Title

Transitions Through Period Doubling Route To Chaos

Abstract

The Duffing driven, damped, "softening" oscillator has been analyzed for transition through period doubling route to chaos. The forcing frequency and amplitude have been varied in time (constant sweep). The stationary 27,47... chaos regions have been determined and used as the starting conditions for nonstationary regimes, consisting of the transition along the (Equation presented). The results are new, revealing, puzzling and complex. The nonstationary penetration phenomena (delay, memory) has been observed for a single and two-control nonstationary parameters. The rate of penetrations tends to zero with increasing sweeps, delaying thus the nonstationary chaos relative to the stationary chaos by a constant value. A bifurcation discontinuity has been uncovered at the stationary 27 bifurcation: the 27 bifurcation discontinuity drops from the upper branches of (a, Ω) or (a, f) curves to their lower branches. The bifurcation drops occur at the different control parameter values from the response x(t) discontinuities. The stationary bifurcation discontinuities are annihilated in the nonstationary bifurcation cascade to chaos-they reside entirely on the upper or lower nonstationary branches. A puzzling drop (jump) of the chaotic bifurcation bands has been observed for reversed sweeps. Extreme sensitivity of the nonstationary bifurcations to the starting conditions manifests itself in the flip-flop (mirror image) phenomena. The knowledge of the bifurcations allows for accurate reconstruction of the spatial system itself. The results obtained may model mathematically a number of engineering and physical systems.

Publication Date

1-1-1991

Publication Title

Proceedings of the ASME Design Engineering Technical Conference

Volume

Part F168436-4

Number of Pages

135-146

Document Type

Article; Proceedings Paper

Personal Identifier

scopus

DOI Link

https://doi.org/10.1115/DETC1991-0328

Socpus ID

85104711465 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/85104711465

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