# Node, Stable

# Node, Stable

If all the paths of equilibrium noncyclically converge to it, the equilibrium is a *stable node*. The conditions for the equilibrium to be a stable node are a function of the system being evaluated, in particular, whether the issue of concern involves discrete versus continuous change or is a linear versus nonlinear system. If the dynamic system is discrete and linear,

*y* ̇ = *ay* + *b*

The equilibrium is stable if and only if *a* > 0.

For continuous linear dynamic systems, an equilibrium is defined as a stable node if the slope of the differential equation in the neighborhood of the equilibrium is negative. For nonlinear discrete or continuous dynamic systems, the Taylor expansion about , a nonzero equilibrium, must satisfy the following for to be stable: . The concept of a stable node can be extended to simultaneous systems of discrete or continuous equations. The condition for a stable node is the same for discrete and continuous simultaneous systems of equation. If ǀλ_{i} ǀ < 0, where λ_{i} are the eigenvalues that solve equation, then λ_{i} is stable. If the eigenvalues are complex, the real part must be negative for the eigenvalues to be stable.

The concept of a stable node is used to describe the dynamics of (1) an oligopoly—stability of a Cournot solution; 2) the IS-LM—stability after monetary and/or fiscal shocks; (3) the model of inflation and unemployment—stability of fiscal or employment policies; and (4) population models—stability of growth. The characterization of a node as stable is most useful in the qualitative analysis of differential equations using phase diagrams, which describes the paths of a system in and out of equilibrium.

**SEE ALSO** *Matrix Algebra*

## BIBLIOGRAPHY

Hoy, Michael, John Livernois, Chris McKenna, and Ray Rees. 2001. *Mathematics for Economics*, 2nd ed. Cambridge, MA: MIT Press.

Shone, Ronald. 2002 *. Economic Dynamics: Phase Diagrams and Their Economic Application*, 2nd ed. Cambridge, U.K., and New York: Cambridge University Press.

*Rhonda V. Sharpe*

*Idrissa A. Boly*

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