Periodic Perturbation Of Vanderpol Equation Hopf And Its Application

One of the elements of discussing small periodically perturbed differential dynamic systems with delay is the nature of equations which are undergoing hopf bifurcation when the external periodically perturb equations with time delay.In this paper, small periodically perturbed hopf bifurcation to the generalized vanderpol equation with time delay is researched.The problem of periodically perturbed hopf bifurcation to narrowly vanderpol equation with delay has been discussed by Yue Xiting. On that basis, periodically perturbed hopf bifurcation to generalized vanderpol equation with time delay is discussed in the paper.Firstly the error of the formula put forward in researching vanderpol equation’s hopf bifurcation with delay by Yue Xiting in 1992 which is unrecognized for a long time is noted.Then the correct conclusion is proofed. Secondly periodically perturbed hopf bifurcation to generalized vanderpol equation with time delay is explored.The results show that there is still subharmonic branch,and when parameters met certain conditions the hopf bifurcation is stable.So we promote the conclusion researched by Yue Xiting.The idea to research this paper is transforming the ordinary differential equations to functional differential equations and then decomposing the spacean,and transforming the problem of hopf bifurcation to the discussion of non-trivial constant solution by integral average theorem.Li Li applid vanderpol equation without varying to psychology for simulating human’s emotional changes. On that base, this paper considers the impact of the periodical external environment.Firstly, emotional model of vanderpol equation with periodical perturbation is established.Then,the periodical solution is proved when the original value of the model is determined.Results of simulation show that:the normal emotions always fluctuate up and down near the equilibrium line,while abnormal emotions always hover near two extremes.Finally, changes of mood also has a relationship with the number of accepting events,so we improve the model.The simulated image show the chaotic nature,which indicate that the mood is very easy to change,and prove the emotion is complex.