Dynamical System State Need Not Have Spectrum

S. Boyd and L. O. Chua

IEEE Transactions on Circuits and Systems, CAS-32(9):968-969, September 1985.

Recently there has been much study of nonlinear dynamical systems (differential equations) which have chaotic solutions. While there is not precise definition of what a chaotic solution is, it is generally agreed upon that a chaotic trajectory should have a continuous spectrum, in particular, it should not be almost periodic. A natural question is, therefore, does a (bounded) trajectory of a dynamical system always have a spectrum? In this short note we give a simple example which shows that it need not.