Synchronization Dynamics Of Conjectural Variation Duopoly Models With Bounded Rationality

The dynamical duopoly game models with bounded rationality based on conjectural variation are researched.Based on the symmetry of the models,the complicated synchronous dynamical behaviors of the systems caused by the change of adjustment speed parameter,gradient adjustment parameter and the measurement of price parameter are studied.Combining with the oligarch game theory,nonlinear dynamics theory and dynamical economic theory,the dynamical phenomena of synchronization,on-off intermittency,coexistence of attractors and global bifurcation are researched from theoretical analysis and numerical simulation.The main contents of the paper are as follows:1.The fixed points and their local asymptotic stability of duopoly models are studied through the theoretical analysis.Based on the symmetry of the models,the invariant sets in the models are given,and the transverse stability of the models restricted on the invariant sets is studied.And the condition of transversally attracting in the systems at the Nash equilibrium point is given.2.In the parametric bifurcation diagrams,when the other parameters are fixed,it is studied that the routes to chaos are different,the stability region of Nash equilibrium point changes and the systems also present the change from cycle to chaos,with the increase of the adjustment speed parameter and the gradient adjustment parameter in the systems through the numerical simulation.3.When the adjustment speed parameter and gradient adjustment parameter change,the Milnor attractor appears on the diagonal.At this time,the attractor is no longer stable in the sense of transverse Lyapunov,but the stability in the sense of Milnor.And the phenomenon of synchronization also occurs.When the transverse stability is considered,synchronization occurs along the invariant sets accompanied by the on-off intermittency.When on-off intermittency occurs,the trajectory is a synchronous trajectory or converges to other different attractors.By characterizing the transverse Lyapunov exponents,the conditions for the occurrence of synchronous dynamics in the system are studied.And the time series diagram is used to simulate the on-off intermittency when synchronization occurs in the system.When the adjustment speed parameter varies,the attractor on the diagonal is coexisting with the attractors off the diagonal,and at the same time the phenomenon of the multi-stability occurs.With the increase of the adjustment speed,the range of attractor changes by describing the structure of the basins of attraction.The changes of the structure of the attractor and the basins of attraction indicate the complex dynamics of the system with the variety of parameter.4.By studying the critical curves,the noninvertibility property of the map in the model is characterized.When the parameter about the measurement of price changes in a certain range,the chaotic attractor is on the diagonal,which is Milnor attractor.There are also attractors off the diagonal,which shows the coexistence of multiple attractors occurs with the phenomenon of synchronization.The coexistence of attractors means the phenomenon of multi-stability.With the change of parameters,the attractor also presents the evolution between period and chaos,and the basins of attraction on the diagonal and off the diagonal also change correspondingly.The global bifurcation leads to the change of the structure of the basins of attraction.Combining with the critical curves,the global bifurcation of the system is researched.