Continuous Dynamical Systems

In many economic and financial models it is important to study the changes over time of the variables involved in the models. In a sense, the static models examined in the previous chapters can be regarded as a “photo” of a phenomenon at a certain time, whereas in the present chapter and in the next chapter we are concerned with the “film” of a phenomenon during a running time period. If time is considered as a continuous variable, we have differential equations and systems of differential equations; if time is considered as a discrete variable, we have finite difference equations and systems of finite difference equations, these ones presented in the next chapter. This chapter treats the basic topics on differential equations and systems of differential equations. Not only these topics have revealed their utility in science and engineering, but also in economic and financial models. Particularly important are the notions of “equilibrium solutions” and the notions of “stability” of the said solutions. These subjects are central in the theory of competitive (Walrasian) equilibrium, in business cycle theory, in growth theory, etc. We give only the basic notions of all these topics, as a complete treatment would require an entire book. The reader is referred to the References for further developments.