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  "body": "\n## Summary\n\nThe Lyapunov–Schmidt reduction turns a nonlinear equation $F(x, \\lambda) = 0$ between infinite-dimensional spaces into an equivalent system of finitely many equations in finitely many unknowns.\nIt works near a solution where the derivative of $F$ is a Fredholm operator, which is close to being invertible.\nThe part of the equation where the derivative is invertible is solved by the [[implicit function theorem|0BXBADR94C]].\nWhat remains is a small bifurcation equation, which describes how solutions branch as the parameter $\\lambda$ varies.\nSee the [[definition|CC666JT89X]].\n\n## Explanations\n\n### L1 human\n\nMany problems ask for the solutions of an equation that depends on a parameter, such as the shape of a beam under a load.\nUsually the solution changes smoothly with the parameter, but at special values several solutions can appear.\nThe Lyapunov–Schmidt method splits the unknown into a few \"dangerous\" directions and all the others.\nThe other directions can be solved for, so only the few dangerous directions are left.\nThe problem then becomes a question about a small number of equations, which can often be solved by hand.\n",
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