# Who’s Afraid of Fractional Order Laplace?

We are now in the pioneering position where clinicians with their stethoscopes poised over the healthy heart, radiologists tracking the blood flow, and physiologists probing the nervous system, are all exploring the frontiers of chaos and fractals. Two concepts are necessary to be introduced: a) chaos theory says that a very minor disturbance in initial conditions leads to an entirely different outcome; and b) fractals are self-similar structures on many or all scales (i.e. the principle of regularity and order) [1,2,3]. These topics are central concepts in the new discipline of nonlinear dynamics developed in physics and mathematics – see Figure 1.

However, the most compelling applications of these abstract concepts are not in the physical sciences [4], but in medicine, where fractals and chaos may change radically long-held views about order and variability in health and disease [5]. A transition to a more ordered or less complicated state may be an indicative of disease (or equivalent a change in the nominal activity). Investigators have, only in the past 5 years or so, discovered that the heart and other physiological systems may behave most erratically when they are young and healthy (i.e. random fractal properties, power law dynamics, can be well characterized by cascaded impedance models). Counter-intuitively, increasingly regular dynamic patterns accompany aging and disease (i.e by using Fourier analysis tools one can detect these locked dynamics) [6,7,8].

The last decades have shown an increased interest in the research community to employ parametric model structures of fractional-order for analyzing nonlinear biological systems [8]. The concept of fractional-order (FO)—or non-integer order—systems refers to those dynamical systems whose model structure contains arbitrary order derivatives and/or integrals [9,10,11]. The dynamical systems whose model can be approximated in a natural way using FO terms, exhibit specific features: viscoelasticity, diffusion and fractal structure [12,13,14].

However, the theoretical concepts of fractals, chaos and multiscale analysis have not yet been enabled breakthrough mainly due to a lack of awareness within the research community.

For further details of the speaker and details of referenced work, see http://www-control.eng.cam.ac.uk/Main/ControlSeminarSlides or contact Tim Hughes.

This talk is part of the CUED Control Group Seminars series.