Periodic Solutions

Existence of periodic solutions and stability of fractional order dynamic systems are two important and difficult issues in fractional order systems (FOS) field. In this paper, the relationship between integer order systems (IOS) and fractional order systems is discussed. A new proof method based on the above involved relationship for the non existence of periodic solutions of rational fractional order linear time invariant systems is derived. Rational fractional order linear time invariant autonomous system is proved to be equivalent to an integer order linear time invariant non-autonomous system. It is further proved that stability of a fractional order linear time invariant autonomous system is equivalent to the stability of another corresponding integer order linear time invariant autonomous system. The examples and state figures are given to illustrate the effects of conclusion derived.

We study the dynamics of a feedback system with hysteresis. The identification of switching surfaces in state space naturally gives rise to a Poincare map whose fixed points represent the periodic orbits of the system. For linear systems in feedback with a relay hysteresis nonlinearity, necessary and sufficient conditions for the existence of unimodal periodic solutions are obtained and the stability of the periodic orbit is analyzed

For an m m -dimensional differential inclusion of the form \[ x ˙ ∈ A ( t ) x ( t ) + F [ t , x ( t ) ] , \dot x \in A(t)x(t) + F[t,x(t)], \] with A A and F F T T -periodic in t t , we prove the existence of a nonconstant periodic solution. Our hypotheses require m m to be odd, and require F F to have different growth behavior for | x | \left | x \right | small and | x | \left | x \right | large (often the case in applications). The idea is to guarantee that the topological degree associated with the system has different values on two distinct neighborhoods of the origin.