Melnikovʹs criteria and chaos in systems with fractional order deflection

In this paper a qualitative analysis of the dynamic systems described with the second-order differential equation with fractional order deflection function is considered. The existence of fixed points, closed orbits and the unions of fixed points and the trajectories connecting them is shown. The homoclinic orbit which connects a fixed point with itself and the corresponding stable and unstable manifolds are given in the closed analytical form. Melnikovʹs procedure for defining the criteria for transversal intersection of the stable and unstable manifolds is extended for the systems with fractional order deflection function. The critical parameter values for chaos are obtained analytically and proved numerically using the Lyapunov exponents. The bifurcation diagrams are plotted for various values of fractional order and the transition to chaos by period doubling is shown. The phase plane diagrams and the Poincare maps for certain fractional orders are obtained. The control of chaos and the transformation to periodic motion is considered.