In this paper, the Adomian decomposition method (ADM) semi-analytical solution algorithm is applied to solve a fractional-order entanglement symmetrical chaotic system. The dynamics of the system are analyzed by the Lyapunov exponent spectrum, bifurcation diagrams, poincaré diagrams, and chaos diagrams. The results show that the systems have rich dynamics. Meanwhile, sliding mode synchronizations of fractional-order chaotic systems are investigated theoretically and numerically. The results show the effectiveness of the proposed method and potential application value of fractional-order systems.

Inventory management is complex nonlinear systems that are affected by various external factors, including course human action and policy. We study the inventory management model under special circumstances and analyse the equilibrium point of the system. The dynamics of the system is analysed by means of the eigenvalue trajectory, bifurcations, chaotic attractor, and largest Lyapunov exponent diagram. At the same time, according to the definition of fractional calculus, the fractional approximate entropy is used to analyse the system, and the results are consistent with those of the largest Lyapunov exponent diagram, which shows the effectiveness of this method.

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A new 3D offset-boostable symmetric system is proposed by an absolute value function introduced. The system seems to be more fragile and easier to the state of broken symmetry. Coexisting symmetric pairs of attractors get closer and closer, and finally get emerged together. Basins of attraction show how these coexisting attractors are arranged in phase space. All these coexisting attractors can be easily offset boosted in phase space by a single constant when the initial condition is revised accordingly. PSpice simulations prove all the phenomena.