The properties of functional relation between a noninvertible chaotic drive and a response map in the regime of generalized synchronization of chaos are studied. It is shown that despite a very fuzzy image of the relation between the current states of the maps, the functional relation becomes apparent when a sufficient interval of driving trajectory is taken into account. This paper develops a theoretical framework of such functional relation and illustrates the main theoretical conclusions using numerical simulations.

We investigate the dynamical origin of the statistical properties of boiling in a short capillary tube. Two different models are proposed (Markov model and rotational model). We show that the behavior of each model may be related to different properties of the physical phenomenon. We conclude with a suggestion of a new experimental measurement which can help to adjust several aspects of the models. (c) 2001 American Institute of Physics.

We prove, under some general assumptions, that master-slave synchronization implies generalized synchronization, that is we show the existence and continuity of the functional dependence between the "slave" coordinates and the "master" ones. Then, we prove that this function may be Lipschitz continuous and even less "smooth", that is only Holder continuous, depending on the coupling strength. We go beyond the above mentioned assumptions by coupling two identical maps of the interval that are neither continuous nor invertible to prove `almost-everywhere' synchronization instead of global synchronization. Then we relate the Hausdorff dimension and the dimension for Poincare recurrence of the attractor of master and slave systems. We provide some examples illustrating these results.

We state some analytical results on of nonsmooth functional dependence that occurs between the phase space coordinates of chaotic drive and response systems when they are synchronized. In particular, we describe the change of regularity of the synchronization function which ranges from Lipschitz to Hölder continuity when the coupling strength decreases.

The onset of generalized synchronization of chaos in directionally coupled systems corresponds to the formation of a continuous mapping that enables one to persistently define the state of the response system from the trajectory of the drive system. A recently developed theory of generalized synchronization of chaos deals only with the case where this synchronization mapping is a single-valued function. In this paper, we explore generalized synchronization in a regime where the synchronization mapping can become a multivalued function. Specifically, we study the properties of the multivalued mapping that occurs between the drive and response systems when the systems are synchronized with a frequency ratio other than one-to-one, and address the issues of the existence and continuity of such mappings. The basic theoretical framework underlying the considered synchronization regimes is then developed.