Torus-doubling bifurcations and strange nonchaotic attractors in a vibro-impact system

Whether strange nonchaotic attractors (SNAs) can occur typically in dynamical systems other than quasiperiodically driven systems has been an open problem. Here we show that the SNAs can be induced by the interruption of torus-doubling bifurcation in a periodically driven vibro-impact system. It is...

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Bibliographic Details
Published in:Journal of sound and vibration 2013-10, Vol.332 (21), p.5462-5475
Main Authors: Zhang, Yongxiang, Luo, Guanwei
Format: Article
Language:English
Subjects:
Bifurcations
Dynamical systems
Dynamics
Fourier analysis
Lyapunov exponents
Mathematical analysis
Orbits
Vibration
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Summary:Whether strange nonchaotic attractors (SNAs) can occur typically in dynamical systems other than quasiperiodically driven systems has been an open problem. Here we show that the SNAs can be induced by the interruption of torus-doubling bifurcation in a periodically driven vibro-impact system. It is found that the creation of SNAs occurs due to the collision of doubled torus with some unstable periodic orbits. Based on the Poincaré map technique, two types of codimension-3 bifurcations are represented and typical torus-doubling bifurcations can occur in the neighborhood of these bifurcation points. We use the singular continuous Fourier spectrum and its scaling to verify the strangeness (nondifferentiability) of the attractors. The implicitly transcendental map is used to calculate the largest Lyapunov exponent by QR-based algorithm, which ensures that the underlying dynamics (with nonpositive Lyapunov exponents) is nonchaotic. The associated mechanism is described for the creation of SNAs through detecting unstable periodic orbits from Poincaré map series. •The strange nonchaotic attractors (SNAs) are verified in a vibro-impact system.•The interruption of torus doubling bifurcation and genesis of SNAs is investigated.•Two types of codimension-3 bifurcations are represented.•The unstable sets embedded in SNAs are detected.
ISSN:0022-460X
1095-8568
DOI:10.1016/j.jsv.2013.04.046