This letter presents the results of an analysis concerning the global, dynamical structure of a second order phase–locked loop (PLL) in the presence of the continuous wave (CW) interference. The invariant manifolds of the PLL equation are focused and analyzed as to how they are extended from the hyperbolic periodic orbits. Using the Melnikov integral which evaluates the distance between the stable manifolds and the unstable manifolds, the transversal intersection of these manifolds is proven to occur under some conditions on the power of the interference and the angular frequency difference between the signal and the interference. Numerical computations were performed to confirm the transversal intersection of the system–generated invariant manifolds for a practical set of parameters.

A mathematical theory is proposed based on the concept of functional analysis, suitable for operation of network systems extraordinarily complicated and diversified on large scales, through connected-block structures. Fundamental conditions for existence and evaluation of system behaviors of such network systems are obtained in a form of fixed point theorem for system of nonlinear mappings.

In any ill-conditioned information-transfer system, as in long-distance communication, we often must construct feedback confirmation channels, in order to confirm that informations received at destinations are correct. Unfortunately, for such systems, undesirable uncertain fluctuations may be induced not only into forward communication channels but also into feedback confirmation channels, and it is such difficult that transmitters always confirm correct communications. In this paper, two fuzzy-set-valued mappings are introduced into both the forward communication channel and the feedback confirmation channel, separately, and overall system-behaviors are discussed from the standpoint of functional analysis, by means of fixed point theorem for a system of generalized equations on fuzzy-set-valued mappings. As a result, a good mathematical condition is successfully obtained, for such information-transfer systems, and fine-textured estimations of solutions are obtained, at arbitrary levels of values of membership functions.

Let us introduce n ( 2) mappings fi (i=1,2,,n) defined on complete linear metric spaces (Xi-1, ρ) (i=1,2,,n), respectively, and let fi:Xi-1 Xi be completely continuous on bounded convex closed subsets Xi-1(0) Xi-1, (i=1,2,,n 0), such that fi(Xi-1(0)) Xi(0). Moreover, let us introduce n set-valued mappings Fi : Xi-1 Xi (Xi)(the family of all non-empty closed compact subsets of Xi), (i=1,2,,n 0). Here, we have a fixed point theorem on the successively recurrent system of set-valued mapping equations: xi Fi(xi-1, fi(xi-1)), (i=1,2,,n 0). This theorem can be applied immediately to analysis of the availability of system of circular networks of channels undergone by uncertain fluctuations and to evaluation of the tolerability of behaviors of those systems. In this paper, mathematical situation and detailed proof are discussed, about this theorem.

A type of infinite dimensional homotopy method is considered for numerically calculating a solution curve of a nonlinear functional equation being a Fredholm operator with index 1 and an A-proper operator. In this method, a property of so-called A-proper homotopy plays an important role.

A mathematical theory is proposed, based on the concept of functional analysis, suitable for operation of network systems extraordinarily complicated and diversified on large scales, through connected-block structures. Fundamental conditions for existence and evaluation of system behaviors of such network systems are obtained in a form of fixed point theorem for system of nonlinear mappings.