Shu-Tang Liu Professor and Tutor of doctoral student, School of Control Science and Engineering, Shandong University. The research areas are theory and applications of fractal control, nonlinear control, chaos and control, bifurcation control theory and applications, physical oceanography, modeling and analysis of marine ecology and marine environment, etc. One book has been published by Springer (Shu-Tang Liu and Pei Wang, Fractal control theory, Springer, 2018.5). He has published more than 150 papers in journals, such as Nonlinear Analysis: Real World Applications, IEEE Trans. Circuits and Systems, Fractal, IEEE Trans. on Nano Bioscience, IEEE Communications Letters, Complexity, Nonlinear Dynamics, International Journal of Bifurcation and Chaos, and so on. He has presided more than 10 national science projects, including one national key project of natural science foundation of China, one national natural science foundation of China and Shandong joint project, five natural science foundation of China, and two central military commission science and technology commission, national defense science and technology innovation zone military engineering projects, and two natural science foundation of Shandong Province. In 2004, he won the national award for 100 excellent doctoral dissertations.

• Introduction.- Preliminaries.- Universal Mathematical Model of Mesoscale Eddy.- Semi-stable Limit Cycle in Mathematical Model of Mesoscale Eddy.- Semi-stable Limit Cycles and Mesoscale Eddies.- Example Verification.- Spatiotemporal Structure of Mesoscale Eddies: Self-similar Fractal Behavior.- Mesoscale Eddies: Disc and Columnar Shapes.- Fractal Analysis and Prediction of Mesoscale Eddy Spatiotemporal Complexity.- Nonlinear Characteristics of Universal Mathematical Model of Mesoscale Eddy.- Same Solution between Momentum Balance Equations and Mesoscale Eddies.- Momentum Balance Equation Based on Truncation Function and Mathematical Model of Mesoscale Eddies.- Interpolation Prediction of Mesoscale Eddies.- Random Elliptic Curve and Brownian Motion Trajectory of Mesoscale Eddy.- Mathematical Model for Edge Waves of Mesoscale Eddies and Its Spatio-temporal Fractal Structures.

Shu-Tang Liu Professor and Tutor of doctoral student, School of Control Science and Engineering, Shandong University. The research areas are theory and applications of fractal control, nonlinear control, chaos and control, bifurcation control theory and applications, physical oceanography, modeling and analysis of marine ecology and marine environment, etc. One book has been published by Springer (Shu-Tang Liu and Pei Wang, Fractal control theory, Springer, 2018.5). He has published more than 150 papers in journals, such as Nonlinear Analysis: Real World Applications, IEEE Trans. Circuits and Systems, Fractal, IEEE Trans. on Nano Bioscience, IEEE Communications Letters, Complexity, Nonlinear Dynamics, International Journal of Bifurcation and Chaos, and so on. He has presided more than 10 national science projects, including one national key project of natural science foundation of China, one national natural science foundation of China and Shandong joint project, five natural science foundation of China, and two central military commission science and technology commission, national defense science and technology innovation zone military engineering projects, and two natural science foundation of Shandong Province. In 2004, he won the national award for 100 excellent doctoral dissertations.

• Introduction.- Preliminaries.- Universal Mathematical Model of Mesoscale Eddy.- Semi-stable Limit Cycle in Mathematical Model of Mesoscale Eddy.- Semi-stable Limit Cycles and Mesoscale Eddies.- Example Verification.- Spatiotemporal Structure of Mesoscale Eddies: Self-similar Fractal Behavior.- Mesoscale Eddies: Disc and Columnar Shapes.- Fractal Analysis and Prediction of Mesoscale Eddy Spatiotemporal Complexity.- Nonlinear Characteristics of Universal Mathematical Model of Mesoscale Eddy.- Same Solution between Momentum Balance Equations and Mesoscale Eddies.- Momentum Balance Equation Based on Truncation Function and Mathematical Model of Mesoscale Eddies.- Interpolation Prediction of Mesoscale Eddies.- Random Elliptic Curve and Brownian Motion Trajectory of Mesoscale Eddy.- Mathematical Model for Edge Waves of Mesoscale Eddies and Its Spatio-temporal Fractal Structures.