Duffing equation with fifth nonlinear-restoring force, one external forcing and a phase shift is investigated, The conditions of existences for primary resonance, second-order, third-order subharmonics, morder subharmonics and chaos are given by using second-averaging method, Melnikov methods and bifurcation theory. Numerical simulations including bifurcation diagrams, bifurcation surfaces, phase portraits, not only show the consistence with the theoretical analysis, but also exhibit the new dynamical behaviors. We show the onset of chaos, chaos suddenly disappearing to period orbit, one-band and double-band chaos, period-doubling bifurcations from period 1, 2, and 3 orbits, period-windows （period-2, 3 and 5） in chaotic regions.
A discrete predator-prey system with Holling type-IV functional responseobtained by the Euler method is first investigated. The conditions of existence for foldbifurcation, flip bifurcation and Hopf bifurcation are derived by using the center manifold theoremand bifurcation theory. Furthermore, we give the condition for the occurrence of codimension-twobifurcation called the Bogdanov-Takens bifurcation for fixed points and present approximateexpressions for saddle-node, Hopf and homoclinic bifurcation sets near the Bogdanov-Takensbifurcation point. We also show the existence of degenerated fixed point with codimension three atleast. The numerical simulations, including bifurcation diagrams, phase portraits, and computationof maximum Lyapunov exponents, not only show the consistence with the theoretical analysis but alsoexhibit the rich and complex dynamical behaviors such as the attracting invariant circle,period-doubling bifurcation from period-2,3,4 orbits, interior crisis, intermittency mechanic, andsudden disappearance of chaotic dynamic.
Ji-cai Huang Department of Applied Mathematics, College of Science, China Agriculture University
