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- Title
Nonsmooth Pitchfork Bifurcations in a Quasi-Periodically Forced Piecewise-Linear Map.
- Authors
Jorba, Àngel; Tatjer, Joan Carles; Zhang, Yuan
- Abstract
We study a family of one-dimensional quasi-periodically forced maps F a , b (x ,) = (f a , b (x ,) , + ω) , where x is real, is an angle, and ω is an irrational frequency, such that f a , b (x ,) is a real piecewise-linear map with respect to x of certain kind. The family depends on two real parameters, a > 0 and b > 0. For this family, we prove the existence of nonsmooth pitchfork bifurcations. For a < 1 and any b , there is only one continuous invariant curve. For a > 1 , there exists a smooth map b = b 0 (a) such that: (a) For b < b 0 (a) , f a , b has two continuous attracting invariant curves and one continuous repelling curve; (b) For b = b 0 (a) , it has one continuous repelling invariant curve and two semi-continuous (noncontinuous) attracting invariant curves that intersect the unstable one in a zero-Lebesgue measure set of angles; (c) For b > b 0 (a) , it has one continuous attracting invariant curve. The case a = 1 is a degenerate case that is also discussed in the paper. It is interesting to note that this family is a simplified version of the smooth family G a , b (x ,) = (arctan (a x) + b sin () , + ω) for which there is numerical evidence of a nonsmooth pitchfork bifurcation. Finally, we also discuss the limit case when a → ∞.
- Subjects
BIFURCATION diagrams; ANGLES; FAMILIES; SIN
- Publication
International Journal of Bifurcation & Chaos in Applied Sciences & Engineering, 2024, Vol 34, Issue 7, p1
- ISSN
0218-1274
- Publication type
Article
- DOI
10.1142/S0218127424500846