biharmonic surfaces in euclidean space $mathbb{e}^3$ are firstly studied from a differential geometric point of view by bang-yen chen, who showed that the only biharmonic surfaces are minimal ones. a surface $x : m^2rightarrowmathbb{e}^{3}$ is called biharmonic if $delta^2x=0$, where $delta$ is the laplace operator of $m^2$. we study the $l_k$-biharmonic spacelike hypersurfaces in the $4$-dimen...

We classify biharmonic submanifolds with certain geometric properties in Euclidean spheres. For codimension 1, we determine the biharmonic hypersurfaces with at most two distinct principal curvatures and the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudo-umbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curva...

Motion of a biharmonic system under action of small periodic force and small damped force is studied. The biharmonic oscillator is a physical system acting under a biharmonic force like a sin θ b sin 2θ. The article contains biharmonic oscillator analysis, phase space research, and analytic solutions for separatrixes. The biharmonic oscillator performs chaotic motion near separatrixes under sma...