Lyapunov exponents and shear-induced chaos for a Hopf bifurcation with additive noise

• 成果类型: Article
• 署名作者: Baxendale, Peter H.
• 署名单位: University of Southern California
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01301-4
• 发表日期: 2025
• 页码: 375-425
• 关键词: strange attractors; STABILITY; systems; oscillator

摘要

This paper considers the effect of additive white noise on the normal form for the supercritical Hopf bifurcation in 2 dimensions. The main results involve the asymptotic behavior of the top Lyapunov exponent lambda associated with this random dynamical system as one or more of the parameters in the system tend to 0 or infinity. This enablesthe construction of a bifurcation diagram in parameter space showing stable regionswhere lambda<0 (implying synchronization) and unstable regions where lambda>0 (implyingchaotic behavior). The value of lambda depends strongly on the shearing effect of the twistfactorb/aof the deterministic Hopf bifurcation. Ifb/ais sufficiently small then lambda<0regardless of all the other parameters in the system. But when all the parameters exceptbare fixed then lambda grows like a positive multiple of b(2/3)as b ->infinity