In recent years, Fractional Ordinary differential equations, FODEs, became an essential tool for modelling of viscoelasticity, neuron behaviour, fluid dynamics, electrical circuits and more. The distinguishing feature of the FODEs is the use of a fractional derivative, which generalises the classical derivative to a non-integer order.

The fractional derivative is a non-local operator, meaning the whole history of the function affects the value of the derivative at a given point. The non-locality introduces analytical difficulties when extending the standard method from the classical dynamical systems to the FODE framework, similar to the challenges faced with time-delay systems. This is particularly evident in the theory of invariant manifolds. For example, the classical notion of invariance is no longer well-posed for the fractional dynamical systems. The literature presents conflicting results on this topic, some studies claim that stable and invariant centre manifolds exist, and one work disputes that claim.

The aim is to resolve this contradiction and provide a concrete framework for the analysis of the fractional dynamical systems along the way.