Complex dynamics in a generalized Langford system

This paper analyzes complex dynamics of the generalized Langford system (GLS) with five parameters. First, some important local dynamics such as the Hopf bifurcations and the stabilities of hyperbolic and zero/double-zero equilibrium are investigated using normal form theory, center manifold theory and bifurcation theory. Besides, an accurate expression of a periodic orbit and some approximate expressions of bifurcating limit cycles by Hopf bifurcation are obtained. Second, by using averaging theory, the zero-Hopf bifurcation at the origin is analyzed and also the stability of the bifurcating limit cycle is obtained. Of particular interest is that one numerically finds an annulus in three-dimensional space which appears nearby the bifurcating limit cycle under proper conditions. If an amplitude system of GLS has a center, its suspension may lead the GLS to exhibit such annulus. Third, it is proved rigorously that there exist two heteroclinic cycles and the coexistence of such two heteroclinic cycles and a periodic orbit under some conditions. This implies system has no chaos in the sense Shil’nikov heteroclinic criterion. Finally, by further numerical observation, it is shown that three different types of attractors exist simultaneously, such as two kinds of periodic orbits, periodic orbit and invariant torus.