Sentence examples for by using normal form from inspiring English sources

Next we investigate the direction of these bifurcations by using normal form theory.

Furthermore, we investigate the direction of these bifurcations by using normal form theory.

In Section 3, we derive the direction and stability of the Hopf bifurcation by using normal form method and central manifold theorem.

As far as we know, there are essentially three differential ways of obtaining Lyapunov constant for nilpotent critical points in theory: by using normal form theory [14], by computing the Poincaré return map [15] or by using Lyapunov functions [16].

In Section 3 we prove the existence of Neimark-Sacker and period doubling (flip) bifurcation for this system by analysing the characteristic equation, and then in Section 4 we investigate the direction of this bifurcations by using normal form theory. Finally in Section 5 we give numerical simulations to support our theoretical analysis.

Also the dynamic behaviors of the system are investigated, by using normal form theory, center manifold theorem and bifurcation theory, it is shown that the system undergoes a Neimark Sacker bifurcation and a flip bifurcation, on varying step-size in some range.

Nevertheless, some elementary types of nonlinear behavior of control systems can be classified by using normal forms.

The vibrations in the horizontal and vertical directions are analyzed on the center manifold near the double-zero degenerate point by using normal-form method.

By using the normal form method and the center manifold theory, we derive the explicit formulas for determining the direction and stability of Hopf bifurcation.

Properties of Hopf bifurcation such as direction and stability of bifurcating periodic solutions are determined by using the normal form theory and centre manifold theorem.

The analytical conditions for the applicability of a Non-linear Superposition Technique (NST) are established by using the Normal Form Method (NFM).