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The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations

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  • Assume that the unperturbed parabolic equation has a degenerate homoclinic orbit. Under $T$-periodic perturbations, the periodic solutions bifurcated from the homoclinic solution are studied. By Fredholm alternative and Lyapunov-Schmidt reduction, the bifurcation functions defined between two finite-dimensional spaces are obtained. Some solvable conditions for the bifurcation functions are given. It is shown that, for any large $n>0$, the perturbed parabolic differential equation has a periodic solution with period $nT$.
    Mathematics Subject Classification: Primary: 34C23, 34C25; Secondary: 34C45, 34C40.


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