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LinearizationLinearization in mathematics and its applications in general refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations. This method is used in fields such as engineering, physics, economics, and ecology. The analysis of linear functions is well defined, but most representations of actual systems are nonlinear. Linearization allows us to use tools for studying linear systems to analyze the behavior of a nonlinear function near a given point. The linearization of a function is the first order term of its Taylor expansion around the point of interest. For a system defined by the equation - ,
the linearized system can be written as -
where is the point of interest and is the Jacobian of evaluated at . In stability analysis, one can use the eigenvalues of the Jacobian matrix evaluated at an equilibrium point to determine the nature of that equilibrium. If all the eigenvalues are positive, the equilibrium is unstable; if they are all negative the equilibrium is stable; and is the values are of mixed signs, the equilibrium is a saddle point. Any complex eigenvalues will appear in complex conjugate pairs and indicate spiral (or circular if the real components are zero) around the equilibrium.
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