A Theoretical Study of Chaos in Higher Dimensions

We use the computational tools available to us to explore the possibility of creating chaos in four dimensions and in higher dimensions. Even basic physics leads to unexpected results, when we go beyond 3-dimensional space. In four dimensions, we can create Klein bottles, can tape the edges of two Möbius strips together, and can invent sailing knots of stunning complexities: all of the knots that work in 3 dimensions will fall apart in higher dimensions. Our calculations for gravitational force reveal that the force drops as 1/(R^D – 1) , where D is the dimension and R is the distance between the interacting objects. We first review generalizations of the Li and Yorke definition of chaos to difference equations in Rⁿ, and the higher dimensional conditions leading to the existence of chaos. Then we consider many 3-D, 4-D, 5-D, and 6-D Generalized Henon Maps (GHMs). We look for fixed points that are locally stable. We find that for many values of the parameter α, chaotic behavior exists in dimensions D = 4, 5 and 6. We also discuss the possibility of uncovering the existence of some of the higher dimensions, by delineating the 3-D projections of chaotic behavior from higher dimensions.