| The world’s multi-polarization and economic globalization supply conditions for the maturity of the international financial market. Besides the benefits, the international financial markets expand in aspects of speed, depth and breadth unprecedentedly with the help of advanced development in both communications and information technology. However, with the rapid development of economy and increasingly boom of financial markets, potential financial risks of globalization also deepen continuously. In the recent 100 years, financial crisis has erupted many times all over the world, which strongly hit the country’s economy. Meanwhile, under the background of economic globalization, financial crisis exposed multiple effects and problems in the international financial market, which encouraged many countries to re-examine the construction and reform of the financial markets. Government and academic circles also began to focus on the outbreak and contagion of financial crisis. To analysis the formation and transmission process of financial crisis itself has a strong economic theory significance and also provide a scientific basis for real financial contagion.Financial crisis is a deviant phenomenon of economic system. As the outburst of crisis is difficult to predict, and the process of the contagion is unable to recreate, there are still a lot of complicated issues to solve.With the development of related theories and complexity science methods, the understanding of the complexity of financial system becomes clear gradually. Besides, new methods that are developed in recent years have been gradually introduced into the financial markets and financial phenomenon in the study. Compared to traditional economics analysis framework and models, understanding phenomena in financial markets and depicting complexity and dynamic change process from the perspective of complex science are closer to the nature of financial system.First of all, this paper describes the overall thinking of financial contagion process by using nonlinear dynamics theory. From the spatial respective, this article defines financial contagion as geography cross-border contagion and each country’s financial system is regarded as a relatively independent economy. Each country will face equivalent risk, law, and the same objective environmental conditions worldwide. The relationship between countries is reflected through the interaction of various systems. The financial crisis is realized through the coupling of various systems. From the microscopic point of view, the outbreak and contagion of financial crisis are related to the characteristics of financial market participants, the characteristics of financial system, regulatory measures, policy environment and other factorsSecondly, the chaotic characteristics of financial time series in the financial contagion process are tested and the process is divided into stages. Because of complexity and nonlinear of the process of financial contagion, the financial system’s nonlinear dynamic state should be changed accordingly in the different stages of financial contagion. Thus, the infection stage can be quantitatively measured by depicting the change and combine the characteristics of financial contagion process. This article selects maximum Lyapunov index as observation index to test the change nonlinear dynamic characteristics of financial markets during the process of financial contagion. In combination with endogenous structure mutation model, this article test whether the structure mutation of time series of the whole financial contagion happened and calculate the point. This result is used as the basis for quantitative division stage of financial contagion and compared to partitioning results of endogenous prior to improvement structure mutation model and results of qualitative classification stages.Thirdly, in the space dimension, this paper uses the concept of nonlinear dynamic system theory and generalized synchronization to further study financial contagion from the perspective of two markets and multiple markets.Cross market contagion research takes source country as a drive system and infected country is considered as a response system. Based on the framework of dynamic generalized synchronization and nonlinear dynamic similarity method, financial contagion problems are analyzed and measured. From the perspective of nonlinear dynamics system, the concept of financial contagion in the market can be portrayed as constantly closing in nonlinear dynamic similarity degree between infected countries and original countries based on the concept of complex system generalized synchronization. In other words, when the financial crisis spread from one country to another country, synchronization degree between infected country system and original country system would be a change from low to high. Although this definition with limited conditions is very broad, but the time window of financial contagion can be captured. At the same time, the measurement of crisis contagion between two markets will become the basis for study of contagion process in multiple markets.Real financial contagion is not limited to single direction transmission between two countries, especially under the background of global financial integration and regional economic integration. The spread of financial information and quick transfer of assets accelerate contagion of financial crisis, and make infectious process more complexity. Meanwhile, cross-infections between markets are obvious. Thus, this article uses correlation matrix analysis of multi-channel signal processing method to carry on quantitative research on correlations between multiple markets in the process of financial contagion, on the basis of two countries correlation. Correlation matrix is constituted by nonlinear dynamic similarity index between two markets. Firstly, this article creates alternative data to compare with correlation matrix eigenvalue and eigenvector of original data and look for small synchronous cluster that may form in the process of financial contagion. Then, synchronization in index is defined to quantitatively present participation in synchronous cluster of each market.Finally, the 2007-2009 global financial crisis is presented as empirical study and 10 global stock markets are chosen and the period of the times series has been separated into before and after crisis contagion. The empirical results show that, based on nonlinear theory and asset pricing model, characteristics of nonlinear complexity in the process of financial contagion is discussed according to studying foreign investment policy and cross-market flow firstly. Secondly, comparing to economic stability, nonlinear dynamic characteristics of financial markets is obvious during the process of financial contagion and fluctuation present in the different stages. Using endogenous structure mutation model that is improved by maximum Lyapunov index to divide financial contagion stage has stronger theoretical basis and persuasion. Thirdly, the result of using nonlinear dynamic similarity based on generalized synchronization of complex system to measure financial contagion between two markets is significantly, which gives full consideration to nonlinear complexity characteristics of financial system and contagion process. Finally, the results demonstrate that cross-link is wide spread based on system dynamic similarity between two market and research of multi-markets interaction. Besides, super position and obvious in local contagion phenomenon. Europe and Asia respectively forms synchronous cluster. Therefore, the framework and methods based on nonlinear dynamics are put forward by this article, and have obvious advantages in the research of financial contagion process. This attempt has advantages in the quantitative research field of multi markets financial contagion.
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