Soliman, Mohamed S.; Thompson, J. M. T. Stochastic penetration of smooth and fractal basin boundaries under noise excitation. (English) Zbl 0726.70020 Dyn. Stab. Syst. 5, No. 4, 281-298 (1990). (Authors’ summary:) Recent work on the escape of a sinusoidally driven oscillator from a universal cubic potential well has elucidated the complex patterns of attractor and basin bifurcations that govern the escape process. Optimal escape, under a minimum forcing magnitude, occurs at a forcing frequency of about 80 per cent of the small-amplitude linear natural frequency, and at this forcing frequency we have identified a significant and dramatic erosion of the safe basin of attraction, triggered by a homoclinic tangency, that would seriously impair the engineering integrity of a practical system long before the final chaotic instability of the constrained attractor. Introducing a superimposed noise excitation, we here quantify this in terms of a stochastic integrity measure, and correlate this with the geometric changes experienced by the deterministic basin of attraction. Reviewer: P.Smith (Keele) Cited in 4 Documents MSC: 70K50 Bifurcations and instability for nonlinear problems in mechanics 34C23 Bifurcation theory for ordinary differential equations 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 70L05 Random vibrations in mechanics of particles and systems 34C25 Periodic solutions to ordinary differential equations Keywords:escape of a sinusoidally driven oscillator; universal cubic potential; patterns of attractor; basin bifurcations; escape process; Optimal escape; minimum forcing magnitude; small-amplitude linear natural frequency; homoclinic tangency; superimposed noise excitation; stochastic integrity measure PDFBibTeX XMLCite \textit{M. S. Soliman} and \textit{J. M. T. Thompson}, Dyn. Stab. Syst. 5, No. 4, 281--298 (1990; Zbl 0726.70020) Full Text: DOI References: [1] Abraham R. H., Physica 21 pp 394– (1986) [2] DOI: 10.1016/0375-9601(84)90035-5 · doi:10.1016/0375-9601(84)90035-5 [3] Crandall S. H., Random Vibrations in Mechanical Systems (1963) [4] DOI: 10.1016/0370-1573(82)90089-8 · doi:10.1016/0370-1573(82)90089-8 [5] Dykman M. I., Conference on Noise and Chaos in Nonlinear Dynamical Systems · Zbl 0057.44703 [6] DOI: 10.1103/PhysRevLett.56.1011 · doi:10.1103/PhysRevLett.56.1011 [7] DOI: 10.1103/PhysRevLett.57.1284 · doi:10.1103/PhysRevLett.57.1284 [8] Grebogi C., Physica 24 pp 243– (1987) [9] Gwinn E. G., Physica 23 pp 396– (1986) [10] DOI: 10.1103/PhysRevA.33.4143 · doi:10.1103/PhysRevA.33.4143 [11] DOI: 10.1016/0022-460X(87)90535-9 · Zbl 1235.74046 · doi:10.1016/0022-460X(87)90535-9 [12] DOI: 10.1016/0022-460X(89)90699-8 · Zbl 1235.70106 · doi:10.1016/0022-460X(89)90699-8 [13] DOI: 10.1098/rspa.1989.0009 · Zbl 0674.70035 · doi:10.1098/rspa.1989.0009 [14] J. M. T. Thompson, Proceedings of the Royal Society 428 pp 1– (1990) · Zbl 0692.70032 · doi:10.1098/rspa.1990.0022 [15] Thompson J. M. T., Nonlinear Dynamics and Chaos (1986) [16] DOI: 10.1080/02681118908806077 · Zbl 0681.70029 · doi:10.1080/02681118908806077 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.