The non-Hermitian skin effect, nonreciprocity-induced anomalous localization of an extensive number of eigenstates, represents a hallmark of non-Hermitian topological systems with no analogs in Hermitian systems. Despite its significance across various open classical and quantum systems, the influence of nonlinearity has remained largely unclear. Here, we reveal the Hopf bifurcation of the nonlinear skin effect as a critical phenomenon unique to nonlinear non-Hermitian systems. We demonstrate that nonlinearity destabilizes skin states and instead gives rise to the emergence of delocalized states associated with limit cycles in phase space. We also uncover the algebraically localized critical skin effect precisely at the Hopf bifurcation point. We illustrate this behavior in a nonlinear extension of the Hatano-Nelson model in both continuum and lattice. Our Letter shows a significant role of nonlinearity in the skin effect and uncovers rich phenomena arising from the interplay between non-Hermiticity and nonlinearity.

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• Critical phenomena
• Nonlinear optics
• Phase transitions
• Skin effect
• Topological phases of matter
• Non-Hermitian systems
• Nonequilibrium systems
• Nonlinear waves
• Chaos & nonlinear dynamics
• Lattice models in condensed matter