A generalised entropy of curves: An approach to the analysis of dynamical systems

A generalisation of the entropy of a plane curve to ¿ⁿ space is provided and the generalised entropy is used to evaluate the nonlinear behaviour of dynamical systems. The entropy of a curve, first introduced within the theory of thermodynamics of plane curves, has been used to quantify the irregularity of a curve. This paper extends the concept to higher dimensions and provides an algorithmic procedure to evaluate the entropy of a curve evolving in the phase space according to the equations of a dynamical system. The thermodynamic indicator is eventually used to infer some properties about the dynamical system. In particular, according to the proposed indicator, all linear systems evolve at entropy zero, while higher entropies characterise nonlinear systems, leading to an effective criterion for their classification.