The document introduces complex systems and defines a system as a set of interacting parts within a single structure. It provides examples of real-world systems and discusses describing a system from different points of view using models. The document outlines functional and oriented models, abstraction in models, and analyzing dynamic and stationary systems. It discusses representing stationary systems using transition and transformation functions and modeling economic systems with relevant variables and equations. Finally, it presents a case study modeling a school system as a linear, stationary system to calculate total student population over time under changing retention rates.

1. Introduction to complex systems Francesco Fusco

2. Introduction In this unit a brief survey of abstract systems is introduced to understand how (linear) system analysis can be applied to model economic phenomena within the territorial marketing field. A few mathematical notations will be adopted to formally model the systems. A simple study case is then presented to validate some conclusions from the theory.

7. The oriented model If some variable values (output) are changed when modifying some other variables (input) the functional description of a real systems defines an abstract oriented model of the system. S i o where S becomes a black box transforming i in o

8. Abstraction The abstraction process involved during the description allows to reduce the number of variables required to describe a real system and therefore simplifies its analysis Studying the model we can derive the behaviour of the real system only for those properties saved under the equivalence principle While analysing, we must care about those properties only owned by the model itself Complex models can be in turn decomposed in simpler submodels

10. Dynamic systems If the system changes over the time, we have a dynamic abstract system or dynamic system In this case we have to introduce the set of all the times T and the formal notation A T(t0) denoting the set of the functions of time t in A, defined for t  t0  T S = S(t0)  I T(t0) x O T(t0)

11. The (internal) state of the system In dynamic abstract systems w e can’t sometime derive the output from the input but we need an additional information, representing the effect of the history of the system S from starting time t0 to t, where t  t0  T We name it the state X of the system S S,X i o

15. Representation of stationary systems x(t) =  (t-t0, x0, i [t0,t) ) transition o(t) =  (x(t), i(t)) transformation For stationary systems the transition and transformation functions become

16. Linear time discrete systems When time t assumes only integer value we have time discrete systems. Among these we are interested in studying linear systems, defined by the two functions x(t+1) = Ax(t) + Bi(t) o(t+1) = Cx(t) + Di(t) where A,B,C,D are specific matrixes and x, i, o are column vectors

17. Generality principle Any abstract system can be locally described as a linear system Under specific conditions of regularity also economic systems can be locally modelled using linear systems This principle therefore allows linear system analysis applicable to real economic phenomena.

18. The generic schema of linear systems Any linear system S has an intrinsic generating schema In the schema on the left the output is the sum of the current input under D and the current state under C. The current state is obtained by multiplying the previous state (delayers Z) by A and then added to input ubder B. A B Z -1 D C i o + + + + x

19. Economic systems Economic modelling is the schematic representation of an economic phenomenon through a set of variables that are the measures of economic entities and a set of related mathematical equations. A valid description of economic systems implies the right selection of initial hypotheses, the choosing of relevant (independent, dependent) variables and their mutual relations to properly derive conclusions for the systemic behaviour .

20. The school case study Given a 3 years course we want to study the total population o(t) of students at ( discrete ) time t We define a state variable x i (t), (where i=1,2,3) the number of student attending the course year i at time t we indicate i(t) the number of students subscribing at first year We assume the system as a stationary linear system

21. The school model The course model at stationary conditions, i.e. t>>1 can be described by 3 constants  1 the fraction of students registering at course 2  2 the fraction of students registering at course 3  3 the fraction of students repeating year course 3 By definition the output function is o(t) = x 1 (t) + x 2 (t)+ x 3 (t)

22. Test Exercise Calculate the global number of attending students in the case that half of students are fired at year 1 and 40% of students are fired at year 2, under the constant registering rate of i.