Analysis Of Dynamic System Properties Based On Bifurcation Theory And Its Application In Image Processing

Bifurcation is an important nonlinear phenomenon in the field of dynamic system research,and the study of bifurcation theory has a broad practical background and significant theoretical significance.In this thesis,the dynamic properties of the delay feedforward neural network model and the reaction-diffusion FHN model are studied respectively,including stability,Hopf bifurcation,Hopf pitchfork bifurcation and 1:1 resonance Hopf bifurcation,and an image enhancement algorithm and an edge detection algorithm are constructed by using the bifurcation properties of the neural network model.For the phenomenon of "tiger with stripes and leopard with spots" in nature,in this thesis the causes of animal body pattern images are explained by analyzing the Turing’s patterns map of the corresponding reaction-diffusion model.The mathematical mechanism of using the bifurcation theory of dynamical systems for image processing is that the neuron oscillator that outputs the rhythm signal has the function of signal amplification,as well as the effect of diffusion on the reaction substrate,which is reflected in the model,namely the multiple period solution of the feedforward dynamical system and the Turing bifurcation of the reaction diffusion system.The main research contents and methods of this paper are as follows:(1)Based on the effect of delay in neuronal signal transmission,feedforward neural network models with two,three and six neuronal oscillators are constructed in this paper,and the existence of periodic solutions and the normal forms of the system on the central manifold are given by studying various bifurcation properties of the model.Based on the analysis of these bifurcation properties,the important properties of the feedforward coupled oscillator for stepwise amplification of signals and important indicators for characterizing the amplitude enhancement factor are obtained.Using the pixel values of the image to be processed as the initial values,the effect of image contrast enhancement can be achieved through the action of a feedforward neuron oscillator.Quantitative analysis and comparison with other image enhancement algorithms demonstrate the advantages of this method.(2)The application of reaction diffusion equation to edge detection is an important method in image processing.In this thesis,we avoid the complicated mathematical calculation in the traditional model and construct a reaction-diffusion FHN model with constant single diffusion and double diffusion coefficients.We also analyze the Turing instability,Turing bifurcation of the model,and prove that single coefficient diffusion does not induce a Turing pattern.Based on the results of the theoretical analysis,a new algorithm for image edge detection with adaptive thresholding is constructed.(3)The variously shaped markings and color patches on the animal body constitute a colorful biological world.Its mathematical mechanism is a pattern formed by the mutual coupling and diffusion of a pair of chemical substances "formins".In this paper,a coupled reaction diffusion system is established,and the existence of Turing bifurcation and Turing-Hopf bifurcation of the system are analyzed.The influence of time delay on pattern generation is discussed.Through simulation in different planar regions,the conditions for animals to produce different types of stripes are discussed.By constructing a dynamic system model,we successfully applies bifurcation theory to image processing,propose new algorithms for image enhancement and image edge detection,conduct experiments on image processing issues in wild Amur tiger and larch plantation sample plots,and clarify the causes of animal speckle.The research results of this paper expand the application of dynamic system bifurcation theory and provide new ideas for image processing theory.