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The qualitative properties of a predatorprey system with Holling-( n + 1) functional response and a fairly general growth rate are completely investigated. The necessary and sufficient condition to guarantee the uniqueness of limit cycles is given. Our work extends the previous relevant results in the reference. [ABSTRACT FROM AUTHOR]

Bifurcations of a degenerate homoclinic orbit with orbit flip in high dimensional system are studied. By establishing a local coordinate system and a Poincaré map near the homoclinic orbit, the existence and uniqueness of 1–homoclinic orbit and 1–periodic orbit are given. Also considered is the existence of 2–homoclinic orbit and 2–periodic orbit. In additon, the corresponding bifurcation surfaces are given. [ABSTRACT FROM AUTHOR]

Consider the time-periodic perturbations of n-dimensional autonomous systems with nonhyperbolic but non-critical closed orbits in the phase space. The elementary bifurcations, such as the saddle-node, transcritical, pitchfork bifurcation to a non-hyperbolic but non-critical invariant torus of the unperturbed systems in the extended phase space (x, t), are studied. Some conditions which depend only on the original systems and can be used to determine the bifurcation structures of these problems are obtained. The theory is applied to two concrete examples. [ABSTRACT FROM AUTHOR]