A fixed point P for a non-linear autonomous system is of the same nature (nodal, focus, saddle, or centre) as that of the associated linearized system.

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The nature of a fixed point for a non-linear autonomous system can differ from that of its linearized version due to non-linear effects. Linearization approximates the behavior near the fixed point, but it may not capture all the dynamics of the non-linear system, leading to potential differences in stability and type of fixed points.

A Fixed Point P for a Non-Linear Autonomous System Is