Local bifurcation analysis of a four-dimensional hyperchaotic system

Local bifurcation phenomena in a four-dimensional continuoushyperchaotic system, which has rich and complex dynamicalbehaviours, are analysed. The local bifurcations of the system areinvestigated by utilizing the bifurcation theory and the centremanifold theorem, and thus the conditions of the existence ofpitchfork bifurcation and Hopf bifurcation are derived in detail.Numerical simulations are presented to verify the theoreticalanalysis, and they show some interesting dynamics, including stableperiodic orbits emerging from the new fixed points generated bypitchfork bifurcation, coexistence of a stable limit cycle and achaotic attractor, as well as chaos within quite a wide parameterregion.