Rigorous Verification of Hopf Bifurcations via Desingularization and Continuation
Copyright policy )Rigorous Verification of Hopf Bifurcations via Desingularization and Continuation
In this paper we present a general approach to rigorously validate Hopf bifurcations as well as saddle-node bifurcations of periodic orbits in systems of ODEs. By a combination of analytic estimates and computer-assisted calculations, we follow solution curves of cycles through folds, checking along the way that a single nondegenerate saddle-node bifurcation occurs. Similarly, we rigorously continue solution curves of cycles starting from their onset at a Hopf bifurcation. We use a blowup analysis to regularize the continuation problem near the Hopf bifurcation point. This extends the applicability of validated continuation methods to the mathematically rigorous computational study of bifurcation problems.
31 pages, 8 figures
General methods in interval analysis, Hopf bifurcation ¨continuation ¨desingularization ¨computer-assisted proofs Mathematics Subject Classification (2010) 37G15 ¨65P30 ¨65G40 ¨34C25 ¨37C27, 34C25, [MATH] Mathematics [math], Desingularization, Dynamical Systems (math.DS), computer-assisted proofs, Bifurcations of limit cycles and periodic orbits in dynamical systems, Numerical bifurcation problems, computer-assisted proofs AMS subject classifications. 37G15, FOS: Mathematics, Computational methods for bifurcation problems in dynamical systems, Hopf bifurcation, Mathematics - Numerical Analysis, Mathematics - Dynamical Systems, Computer-assisted proofs, 65P30, desingularization, Continuation, Numerical Analysis (math.NA), 37C27, 68V05, 37G15, 37C27, Hopf bifurcation continuation desingularization computer-assisted proofs AMS subject classifications. 37G15 65P30 65G40 34C25 37C27, 65G40, continuation
General methods in interval analysis, Hopf bifurcation ¨continuation ¨desingularization ¨computer-assisted proofs Mathematics Subject Classification (2010) 37G15 ¨65P30 ¨65G40 ¨34C25 ¨37C27, 34C25, [MATH] Mathematics [math], Desingularization, Dynamical Systems (math.DS), computer-assisted proofs, Bifurcations of limit cycles and periodic orbits in dynamical systems, Numerical bifurcation problems, computer-assisted proofs AMS subject classifications. 37G15, FOS: Mathematics, Computational methods for bifurcation problems in dynamical systems, Hopf bifurcation, Mathematics - Numerical Analysis, Mathematics - Dynamical Systems, Computer-assisted proofs, 65P30, desingularization, Continuation, Numerical Analysis (math.NA), 37C27, 68V05, 37G15, 37C27, Hopf bifurcation continuation desingularization computer-assisted proofs AMS subject classifications. 37G15 65P30 65G40 34C25 37C27, 65G40, continuation
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