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A new concept named volatility monotonous persistence duration (VMPD) dynamics is introduced into the research of energy markets, in an attempt to describe nonlinear fluctuation behaviors from a new perspective. The VMPD sequence unites the maximum fluctuation difference and the continuous variation length, which is regarded as a novel indicator to evaluate risks and optimize portfolios. Further, two main aspects of statistical and nonlinear empirical research on the energy VMPD sequence are observed: probability distribution and autocorrelation behavior. Moreover, a new nonlinear method named the cross complexity-invariant distance (CID) FuzzyEn (CCF) which is composed of cross-fuzzy entropy and complexity-invariant distance is firstly proposed to study the complexity synchronization properties of returns and VMPD series for seven representative energy items. We also apply the ensemble empirical mode decomposition (EEMD) to resolve returns and VMPD sequence into the intrinsic mode functions, and the degree that they follow the synchronization features of the initial sequence is investigated.

Based on the multi-color contact process and the oriented percolation, this paper develops a novel statistical physics financial price model with jumps to simulate and understand the price fluctuation characteristics in the real market. In this model, the interaction between investors and the dynamic changes of their investment attitudes and strategies are simulated by the complex interactions between different states of particles in the multi-color contact process. This is considered to lead to the rise and fall of the prices of financial products and affect the financial market price fluctuations. In addition, the herd behavior of investors facing sudden information outside the market also leads to dramatic price jumps. The model uses the oriented percolation combined with the Poisson process to describe those sudden fluctuations, in which the magnitude of the price jump of financial product is reconstructed by the clusters of partial interactions in the system. Therefore, the model constructs a total price process by considering both normal price fluctuations caused by the interaction of investor attitudes and sudden price jump fluctuations caused by the herd behavior. Based on the proposed model, this paper selects different parameter sets to generate the simulated return series and studies some basic descriptive statistics, stylized facts, multi-fractal property and complexity behavior of the simulated return series. To verify the rationality of the model, this paper conducts the same analysis and comparison on the return series of the real financial market. The outcomes of analysis indicate that the model proposed in this paper can reproduce the main fluctuation characteristics in the real financial market to a certain extent.