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This paper investigates some nonlinear dynamical behaviors about domains of attraction, bifurcations, and chaos in an axially accelerating viscoelastic beam under a time-dependent tension and a time-dependent speed. The axial speed and the axial tension are coupled to each other on the basis of a harmonic variation over constant initial values. The transverse motion of the moving beam is governed by nonlinear integro-partial-differential equations with the rheological model of the Kelvin–Voigt energy dissipation mechanism, in which the material derivative is applied to the viscoelastic constitutive relation. The fourth-order Galerkin truncation is employed to transform the governing equation to a set of nonlinear ordinary differential equations. The nonlinear phenomena of the system are numerically determined by applying the fourth-order Runge–Kutta algorithm. The tristable and bistable domains of attraction on the stable steady state solution with a three-to-one internal resonance are analyzed emphatically by means of the fourth-order Galerkin truncation and the differential quadrature method, respectively. The system parameters on the bifurcation diagrams and the maximum Lyapunov exponent diagram are demonstrated by some numerical results of the displacement and speed of the moving beam. Furthermore, chaotic motion is identified in the forms of time histories, phase-plane portraits, fast Fourier transforms, and Poincaré sections.

BACKGROUND This paper introduces a case of recurrent keratoacanthoma (KA). KA is a self-healing disease. Recurrence after surgical resection is rare. In this case, the local application of retinoic acid ointment after the second operation achieved a good prognosis after 2 years of follow-up. CASE SUMMARY A 76-year-old male patient was admitted to the hospital for "lower lip rupture and scab for 3 mo". Treatment: A rectangular incision was made in the healthy tissue about 3 mm outside the periphery of the lower lip mass, and a modified Bernard sliding flap was designed to completely remove the mass. Pathology showed (lower lip) KA. When the patient returned 6 mo after surgery, the middle mucosa of the lower lip had a bulge with a diameter of about 0.5 cm. The boundary was still clear, the surface was ulcerated. A recurrence of lower lip KA was suspected and a fan-shaped incision was performed in the healthy tissue about 5 mm outside the lesion to completely resect. Pathological showed lower lip KA had recurred. Topical application of tretinoin cream was applied once a day for 3 mo. The lower lip wounds were clean at the 2-year postoperative follow-up and the mucosa was normal. CONCLUSION Adjuvant retinoic acid treatment after KA surgical resection can achieve good results.