Sentence examples for manifold argument from inspiring English sources

Using the normal form theory and center manifold argument, the explicit formulae which determine the stability and direction of bifurcated periodic solutions are derived.

Concretely, using the normal form theory and center manifold argument, we will give the explicit formulae which determine the stability and direction of bifurcated periodic solutions.

Using the normal form theory and center manifold argument, the explicit formulae which determine the stability, direction and other properties of bifurcating periodic solutions are derived.

Applying the normal form theory and the center manifold argument, some concrete expressions to judge the properties of the bifurcating periodic solutions are given.

By using the normal form theory and center manifold argument introduced by Hassard [30], we derive the explicit formulae for determining the stability, direction, and other properties of bifurcating periodic solutions.

Using the normal form theory and center manifold arguments, the explicit formulae which determine the stability, the direction, and the period of bifurcating periodic solutions are derived.

In Section 4, a sharp threshold for global existence and blowup of the solutions are derived by utilizing two invariant manifolds, applying potential well argument and the concavity method.

The proof is similar to the argument of Nehari manifold method [16].

If c is negative the dimensions of the stable and unstable manifolds are interchanged, but the argument above still applies.

Lately in [2], Pankov and Rothos employed Nehari manifolds approach and the Mountain Pass argument to demonstrate the existence of solutions in the discrete nonlinear Schrödinger equation with saturable nonlinearity.

Secondly, a sharp threshold for the global existence and blowup of the solutions was derived by introducing some auxiliary functionals and invariant manifolds and applying potential well argument and the concavity method.