| Fractal belongs to the research field of nonlinear science.On the basis of traditional geometry,fractal provides a mathematical tool to recognize the seemingly disorderly geometric objects that exist widely in nature.Chaos theory is also an important branch of nonlinear science,the study of fractal and chaos has long been mature,but few scholars try to use fractal thought to design chaotic systems with rich dynamic behavior.On the one hand,multi-scroll attractors usually have better randomness and wider key space,Therefore,it is a practical work to generate complex multi-scroll attractors with a relatively simple mathematical model.On the other hand,the multi-stability behavior of nonlinear systems can adjust the amplitude of chaotic signal losslessly or perform fast image processing,and it also has a certain application prospect.At present,there is few general design idea to construct a chaotic system with the above-mentioned rich dynamic behavior,which usually requires complicated derivation.Since fractals are good at describing complex features,combining fractals with chaos to design the desired chaotic system is a feasible approach.This thesis focus on designing complex fractal-inspired chaotic systems,and applying various fractal processes in chaotic systems to generate multi-scroll attractors with different topologies and a number of controllable coexisting attractors respectively.The main works of this thesis are as follows:(1)Applying two different complex-valued fractal processes to chaotic systems,the complexity of chaotic systems is effectively increased.Two different fractal processes are combined to construct different systems of fractal processes to generate attractors with complex structures such as separation and nesting.The relationship between attractor separation behavior and parameters is also discussed.The spectral entropy complexity of the multi-ring attractor is compared with other multi-scroll attractors,the result indicates that the sequence of fractal-processed attractors has a higher complexity.An algorithm for image encryption using separated attractors is proposed,the encryption results have proved that fractal-based attractors have better performance in the field of information encryption.Finally,a microcontroller-based hardware platform is developed to verify the physical existence of the separated attractor and the multi-ring attractor.(2)The quaternion fractal process is derived based on the quaternion Julia fractal,so that the number of coexisting attractors can be flexibly controlled by parameter.The quaternion fractal process is employed to a chaotic system without coexisting attractors,corresponding fractal-processed system has bistability behavior.The quaternion fractalbased system is analyzed numerically by means of Lyapunov exponents,bifurcation diagrams,and the basins of attraction,and then the chaotic system is combined with a higherorder Julia fractal to obtain coexisting attractors in arbitrary number.Then,in order to enhance the applicability of the quaternion fractal process,a general form of the method is given based on the imaginary part expansion.Meanwhile,a specified number of coexisting attractors is implemented by a microcontroller,which verifies the feasibility of applying the quaternion fractal process to the engineering field.(3)The fractal process is combined by the system of fractal processes,resulting in a more comprehensive adjustment of attractors.Several typical system of fractal process are designed,one of which can control the number of scrolls and coexisting attractors at the same time.The other can generate lattice-like coexisting attractors of specified size by independent parameters.The system of fractal process provides new ideas for the chaotic systems design. |
