Discontinuous bifurcation and coexistence of attractors in a piecewise linear map with a gap

Coexistence of attractors with striking characteristics is observed in this work, where a stable period-5 attractor coexists successively with chaotic band-ll, period-6, chaotic band-12 and band-6 attractors. They are induced by dif- ferent mechanisms due to the interaction between the discontinuity and the non-invertibility. A characteristic boundary collision bifurcation, is observed. The critical conditions are obtained both analytically and numerically.

Dynamic analysis,FPGA implementation,and cryptographic application of an autonomous 5D chaotic system with offset boosting

An autonomous five-dimensional(5D)system with offset boosting is constructed by modifying the well-known three-dimensional autonomous Liu and Chen system.Equilibrium points of the proposed autonomous 5D system are found and its stability is analyzed.The proposed system includes Hopf bifurcation,periodic attractors,quasi-periodic attractors,a one-scroll chaotic attractor,a double-scroll chaotic attractor,coexisting attractors,the bistability phenomenon,offset boosting with partial amplitude control,reverse period-doubling,and an intermittency route to chaos.Using a field programmable gate array(FPGA),the proposed autonomous 5D system is implemented and the phase portraits are presented to check the numerical simulation results.The chaotic attractors and coexistence of the attractors generated by the FPGA implementation of the proposed system have good qualitative agreement with those found during the numerical simulation.Finally,a sound data encryption and communication system based on the proposed autonomous 5D chaotic system is designed and illustrated through a numerical example.

Complex Dynamical Behaviors in a Dual Channel Supply Chain Model with Retailer's Service Input

Based on bounded rationality,this paper established a price game model of dual channel supply chain composed of manufacturers and retailers.According to the eigenvalue of Jacobi matrix and Jury criterion,the stability of the equilibrium point is analyzed,and then the dynamic evolution process under the parameters of price adjustment speed and retailer's service input is studied through stability region,bifurcation diagram,maximum Lyapunov exponent diagram and attraction basin.The results show that the system enters chaos through flip and Neimark-Sacker bifurcation,and increase of price adjustment speed and service input value will make the system produce more dynamic behavior.In addition,it can be found that the impact of service input value on itself is much greater than that on manufacturers.Secondly,when adjustment speed is selected as bifurcation parameter,the change curve of sales price is inconsistent,in which the change of retailers mainly remains in periodic state,while manufacturers will gradually enter chaos.Finally,studies the evolution of attraction basin in which three kinds of attractors coexist.In particular,coexistence of boundary attractors and internal attractors increases the complexity of system.Therefore,enterprises need to carefully adjust parameters of the game model to control the stability of system and maintain the long-term stability of market competition.