Complexity Analysis of a Four-Dimensional Energy-Economy-Environment Dynamic System

Abstract

A novel four-dimensional energy, economic, and environmental (3E) under energy reduction constraints chaotic system is proposed. The acquisition of environmental quality data is the key to this paper. During the course of the study, we used Bayesian estimation algorithm to calibrate the environmental quality. Based on the official data, the Levenberg–Marquardt backpropagation neural network method was optimized by genetic algorithm to effectively identify the parameters in the 3E system. The research results show that although increasing energy reduction inputs can improve environmental quality, the effect on energy intensity and overall stability of the system is not obvious. When polluting input in the ecological environment system affects its maximum capacity, the environmental system collapses (i.e., the ecological system can no longer purify the environment through the self-circulation process and will eventually die out). Therefore, it is necessary to correctly grasp the ecological environment protection and the relationship between economic developments and explore synergies to promote ecological priorities and green development new ideas.

Fractional-Order Financial System and Fixed-Time Synchronization

“…where q ∈ (0, 1] denotes the order of the derivative. Evidently, system (14) degenerates to model ( 13) when q = 1.…”

“…Here, we let the system (14) parameter a 2 = a 3 = 0.1, and the present work focuses on the impact of the saving quantity on the financial system (14); therefore, we choose different values of parameter a 1 and order q in the equations to perform parameter sensitivity analysis. An approximate solution to the fractional-order financial system (14) can be expressed as follows:…”

“…In this section, the numerical solutions given by ADM of fractional-order financial system ( 14) are obtained, then the sensitivity analysis of parameter a 1 and the fractional-order q is conducted to study their influence and effectiveness in relation to the financial system (14). Bifurcation diagrams, maximum Lyapunov exponent diagrams, complexity diagrams, phase portraits, p − s plots and time series are drawn to show the rich dynamic behaviors, including periodic and chaotic motions.…”

“…In order to study the influence of parameter a 1 on the dynamic behaviors of the financial system ( 14), we take the parameter values a 2 = a 3 = 0.1, fractional-order q = 0.95 and initial state (x 0 , y 0 , z 0 )= (1, 3, 2), and choose parameter a 1 as the critical variable. Figure 1 is the bifurcation diagram, which shows rich dynamic behaviors as a 1 changes within the interval [0, 1]; the diagram implies that system (14) shows inverse perioddoubling bifurcation, that is, the system goes from periodic states to chaotic states with the period-doubling bifurcation process, as the parameter a 1 decreases within the interval [0, 1]. Specifically, the system is in period one when a 1 ∈ [0.9, 1], and then period-doubling bifurcation occurs for a 1 = 0.9; period-two appears when a 1 ∈ [0.83, 0.9), and perioddoubling bifurcation occurs for a 1 = 0.83.…”

“…Therefore, it is necessary to investigate the dynamic characteristics and analyze chaotic effects of such complex financial systems in depth. In order to accurately grasp the operation laws of financial systems, researchers [10][11][12][13][14][15][16] have established various integer-order financial models to study the complex dynamic behaviors of the economy and society, revealing the intrinsic characteristics of economic development. However, studies have shown that economics and finance are extremely complex nonlinear systems involving many subjective factors, and there are many characteristics that cannot be described by the theory of integer calculus.…”