On The Shape Of Limit Cycles Hopf Bifurcated From Two Classes Of High Dimensional Differential Systems

This paper studies the asymptotic expressions of the limit cycles for two classes of high dimensional differential systems on Hopf bifurcation. To solve this problem, by reducing the original systems to a simplified two-dimensional system with the help of center manifold theory firstly. According to the stability of the focus and Poincaré-Bendixson Theorem, we prove the existence of limit cycles. By applying the method of Friedrich to get the first several terms of the asymptotic expansions of the limit cycles and plot the graphs of limit cycles with Maple 17.Firstly, this paper studies the asymptotic expressions of the limit cycle for a new modified four-dimensional Lü system, we study the shape of its limit cycles at equilibrium O(0,0,0,0).Then get the first several terms of the asymptotic expansions of the limit cycles and plot its graph.In the next place, this paper considers the Moon-Rand system. By noting that the trace of Jacobian matrix of the Moon-Rand system at the equilibrium is always equal to zero, the first Lyapunov constants of the reduced Moon-Rand system on the center manifold are computed,and the condition of existence of the limit cycle of the Moon-Rand system is given. Then we compute the asymptotic expansion of the period solutions of the generalized Moon-Rand system and plot its graph.