Non-Linear Dynamics in a Small-Open-Economy Model in the Euro Area Panayotis G. Michaelides and Angelos T. Vouldis National Technical University of Athens Athens, Greece pmichael@central.ntua.gr and avouldis@biosim.ntua.gr Abstract— This article focuses on the dynamics of the commodity and money market. The purpose of the article is to study a two-equation linear dynamic model in order to examine its behavior in complex conditions and its dependence on the parameters. Stability conditions are examined by means of the Ruth - Hurwitz criterion. The system and its stability are also examined when non-linear dynamics is introduced. The linear system was stable for any economically admissible values of the parameters. Also, non-linear perturbations were applied in two cases which both led to a stable system. From a theoretical perspective in economics this implies that for any small open economy operating in the Euro area without exchange rate dynamics and operating under the specific conditions, regardless of the possible non-linearity in the investment function, the resulting economic system is asymptotically stable. Keywords— small open economy; dynamical system; non- linear; numerical. I. INTRODUCTION The model presented in this paper is primarily of Keynesian inspiration and is based on the Mundell-Fleming theory using IS-LM model [1], [2]. This article models a small open economy without exchange rate dynamics as a two- equation dynamic model in order to examine its behavior. This model could be suitable for modeling the economy of a country such as Greece operating in the Euro area, where exchange rates are absent. In the commodity market, demand for investment and net export is related to savings. Total demand and savings are said to be in equilibrium. The investment demand function depends on the interest rate. The net export demand is considered to be constant. First, we begin with an explanation of the linear model where the economic nature of the parameters results in stability. II. THE MODEL Production Y is described by the equation: [ ] ( , ) ( , ) ( , ), 0 Y aIRY XY Z SY Z a ¢= + - > (1) Equation (1) simply expresses the difference between aggregate demand and aggregate supply. Aggregate demand is equal to C + I + E where C, I, E are investment, consumption and exports, respectively, and aggregate supply is equal to C + S + M where C, S, M, are consumption, savings and imports, respectively. Subtracting aggregate supply from aggregate demand we get I + X – S where X = E – M denotes net export [3]. The difference between aggregate demand and supply causes the increase of production. If the difference is positive, this implies an increase in production and if the difference is negative this implies a decrease in production [4]. Dividing equation (1) by Y we get: ( , ) ( ) ( ) I Y R XY SY Y a Y Y Y Y ¢ é ù ê ú = + - ê ú ë û (2) The investment function has the form: 0 2 0 2 ( ) , , 0 I i iRY i i = - > (3) The net exports function has the form: 0 0 , 0 X xY x = > (4) The savings function has the form: 0 1 0 1 ( ln ) , , 0 S s s YY s s = + > (5) Also, let: ln y Y = (6) ln x X = (7) Substituting equations (3)-(7) into equation (2) we get: [ ] 0 2 0 0 1 ( ) y ai iR x s sy ¢= - + - + (8) For the money market we assume that: