Discontinuous Dynamical Systems with Applications in Economics DUMITRU BĂLĂ Faculty of Economics and Business Adminiastation University of Craiova- University Centre of Drobeta Turnu Severin Number 1, Călugăreni Street, Drobeta Turnu Severin ROMANIA dumitru_bala@yahoo.com Abstract: - In this paper we intend to study two dynamical systems with applications in economics. The dynamical systems are nonlinear and that’s why the difficulty and even the impossibility of obtaining right solutions. We are studying the stability in perturbations and that’s where the issue of economic equilibrium occur. We study the stability by using the Liapunov function. The originality of the work is how the Liapunov function is built and how to make qualitative analyses on dynamical systems. Key-Words: - dynamical system, Liapunov function, discontinuous dynamical system, equilibrium points, stable system, full first 1 Introduction The evolution in time of the phenomena and of the processes can be described by differential equations and systems of differential equations. Whether 0 x the value of an economic size at the time moment 0 t and 1 x the value of the same economic size at the 1 t time moment. We note 0 1 1 x x x − = Δ and we note 0 1 1 t t t − = Δ . When the evolution in time is continuous, the variation speed of the economic size x is: 1 1 1 t x v Δ Δ = Going to the limit : i i i x dt dx v & = = In the speciality literature there are many economical, technical, computer, biological processes which are described by differential equations and by systems of differential equations [1], [10], [13]. Usually the ones with applicability in economics are called dynamic systems with applicability in economics. The differential equations which are used can be ordinary or with partial derived. About the dynamic systems with applicability in economics we sometimes know some initial conditions, some final conditions, some limit conditions or other conditions (connections, extremes, optimum) [24]. Solving these dynamic systems, that is to obtain accurate solutions is often difficult. For this reason the qualitative and aproximate methods appeared. If we can get the right solution for the differential equation, for the system of differential equation,for the equation with partial differentials, for the system of equqtions with partial differentials of the dynamic system, the issue is completely solved. Otherwise, the approximate methods must be combined with the qualitative ones [15], [16], [18], [20]. There are sometimes situations when mathematically a dynamic system may not have a solution and yet we may find an approximate solution. This is not right. For the approximate methods it is important how well they approximate the right solution (there are methods where the error is as ) 10 9 − , as it is in [15], [16]. In conclusion, generally speaking, the dynamic systems with applicability in economics are nonlinear, therefore it is hard to find the right solution and usually we use approximate solutions and qualitative methods. Below we’ll refer to qualitative methods of solving the dynamic systems with applicability in economics, that is the study of the stability of some dynamic systems. In the last 20 years we have been studying on concrete examples [1], [2], [5], [6], [7] the stability of some dynamics with applicability in economics using the method of the Liapunov function. Depending on the form in which the time appears in the dynamic system with applicability in economics we can classify the dynamic systems in autonomous systems and non-autonomous systems. For both cases there are three theorems in the speciality literature (six all in all) , which tell us whether there is a function , usually noted with V which fulfils the conditions in the theorems , and then we can say if the solution is stable, asymptotic stable or instable. So, in this paper we will also deal the Liapunov stability or the stability in Recent Researches in Tourism and Economic Development ISBN: 978-1-61804-043-5 119