3 Calculation of NS bifurcation

As an application of the method mentioned in previous section, we propose a new method of the NS bifurcation parameter values. Let $r=1$ (on the unit circle). We use Eq.(5) again for calculation of NS bifurcation by choosing new variables $(\mbox{\boldmath$\ u $}, \lambda_1, \lambda_2) \in \mbox{\boldmath$\ R $}^{n+2}$. The Jacobian matrix of the Newton's method is as follows:

$$D\mbox{\boldmath$ F $} = \left( \begin{array}{ccc} \displa... ...frac{\partial \Im\chi}{\partial \lambda_2} \end{array}\right)$$ (7)

It is noticed that only the third column is exchanged from Eq.(6).

Several methods are available to detect the NS bifurcation; Kuznetsov summarize the related algorithms in Ref.[3]. These methods basically rescale the system into a boundary valued problem by using unknown period of the limit cycle. Numerical differences and collocation method are also used to have the Jacobian matrix for limit cycles. However the dimension of th equation corresponding Eq.(5) in their methods becomes too huge. As against them, our method uses solutions of the variational equations obtained from numerical integration, and it is only $n+2$ dimensional.

We previously proposed a method to obtain NS bifurcation by using similar way [4]. In this reference, it chooses a location of the fixed point $\mbox{\boldmath$\ u $}_0$, a parameter $\lambda_1$, and the argument $\theta$ are independent variables for Newton's method. Thus to show 2-parameter bifurcation diagram, we have to use a suitable predictor-corrector method complementally since basically it is one of continuation problems.

In this our new method, the argument $\theta$ can be chosen as an incremental parameter, i.e., it is not a variable. This method utilizes a fact that the NS bifurcation curve is parameterized by $\theta$, thus any two parameters can chosen as variables for Newton's method. The algorithm is not sensitive for the curvature of the NS bifurcation curve, it is not necessary to control the incremental parameter with predictor-corrector method. Moreover, Newton's method would converge with any argument even if it approach a codimension-2 bifurcation point since the Jacobian matrix (7) is not singular for any parameter values.

Although our method relies on the numerical integration method, e. g., Runge-Kutta method, numerical solution of variational equation can be obtained accurately. Therefore the Newton's method can gains good estimation for every iteration. We also emphasize that this method does not depend on the stability of the target limit cycle, namely, the method can calculate a bifurcation point normally even if the periodic solution has a very strong unstability ($\vert\mu_j\vert \gg 1$).