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| Recognition of Pitchfork Bifurcation in Banach Spaces |
| Xu Jingbo, Zhang Shiqi
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| Jilin Normal University, Siping 136000, China |
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Abstract Based on the orthogonality of Banach spaces and analogous to the method of simplifying operator equations using Liapunov Schmidt, a Liapunov Schmidt scheme is derived for converting bifurcation equations into finite dimensional or low dimensional algebraic equations. Determine the bifurcation point of the simplified bifurcation equation, and calculate the partial derivative value at the bifurcation point to construct a smooth function strongly equivalent to the bifurcation equation .When the constructed smooth function satisfies the same condition as the partial derivative of the Pitchfork bifurcation equation at the bifurcation point, the smooth function is strongly equivalent to the Pitchfork bifurcation equation. The condition satisfied by the partial derivative is the necessary and sufficient condition for identifying Pitchfork bifurcations, enriching the identification methods of bifurcations in Banach spaces.
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Published: 21 December 2025
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