Fractal Analysis Of Controlled M-J Sets Topology Structure Under Noise Perturbations

Fractal is a very active branch in the field of nonlinear scientific research.It is essentially a new world ideology and methodology.It also reveals the unity of order and disorder,the unity of determinism and randomness.It has developed rapidly,and has become an effective mathematical tool and method.It has been widely used in fields such as biology,earth science,medicine,mathematics,philosophy,oil exploration,chemical production,and arts,which has gained huge value.The M-J set produced by the complex system is the most classic two types of fractal sets.It has been the research hotspot of scholars in the field of fractal,and has achieved quite rich results.There are a wide variety of noise perturbations in nature,which have various effects on the movement and state of things.As a random variable,noise affects the evolution of a dynamic system in two ways.One is to add a random variable outside the system,called additive dynamic noise;the other is to affect the dynamic equations inside the system directly,called multiplicative dynamic noise.Random numbers can be divided into two categories according to the way they are generated.True random numbers generated by physical methods and pseudo-random numbers generated by mathematical algorithms.People have established mathematical models such as Gaussian noise,sine noise,and time-delay systems to simulate noise perturbations characteristics for dynamic system evolution.This paper mainly studies the noise perturbations added by the M-J set in a controlled state,and analyzes the change of its topological structure.The major contents as follows:1.Fractal analysis of the topological structure of Mandelbrot set and Julia set with noise perturbations under controlled state.In order to analyze and observe the changes in the shape of Mandelbrot set and Julia set,this paper introduces the two tools of deviation distance and deviation plot into the controlled Mandelbrot set and Julia set.Random numbers are used to simulate the system under noise perturbations.From the deviation distance and deviation plot,it can be seen that compared with the original state,under the controlled state,the shape change of the Mandelbrot set and the Julia set is small,and the stability is significantly improved.Forever,the size of the deformation of Mandelbrot set and Julia set is quantified,and the position of the deformation of Mandelbrot set and Julia set is clearly positioned.In addition,additive dynamic noise and multiplicative dynamic noise have different damage strengths to Mandelbrot and Julia sets.For simulation,it can be seen that multiplicative dynamic noise is more destructive.2.As the noise intensity changes,the deviation distance will change,and the two will have a positive correlation.Therefore,in this paper,Julia deviation distance is used as a function of the noise intensity,and the additive dynamic noise and multiplicative dynamic noise under controlled state are obtained.The Julia deviation distance function also shows that the multiplicative dynamic noise is more destructive,and after a certain critical value,Julia deviation distance remains constant and the critical noise intensity value is obtained.In addition,when calculating Julia deviation distance and making Julia deviation plot,the division of spatial lattice is a fixed matrix.Therefore,by comparing different lattice matrix divisions,it can be concluded that the critical noise intensity value is independent of the number of lattice points,and the ratio of the number of points in the Julia set to the number of lattice points calculated in the Julia set is independent of the number of lattice points.