Research On The Construction And Fractal Structures Of Quaternion M-J Sets

Mandelbrot sets(abbreviated as M sets) and Julia sets are classic fractal sets in fractals. People have made deep investigation on M sets and Julia sets utilizing the computation and simulation of computer,which realize many supposes of dynamic systems.These researches involve many interdisciplinary subjects in science,such as Klein group,quasi-conformal mappings,Teichm(u|¨)ller spaces theory,analysissitus,complex analysis,numerical computation method,ergodic theory and symbolic dynamics,etc.At present high-dimensional fractal is a focus in fractal field.However,due to the complexity of high-dimension space and difficulty in shape representation,high-dimensional fractal still has much work to be done.The disseriation constructs the general M sets and Julia sets on the quatemion mapping f:z←z~α+c(α∈Z),researches their properties and explores the fission evolution rules of quaternion M-J sets.The disseriation describes the general hypercomplex M sets and Julia sets utilizing parametric n-dimensional L systems.The algorithm and the fractal images produced are presented,and the dynamic properties of the general quatemion M-J sets are discussed.It is found that the quatemion M sets have special topology structures and quaternion Julia sets have invariability in the selection of parameter,which extends the conclusion of Bogush.It can be concluded that n-dimensional L systems,which have pithily alphabet but convey plentiful information,could depict such fractals as general hypercomplex M-J sets well.The disseriation presents the 3-D projections of general quaternion M-J sets and constructs the cycle regions of quaternion M-J sets utilizing the time-escape algorithm or Lyapunov exponent method combining cycle detecting method.The boundary of the cycle region is calculated,and their dynamic characters are theoretically analyzed.The quaternion Julia sets are constructed with the parameter selected from quatemion M sets,which establishes relationship between the whole portray of the quaternion Julia sets and the point coordinates of the quaternion M sets qualitatively.The disseriation constructs the quaternion M sets on the mapping f:z←z~2+c utilizing the time-escape algorithm combining cycle detecting method,which has multiple critical points.The new M sets' topology structures and fission evolution rules are investigated;the center positions and the boundaries of the cycle regions are calculated.An improved time-escape algorithm is presented,by which the quatemion M sets are constructed.It can be observed that the center of the cycle region appear displacement(even differentiation) under different critical point.The bifurcation diagrams are constructed and the box-dimension is calculated for observing the changes which take place on the M sets.Plenty of experiments show that the collection of the quaternion M sets with different critical points make up of the complete M sets on the mapping f:z←z~2+c.The disseriation constructs the quaternion M sets perturbed by the dynamic and output noises and analyses the topology structures affected by the noises.The cycle regions and bifurcation diagrams are constructed for observing the influence which the noises bring to the M sets.The experimental results show that the additive dynamic noise does not change the M sets essentially,which effects displacements of the stability regions.The multiplicative dynamic noise makes the stability regions of the M sets shrink acutely with the proportion determined by the strength of the noise and the perturbed M sets keep symmetrical around the real axis.The output noise does not change the area of the stability regions of the M sets but affects the inner structure substantially.Both the additive and multiplicative output noises bring the absence of the stability areas.The M sets with multiplicative output noises keep symmetric around the real axis while the additive output noises destroy the M sets' symmetries completely.