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: θ θ 2 sin sin b a + . The article contains biharmonic oscillator analysis, phase space research, and analytic solutions for separatrixes. The biharmonic oscillator performs chaotic motion near separatrixes unde...

This paper studies conformal biharmonic immersions. We first study the transformations of Jacobi operator and the bitension field under conformal change of metrics. We then obtain an invariant equation for a conformal biharmonic immersion of a surface into Euclidean 3-space. As applications, we construct a 2-parameter family of non-minimal conformal biharmonic immersions of cylinder into R and ...

In this paper, non-geodesic biharmonic curves in ̃ SL(2, R) space are characterized and the statement that only proper biharmonic curves are helices is proved. Also, the explicit parametric equations of proper biharmonic helices are obtained. AMS subject classifications: 53A40

In this paper, we study biharmonic maps into Sol and Nil spaces, two model spaces of Thurston's 3-dimensional geometries. We characterize non-geodesic biharmonic curves in Sol space and prove that there exists no non-geodesic biharmonic helix in Sol space. We also show that a linear map from a Eu-clidean space into Sol or Nil space is biharmonic if and only if it is a harmonic map, and give a c...

in this paper, we study spacelike dual biharmonic curves. we characterize spacelike dual biharmonic curves in terms of their curvature and torsion in the lorentzian dual heisenberg group . we give necessary and sufficient conditions for spacelike dual biharmonic curves in the lorentzian dual heisenberg group . therefore, we prove that all spacelike dual biharmonic curves are spacelike dual heli...

We study the eigenvalues of the biharmonic operators and the buckling eigenvalue on complete, open Riemannian manifolds. We show that the first eigenvalue of the biharmonic operator on a complete, parabolic Riemannian manifold is zero. We give a generalization of the buckling eigenvalue and give applications to studying the stability of minimal Lagrangian submanifolds in Kähler manifolds. MSC 1...

In this paper, we first establish regularity of the heat flow of biharmonic maps into the unit sphere S ⊂ R under a smallness condition of renormalized total energy. For the class of such solutions to the heat flow of biharmonic maps, we prove the properties of uniqueness, convexity of hessian energy, and unique limit at t = ∞. We establish both regularity and uniqueness for Serrin’s (p, q)-sol...

Chen's biharmonic conjecture is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we consider an advanced version of the conjecture, replacing $Delta$ by its extension, $L_1$-operator ($L_1$-conjecture). The $L_1$-conjecture states that any $L_1$-biharmonic Euclidean hypersurface is 1-minimal. We prove that the $L_1$-conje...

We prove an apriori estimate in Morrey spaces for both intrinsic and extrinsic biharmonic maps into spheres. As applications, we prove an energy quantization theorem for biharmonic maps from 4-manifolds into spheres and a partial regularity for stationary intrinsic biharmonic maps into spheres. x

In an arbitrary bounded 2-D domain, a singular perturbation approach is developed to analyze the asymptotic behavior of several biharmonic linear and nonlinear eigenvalue problems for which the solution exhibits a concentration behavior either due to a hole in the domain, or as a result of a nonlinearity that is non-negligible only in some localized region in the domain. The specific form for t...