Some Mathematics for Complex Economic Systems

The term ‘dynamical system’ describes a system that evolves in time according to a well-defined rule. More mathematically, a dynamical system is characterized by the fact that the future values of its observable variables can be given as a function of the values of the variables at the present time. The space ∑ where such variables are defined is called the phase space. The behaviour of dynamical systems is represented in a multidimensional phase space, where the state of the system at any time is represented by a point. The dynamical system describes the change in the states with time; that is, a transformation acting on ’ is given and is called a flow. More specifically, associated with each t ∈ R (or t ∈ R+) there is a mapping f t : ∑ → ∑ such that the group property holds (that is f0 = Id(∑) and f t+s = f t . f s for all t, s). The system moves from the state x ∈ ∑ to the state f t x after time t. A cascade differs from a flow in that the maps f t are defined only for integer t. The evolution of a system starting from a given initial state — that is, from a given point x in the phase space — is represented by a trajectory in this space; that is, the trajectory of x is the set {ftx}. If t > 0 we use the prefix semi- for flows, trajectories, etc. If ftx = x for all t, then x is an equilibrium point. If ft+Tx=ftx for all t and for some T ≠ 0, then the trajectory { t x} is said to be periodic. Periodic trajectories are closed and are often called cycles. If a set A ⊂ ∑ is such that f t A=A for all t, then A is said to be an invariant set.