CSB −Synergetics: Escape from Chaos

Biomechanical Chaos

The so–called deterministic chaos represents irregular and unpredictable time evolution of many simple nonlinear deterministic systems [GOY87, II07a], like a single–joint musculo–skeletal movement. Its central characteristics is that the system does not repeat its past behavior (even approximately). If we now the forcing amplitude A and the frequency w of a muscle–actuator, as well as the linear damping b in the uniaxial joint, then the motion of the flexion–extension type can be described in canonical q − p–coordinates as

• 1 as a linear harmonic oscillator

  $$ \dot{q}=p,\qquad \dot{p}+b\,p+q=A\,\\cos (w\,t) $$

  or,

• 1 as a nonlinear Hamilton’s oscillator

  $$ \dot{q}=p,\qquad \dot{p}+b\,p+\\sin q=A\,\\cos (w\,t) $$

The difference between these two muscle–joint models is just in the q versus

sinq term, but the produced dynamics is qualitatively different. However, for a very short time, or a small change of the joint angle q, the difference could be neglected, and nonlinear model could be satisfactorily approximated by a linear one.