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Transactions of the American Mathematical Society

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Characteristic multipliers and stability of symmetric periodic solutions of $ \dot x(t)=g(x(t-1))$


Authors: Shui-Nee Chow and Hans-Otto Walther
Journal: Trans. Amer. Math. Soc. 307 (1988), 127-142
MSC: Primary 34K20; Secondary 34C25, 58F14
DOI: https://doi.org/10.1090/S0002-9947-1988-0936808-2
MathSciNet review: 936808
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Abstract: We study the scalar delay differential equation $ \dot x(t) = g(x(t - 1))$ with negative feedback. We assume that the nonlinear function $ g$ is odd and monotone. We prove that periodic solutions $ x(t)$ of slowly oscillating type satisfying the symmetry condition $ x(t) = - x(t - 2)$, $ t \in {\mathbf{R}}$, are nondegenerate and have all nontrivial Floquet multipliers strictly inside the unit circle. This says that the periodic orbit $ \{ {x_t}:t \in {\mathbf{R}}\} $ in the phase space $ C[ - 1,\,0]$ is orbitally exponentially asymptotically stable.


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DOI: https://doi.org/10.1090/S0002-9947-1988-0936808-2
Article copyright: © Copyright 1988 American Mathematical Society

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