A proof of Wright's conjecture |
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Authors: | Jan Bouwe van den Berg Jonathan Jaquette |
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Institution: | 1. Department of Mathematics, VU Amsterdam, de Boelelaan 1081, 1081 HV Amsterdam, The Netherlands;2. Department of Mathematics, Rutgers, The State University Of New Jersey, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA |
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Abstract: | Wright's conjecture states that the origin is the global attractor for the delay differential equation for all when . This has been proven to be true for a subset of parameter values α. We extend the result to the full parameter range , and thus prove Wright's conjecture to be true. Our approach relies on a careful investigation of the neighborhood of the Hopf bifurcation occurring at . This analysis fills the gap left by complementary work on Wright's conjecture, which covers parameter values further away from the bifurcation point. Furthermore, we show that the branch of (slowly oscillating) periodic orbits originating from this Hopf bifurcation does not have any subsequent bifurcations (and in particular no folds) for . When combined with other results, this proves that the branch of slowly oscillating solutions that originates from the Hopf bifurcation at is globally parametrized by . |
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Keywords: | Delay differential equation Hopf bifurcation Wright's conjecture Supercritical bifurcation branch Newton–Kantorovich theorem |
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