چکیده:
In recent years, it has become increasingly important to incorporate explicit
dynamics in economic analysis.
These two tools that mathematicians have developed, differential equations and
optimal control theory, are probably the most basic for economists to analyze
dynamic problems.
In this paper I will consider the linear differential equations on the plane
(phase diagram) and elements of nonlinear systems, when we have unequal real
roots of the same signs and opposite signs of characteristic roots, and the
applications of the theory of differential equations to certain macroeconomic
problems. The basic tools for discussion are phase diagram techniques.
خلاصه ماشینی:
"In this paper I will consider the linear differential equations on the plane (phase diagram) and elements of nonlinear systems, when we have unequal real roots of the same signs and opposite signs of characteristic roots, and the applications of the theory of differential equations to certain macroeconomic problems.
1- The Phase Plane: Linear Systems Since many differential equations cannot be solved conveniently by analytical methods, it is important to consider what qualitative information can be obtained about their solutions without actually solving the equations.
Since Liapunov’s method is not confined to the two-dimensional case, the system of n first-order nonlinear differential equations can be considered as follows: (14) The system (14) is a compact form of Let the origin be an equilibrium point of (13), so The heart of Liapunov’s direct method is to construct a real-valued function V(x) in which x is governed by (13).
L. (2001) were considered a discrete-time economic model where the savings are proportional to income and investment demand depends on the difference between the current income and its exogenously assumed equilibrium level, through a nonlinear S-Shaped increasing function and they also analyzed how changes in the parameters’ values modify both the asymptotic dynamics of the system and the structure of the basins of the different and often coexisting attractors in the phase-plane.
In this article at first, I review the linear differential equation on the plane (phase diagram) and nonlinear systems, when we have unequal real roots of the same signs and opposite signs of characteristic roots."