonant zones are well-separated. As the energy increases, the zones overlap and a "macroscopic zone of instability" appears. When the overlap starts, many higher-order resonances are also involved so fairly large areas of phase space have their tori destroyed and the ensuing chaos is "widespread" since trajectories are now..

### Attractor

An attractor is a set of states (points in the phase space), invariant under the dynamics, towards which neighboring states in a given basin of attraction asymptotically approach in the course of dynamic evolution. An attractor is defined as the smallest unit which cannot be itself decomposed into two or more attractors with distinct basins of attraction. This restriction is necessary since a dynamical system may have multiple attractors, each with its own basin of attraction.Conservative systems do not have attractors, since the motion is periodic. For dissipative dynamical systems, however, volumes shrink exponentially so attractors have 0 volume in -dimensional phase space.A stable fixed point surrounded by a dissipative region is an attractor known as a map sink. Regular attractors (corresponding to 0 Lyapunov characteristic exponents) act as limit cycles, in which trajectories circle around a limiting trajectory which they..

### Mode locking

A phenomenon in which a system being forced at an irrational period undergoes rational, periodic motion which persists for a finite range of forcing values. It may occur for strong couplings between natural and forcing oscillation frequencies.The phenomenon can be exemplified in the circle map when, after iterations of the map, the new angle differs from the initial value by a rational numberThis is the form of the unperturbed circle map withthe map winding numberFor not a rational number, the trajectory is quasiperiodic.

### Arnold tongue

Consider the circle map. If is nonzero, then the motion is periodic in some finite region surrounding each rational . This execution of periodic motion in response to an irrational forcing is known as mode locking. If a plot is made of versus with the regions of periodic mode-locked parameter space plotted around rational values (the map winding numbers), then the regions are seen to widen upward from 0 at to some finite width at . The region surrounding each rational number is known as an Arnold tongue.At , the Arnold tongues are an isolated set of measure zero. At , they form a general cantor set of dimension (Rasband 1990, p. 131). In general, an Arnold tongue is defined as a resonance zone emanating out from rational numbers in a two-dimensional parameter space of variables...

### Peano curve

A number of fractal curves are associated with Peano.The Peano curve is the fractal curve illustrated abovewhich can be written as a Lindenmayer system.The th iteration of the Peano curve illustrated above curve is implemented in the Wolfram Language as PeanoCurve[n].

### Cantor set

The Cantor set , sometimes also called the Cantor comb or no middle third set (Cullen 1968, pp. 78-81), is given by taking the interval (set ), removing the open middle third (), removing the middle third of each of the two remaining pieces (), and continuing this procedure ad infinitum. It is therefore the set of points in the interval whose ternary expansions do not contain 1, illustrated above.The th iteration of the Cantor is implemented in the Wolfram Language as CantorMesh[n].Iterating the process 1 -> 101, 0 -> 000 starting with 1 gives the sequence 1, 101, 101000101, 101000101000000000101000101, .... The sequence of binary bits thus produced is therefore 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, ... (OEIS A088917) whose th term is amazingly given by (mod 3), where is a (central) Delannoy number and is a Legendre polynomial (E. W. Weisstein, Apr. 9, 2006). The recurrence plot for this..