Chaotic Motions in Linear/Nonlinear and Quasi-Periodical Dynamic Systems Revealed by Energy-Flow Investigations

By using 1~2 degrees of freedom (DOF) examples, linear/nonlinear and quasi-periodical systems (QPS) are investigated based on the energy-flow theory (EFT), which reveals the following new findings on chaotic motions. (a) A 1-DOF linear non-damped-forced motion is chaotic if there is no least common multiple (LCM) for the periods of natural and force frequencies. (b) For a 1-DOF nonlinear non-damping-forced system with nonlinear stiffness of γx2,(γ>0), both the free vibration by initial conditions and the composed motion with forced one excited by a force of given frequency are chaotic; while, when damping is added, the free vibration is damped to zero, but the forced vibration consists of different frequency components showing chaotic characteristics. (c) For a 1-DOF system of natural frequency 1 with nonlinear damping 2ε(x˙)=−2+x˙+x˙2, ε(x˙)