Clifford A. Reiter

Chaotic attractors with discrete planar symmetries

Abstract Chaotic behavior is known to be compatible with symmetry and illustrations are constructed using functions equivariant with respect to the desired symmetries. Earlier investigations determined families of equivariant functions for a few of the discrete symmetry groups in the plane; those results are extended to all the discrete symmetry groups of the plane. This includes consideration of the all the frieze and two-dimensional crystallographic groups.

Fuzzy automata and life

Boolean cellular automata may be generalized to fuzzy automata in a consistent manner. Several fuzzy logics are used to create 1-dimensional automata and also 2-dimensional automata that generalize the game of life. These generalized automata are investigated and compared to their Boolean counterparts empirically and using rule entropy and repeated input response functions. Fuzzy automata offer new mechanisms for classification of classical automata and can be used for insight into their qualitative behavior.

Symmetric attractors in three-dimensional space

Abstract General forms for polynomial functions that yield attractors having the symmetry of the cube are developed. Finite generating sets for these functions are also determined. These functions are applied experimentally to produce attractors that appear as points, curves, clouds, surfaces and strange attractors.

Fuzzy hexagonal automata and snowflakes

Abstract There is a common perception that snowflakes are six-sided and a common quip that “no two snowflakes are alike”. The 6-fold symmetry suggests that the growth is deterministic, while the differences suggest the growth is so sensitive to conditions that remarkable variety appears. We investigate fuzzy automata that give a great diversity of growth patterns, have sensitivity to background conditions, and which maintain the symmetry of snowflakes.

Visualizing generalized 3x+1 function dynamics

Abstract The function that results in 3x+1 for odd integers x and half of x for even x has led to intriguing questions. The 3x+1 conjecture states that iteration of that function on positive integers eventually results in the value 1. We investigate many generalizations of that function to the complex domain and visualize the resulting dynamics using escape time, stopping time, and basin of attraction images. We will see beautiful, rich dynamics consistent with the conjecture. We see that some generalizations have relatively simple real dynamics, which may make them useful for analysis and we see a complex generalization where sequences of stable egg shaped regions appear in coefficient stopping time images that suggest remarkable patterns for such stopping time.

n-Dimensional chaotic attractors with crystallographic symmetry

Abstract Chaotic attractors with planar symmetries have been the focus of much recent study. We establish a general method to create attractors with crystallographic symmetry in R n . Using this technique we provide a uniform approach to creating chaotic attractors in R 2 and generate provocative illustrations in R 3 with crystallographic symmetry.

Iterated function systems with symmetry in the hyperbolic plane

Abstract Images are created using probabilistic iterated function systems that involve both affine transformations of the plane and isometries of hyperbolic geometry. Figures of attractors with striking hyperbolic symmetry are the result.

Chaotic attractors with cyclic symmetry revisited

Abstract Chaotic attractors generated by the iteration of polynomial functions with cyclic symmetry have been the subject of recent study. A new formulation is investigated which generates cyclic symmetry using arbitrary functions. This allows the use of diverse function classes including functions with various types of singularities. The resulting images have significantly more diversity than those arising from polynomials, yet the cyclic symmetry of the chaotic attractor is preserved.

Chaotic attractors with the symmetry of a tetrahedron

Abstract Functions that are equivarian with respect to the symmetries of a tetrahedron are determined. Linear combinations of these functions that give rise to chaotic attractors are used to create images in 3-space of attractors with the symmetry of the tetrahedron. These attractors are visually appealing because of the tension between the pattern forced by their symmetry and the randomness arising from their chaotic behavior.

Chaos and Newton's method on systems

Abstract The chaotic behavior of multivariable Newtons method is explored. Initial conditions are varied by selection of complex parameters corresponding to screen position in two ways. Some examples illustrate that basins of attraction may be globally dominant in the multivariable case.