Bonnie Steves

Generalization of the Mandelbrot set: Quaternionic quadratic maps

Abstract The iterated map Q → Q2 + C, where Q and C are complex 2 × 2 matrices representing quaternions, provides a natural generalisation of the Mandelbrot set to higher dimensions. Using the well-known expansion of the quaternion in terms of the generators of SU(2), the Pauli matrices, it is shown that the fixed point Q = Q2 + C is stable for C inside a cardioidal surface M3 in R 4 and the boundary set ∂M3 sprouts domains of stability of multiple cycles. Stability calculations up to 3-cycle leading to explicit expressions for the associated Mandelbrot domain in R 4 are presented here for the first time. These analyses lay down the theoretical frame work for characterizing the stability domain for general k-cycles.

Generalized Mandelbrot sets for meromorphic complex and quaternionic maps

The concepts of the Mandelbrot set and the definition of the stability regions of cycles for rational maps require careful investigation. The standard definition of the Mandelbrot set for the map f : z → z2+ c (the set of c values for which the iteration of the critical point at 0 remains bounded) is inappropriate for meromorphic maps such as the inverse square map. The notion of cycle sets, introduced by Brooks and Matelski [1978] for the quadratic map and applied to meromorphic maps by Yin [1994], facilitates a precise definition of the Mandelbrot parameter space for these maps. Close scrutiny of the cycle sets of these maps reveals generic fractal structures, echoing many of the features of the Mandelbrot set. Computer representations confirm these features and allow the dynamical comparison with the Mandelbrot set. In the parameter space, a purely algebraic result locates the stability regions of the cycles as the zeros of characteristic polynomials. These maps are generalized to quaternions. The powerful theoretical support that exists for complex maps is not generally available for quaternions. However, it is possible to construct and analyze cycle sets for a class of quaternionic rational maps (QRM). Three-dimensional sections of the cycle sets of QRM are nontrivial extensions of the cycle sets of complex maps, while sharing many of their features.

The Caledonian Symmetrical Double Binary Four-Body Problem I: Surfaces of Zero-Velocity Using the Energy Integral

The Caledonian four-body problem introduced in a recent paper by the authors is reduced to its simplest form, namely the symmetrical, four body double binary problem, by employing all possible symmetries. The problem is three-dimensional and involves initially two binaries, each binary having unequal masses but the same two masses as the other binary. It is shown that the simplicity of the model enables zero-velocity surfaces to be found from the energy integral and expressed in a three dimensional space in terms of three distances r 1, r 2, and r 12, where r 1 and r 2 are the distances of two bodies which form an initial binary from the four body system’s centre of mass and r 12 is the separation between the two bodies.

The hierarchical stability of quadruple stellar and planetary systems using the Caledonian Symmetric Four-Body Model

The hierarchical stability and evolution of a symmetrically restricted four-body model called the Caledonian Symmetric Four-Body Problem (CSFBP) is studied. This problem has two dynamically symmetric pairs of masses m and M with mass ratio µ = m . The analytical stability criterion derived for the CSFBP by Steves & Roy (2001) is verified numeri- cally for the coplanar case. It is shown numerically that there exists a direct relationship between the hierarchical stability of the system and the Szebehely constant C0, a combination of the total energy and angular momentum of the system. For C0 > Ccrit2, where Ccrit2 is a critical value dependent only on µ, the system undergoes no change in its hierarchical arrangement, and is therefore considered to be hierarchically stable. It is also shown that for large mass ratios µ the double binary configurations are the dominant hierarchical configuration, while for smaller mass ratios µ it is the configuration containing a single binary with two outer bodies that is the dominant configuration.

Numerical escape criteria for a symmetric four-body model

Numerical escape criteria is presented for the Caledonian Symmetric Four-Body problem (CSFBP). The numerical experiments show that escapes can be detected very early with the help of the method. Integrating a huge amount of orbits of the symmetric four-body system we found that for the equal mass case the double binary escape, and in the planetary case the single bodies escape are the most likely outcome of the disintegration of the system.

Some Equal Mass Four-Body Equilibrium Configurations: Linear Stability Analysis

A linear stability analysis of the coplanar, rigid motion of 4 equal masses about their centre of mass is presented. We consider cases where particles are located at (A) the centre of a square, (B) the vertices and centroid of an equilateral triangle and (C) designated points on a line. The variational equations are analysed in the rotating frame. These equations decompose into invariant subspaces of perturbation evolution in the orbital plane and in the normal direction. All the three cases, A, B and C are linearly unstable. It is worth pointing out that, for cases A and B, perturbations normal to the orbital plane have amplitudes which grow as a power of time t.

Quaternionic Generalisation of the Mandelbrot Set

The complex quadratic map leading to the celebrated Mandelbrot set is generalised in a natural way to R4 using the framework of quaternions. Calculations are presented with the aid of Pauli spin matrices. Stability analyses of 3-cycles are summarised, with explicit expressions for generalised Mandelbrot domains in R4. It is conjectured that similar expressions hold for stability domains of k-cycles.