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Limiting profiles for periodic solutions of scalar delay differential equations

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Let f(ยท,) : R → R be given so that f(0,) = 0 and f(x,) = (1 + )x + ax2 + bx3 + o(x3) as x → 0. We characterize those small values of > 0 and ∈ R for which there are periodic solutions of periods approximately 1/k with k ∈ N of the delay equations

xdot(t) = -x(t) + f(x(t - 1),).

When a = 0, these periodic solutions approach square waves if b < 0 or pulses if b > 0 as → 0. These results are similar to those obtained by Chow et al. and Hale and Huang, where the case of f(x,) = -(1 + )x + ax2 + bx3 + o(x3) as x → 0 is considered. However, when a ≠ 0, all these periodic solutions approach pulses as → 0; an interesting phenomenon that cannot happen in the case considered by Chow et al. and Hale and Huang.

Document Type: Research Article

Publication date: August 17, 2001


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