A numerical method is presented for calculating transverse and non-transverse (or tangent) types of homoclinic points of a two-dimensional noninvertible map having an invariant set that reduces to a one-dimensional noninvertible map. To illustrate bifurcation diagrams of homoclinic points and transitions of chaotic states near the bifurcation parameter values, three systems including coupled chaotic maps are studied.

We investigate chaotic dynamics due to the homoclinic points observed from a widely used phase–locked loops operating as a frequency–modulated demodulator. Our purpose is to obtain parameter region of the homoclinic points using Melnikov method in a periodically-forced second–order nonlinear nonautonomous equation representing phase–locked loops. If the PLL equation has large damping (actually, this is the case of standard PLL), the unperturbed system becomes non–Hamiltonian. Therefore, one cannot obtain the saddle loop analytically in general, and hence it is very difficult to apply Melnikov method to such a system. Since the current PLL used in practice has a triangular phase detector (i.e., a periodic triangular shaped function) as its nonlinearity, we can use piecewise–linear method, and thus we are succeeded in deriving both the saddle loop and the Melnikov integral analytically even in the PLL equation with practical large damping. We have obtained many boundary curves for homoclinic tangency for a wide range of damping coefficients and modulation frequency. In particular, we treat the general case of β2ζ in this paper where β denotes the normalized natural frequency and ζ denotes the damping coefficient. We compare this results with our previous totally numerical results and have found that this method gives more accurate boundary curves than our previous method.