Bifurcation and chaotic threshold of Duffing system with jump discontinuities Ruilan Tian, Yufeng Zhou, Qiubao Wang and Lili Zhang Eur. Phys. J. Plus, 131 1 (2016) 15 Published online: 18 January 2016 DOI: 10.1140/epjp/i2016-16015-9