Abstract

In this article, we have discussed the stability of second order linear and non-linear systems by characteristic roots. In the case of non-linear system, we linearize the nonlinear system under certain specified conditions and study the stability of critical points of the linearized systems. Necessary theories have been presented, applied, and illustrated with examples. A self-contained theory for a homogeneous linear system of third order is built by using the basic concept of the differential equation.

Highlights

  • Stability means a situation in which something is not likely to move or change that is the state or quality of being stable.2

  • A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the dx =P (x, y), dt function itself and its derivatives of various orders

  • The simplest differential equations admit solutions dt given by explicit formulas

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Summary

STABILITY OF SECOND ORDER SYSTEMS BY CHARACTERISTIC ROOTS

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the dx =P (x, y), dt function itself and its derivatives of various orders. Let second and third-order linear and nonlinear ordinary differential equations. The critical point (0, 0) is called stable if for every number valuable results Such a “linearization” is not always feasible; and when ε >0, there exists a number δ >0 such that the following is true: Every it is not original nonlinear equation itself must be considered. Λ2 − 2λ + 1 = 0 If one of the two real roots of the characteristic equation (3) ⟹ (λ − 1)2 = 0 of the linear system (2) is positive or two conjugate complex ∴ λ =1, 1 with positive real parts, (0, 0) is an unstable critical point of (2), and (0, 0) is an unstable critical point Here the roots are real, equal and both positive and the system (1) is of (1).

Dissimilar Stability of Non-linear Systems
A A2 A
CONCLUSIONS
CONFLICT OF INTERESTS

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