HOPF BIFURCATION OF FPSO MOORED SYSTEMS WITH TERMS OF TYPE V | V|: AN ANALYSIS USING THE CENTRAL MANIFOLD THEOREM AND INTEGRAL AVERAGING TECHNIQUES José de França Bueno University of São Paulo, Escola Politécnica, Department of Mechanical Engineering francabj@terra.com.br Celso Pupo Pesce University of São Paulo, Escola Politécnica, Department of Mechanical Engineering ceppesce@usp.br Clodoaldo Grotta Ragazzo University of São Paulo, Mathematics and Statistics Institute, Department of Applied Mathematics ragazzo@ime.usp.br Abstract: This work treats the problem of dynamic equilibrium bifurcation of systems that exhibit terms of t ype x x & & . Examples of this kind are common in mechanics, particularly in those involving viscous fluid forces that appear in mechanical, aeronautical and ocean engineering. Nowadays the current use of computer simulators enables the analyst to address the problem under an exhaustive non-linear time domain approach. Alternatively, some other techniques of Applied Mathematics as the Central Manifold Theorem (CMT) and the Integral Averaging Method (IAM) may be proper to qualify and quantify Hopf bifurcation scenarios. For systems with terms of type x x & & a direct application of such techniques are not straightforward, though. The analyst must treat the low differentiability, of order one, at 0 = x & . In this paper a simple regularization technique, based on polynomial approximations is proposed, enabling a local treatment of the post-critical behaviour for this kind of problems. Examples extracted from engineering, as the fishtailing instabilities of a moored offshore vessel can be viewed as practical applications. Keywords : Hopf Bifurcation, Central Manifold Theorem, Integral Averaging, Dynamical Systems, FPSO moored systems. 1. Introduction Differential equations modeling systems, which involve viscous fluid forces, as those appearing in mechanical, aeronautical and ocean engineering, usually present terms of type x x & & . One possible approach to study the problems of static and dynamic equilibrium bifurcation related to this kind of system is the use of computer simulators, through exhaustive non-linear time domain simulations. Another possible one is based on some more recent techniques of Applied Mathematics, as the Central Manifold Theorem (CMT) and Integral Averaging Method (IAM), to quantify and qualify static and dynamic bifurcation scenarios in analytical form. The main advantages of these latter techniques are: (I) The associated simplification of the dynamical system, near the bifurcation point, with no loss of relevant information on dynamics. This simplification may be seen in two ways. Firstly, by the reduction of dimensionality of the dynamical system. In most examples the analyst can reduce the dimensionality of the dynamical system near the bifurcation point from N to 1 or 2 dimensions. Secondly, this simplification implies the elimination of a number of non-linear terms of the dynamical system. With such a simplification the analyst can obtain, in a proper way, analytical post-critical scenarios to the static or dynamic bifurcation as well as can study the dependence of the system on the parameters that are relevant to the problem. (II) These techniques present an intrinsic geometrical appeal, in the sense that they enable to re-write the system into new coordinates (on the Central Manifold) associated, in the linear part, with the eigenvalues with zero real part. (III) Another important point is that the Central Manifold is an invariant manifold. So, once an orbit enters the Center Manifold, this orbit will not escape from it. (IV) With this Technique it is possible to extend (in an analytical form) the results from Bernitsas (1999), including nonlinear terms in the analysis. The results we see in Bernitsas (1999) take into account information coming only from linear terms of Taylor expansion of the equations of motion. Considering only linear terms it would be eventually impossible to correctly qualify (analytically and a priori) any Dynamic Bifurcation and, certainly, it would be impossible to quantify the amplitude of the resulting cycle limits.