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Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright’s equation. (English) Zbl 1200.34078
A Fourier expansion for periodic solutions of Wright’s equation \[ \dot{x}(t)=-\alpha x(t-1)(1+x(t)) \]
leads to a system of equations for the Fourier coefficients, with convolution terms. For \(\alpha\in[\frac{\pi}{2}+\epsilon, 2.3]\) \((\epsilon\approx 10^{-4})\), it is shown with computer assistance that this system can be solved uniquely (using the contraction principle) in a space of polynomially decaying sequences. Thus, a branch of periodic solutions parametrized with \(\alpha\) is obtained.
The method is applied a second time to a homotopy between Wright’s equation and the linear equation
\[ \dot{x}(t)=-\frac{\pi}{2}x(t-1), \]
to show that the constructed branch belongs to the continuum of the Hopf bifurcation at \(\alpha=\pi/2\).

MSC:
34K13 Periodic solutions to functional-differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
65L03 Numerical methods for functional-differential equations
Software:
INTLAB
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References:
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