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We study entanglement in two coupled quartic oscillators. It is shown thatthe entanglement, as measured by the von Neumann entropy, increases with theclassical chaos parameter for generic chaotic eigenstates. We consider certainisolated periodic orbits whose bifurcation sequence affects a class of quantumeigenstates, called the channel localized states. For these states, theentanglement is a local minima in the vicinity of a pitchfork bifurcation butis a local maxima near a anti-pitchfork bifurcation. We place these results inthe context of the close connections that may exist between entanglementmeasures and conventional measures of localization that have been much studiedin quantum chaos and elsewhere. We also point to an interesting near-degeneracythat arises in the spectrum of reduced density matrices of certain states as aninterplay of localization and symmetry.

Classical bifurcations and entanglement in smooth Hamiltonian system