Heteroclinic bifurcations in four-dimensional vector fields are investigated by setting up local coordinates near a heteroclinic loop. This heteroclinic loop consists of two principal heteroclinic orbits, but there is one stable foliation that involves an inclination flip. The existence, nonexistence, coexistence and uniqueness of the 1-heteroclinic loop, 1-homoclinic orbit, and 1-periodic orbit are studied. Also, the nonexistence, existence of the 2-homoclinic and 2-periodic orbit are demonstrated.

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This paper focuses on the initial-value problem of a linear plate equation with memory in multi-dimensions (n ≥ 1), the decay structure of which is of regularity-loss property. By making use of the point-wise estimate of solutions in the Fourier space, the energy estimates and decay estimates of solutions to the linear problem are obtained.

<abstract><p>In this paper, we mainly investigate the radial distribution of Julia sets of difference operators of entire solutions of complex differential equation $ F(z)f^{n}(z)+P(z, f) = 0 $, where $ F(z) $ is a transcendental entire function and $ P(z, f) $ is a differential polynomial in $ f $ and its derivatives. We obtain that the set of common limiting directions of Julia sets of non-trivial entire solutions, their shifts have a definite range of measure. Moreover, an estimate of lower bound of measure of the set of limiting directions of Jackson difference operators of non-trivial entire solutions is given.</p></abstract>