Chaotic phenomenon widely exists in the nonlinear dynamic systems. Such phenomenon is described to be highly related the initial conditions and intrinsic system properties. Traditionally, large effort has been put on evaluation of the existence of chaos, such as Lyapunov exponent or power spectrum methods. In addition, iteration methods are usually employed to explore the states of system. In this work, we will study the chaotic system in a different point of view. The system properties of physics quantities will be considered firstly. Starting from these physics quantities, it is expected to potentially determine the final states of the system. Then a relationship between the initial conditions and final states will be established. The objective of this work is to preliminary study the feasibility of employing such method for predetermine the final state of reappear initial conditions. The main approach is building a strong relationship between each initial conditions and final states within limited computation. Once the relationship is established, we could know the system states distribution and achieve the exploitation on the system. In this work, we will focus on the deterministic chaotic systems that the final states are deterministic and non-periodic.
In order to perform the proof-of-concept, a typical chaotic system, the magnetic pendulum system, will be employed for demonstration. A program written in Java will display the pendulum movement in real-time and also the basin diagram that shows the relationship between initial conditions and final states. All of code for this program are constructed from scratch.
After the implementation of our program, we will compare results from it and that of the separate numerical simulation of the same system. The numerical simulation will be performed through the commercially available software Mathematica. Further investigation will also be discussed with the results from numerical simulation.
Three conclusions could draw from this work. The first is a proof-of-concept has been established for the opportunity to develop a new approach to study the deterministic chaotic system based on the physical properties methods. The second is the relationship between the initial condition and final states are proved to be work in states predetermination. The third is that relationship between the intrinsic system parameter and the final system states distributions has been found, which could be a guiding line for the similar deterministic chaotic system study in the future. It is also worthy to mention that herein the magnetic pendulum is only an example to testify our approach. And this approach should also be valid for other general deterministic chaotic systems.
In this work, we only perform the preliminary study on this approach. In the future, our study will be extended to more systems and even the system in real world. It is expected such approach would build a new viewpoint on understanding the chaotic system, and a potentially new method to understand data
