Lyapunov method: a tool to describe fabric attractor in non - linear and heterogeneous flows with application to shear zones

In this contribution we review and expand the concept of attractor to non-linear flow sys-tem, introducing the concept of stability analysis to unravel some simple heterogeneous flow system. A semi-quantitative tool (using Lyapunov method) is suggested to predict the fabric attractor within some simple and general shear zones defined by non-linear flow tensor. The theory is briefly explained and it is shown the potentiality of this approach on describing the flow in heterogeneous shear zones. An example is discussed and a kinematic classification of some possible non-linear flow systems, using Lyapunov exponent analysis, is proposed. Deformation is generally treated as homogeneous and steady state (i.e. the kinematics of the flow at a given material particle is not varying with time and space) because the mathematical description becomes too complex otherwise. However, several field, experiment and theoretical oriented works showed that strain rate, as well as rheological properties of rocks, generally change during the deformation history. This implies that the deformation could vary non-linearly along space and with time (Fossen and Tikoff, 1997; Jiang and Williams, 1999; Trepmann and Stockhert, 2003). Non-linear flow implies that the principal asymptoti-cally stable directions of flow, that behave as attractor or repulsor (Ruelle, 1981; Tabor, 1989), are expected to control the final orientations of the principal strain axes as well as the final fabric distribution, that could vary in time and in space. Heterogeneous deformation represents a classical example of autonomous non-lin-ear system as it implies that the strain varies following a non-linear function (Ramsay, 1980), e.g. showing a vorticity or strain gradient along a specific direction (Jiang and Williams, 1999). Solutions of differential equations describing non-linear autonomous system are not obvious and if exist are not deterministic. These non-linear properties strongly limit any attempt to reconstruct the flow history in a unique way as dif-ferent kinematics histories could produce the same results. As a consequence, geological structures cannot be generally described in terms of unique flow pattern. The aim of this extended abstract is to contribute to the knowledge of the possible flow pattern and related structures produced in some non-linear flow in order to understand or predict similarly to the linear case if some fabric attractors (McKenzie, 1979; Passchier, 1997) could be expected. With this purpose, we intro-duce the concept of stability analysis (using Lyapunov method) as a criterion to describe the flow pattern where the flow path cannot be derived by simple inte-gration, and we discuss the properties of some non-lin-ear flow that could be analysed with this method.