saddle point theory

Saddle point theory involves analyzing the behavior of functions that describe a system’s state, especially where a point is neither a maximum nor a minimum but instead has properties of both—like a saddle on a horse. In mathematics and optimization, a saddle point is a point where the function's slope is zero, but the point isn't a local maximum or minimum. It’s a critical point where the system can be stable in some directions and unstable in others. This concept helps in understanding complex systems, from economics to physics, by identifying points where small changes can lead to different outcomes.