Yuu Miino, Tetsushi Ueta, Hiroshi Kawakami, Bending angles of a broken line causing bifurcations and chaos, Institute of Electronics, Information and Communication Engineers, Nonlinear Theory and Its Applications, IEICE, Vol.11, No.3, 359-371, July 2020.
Abstract: We replace the cubic characteristics in the Duffing equation by two line segments connected at a point and investigate how an angle of that broken line conducts bifurcations to periodic orbits. Firstly we discuss differences in periodic orbits between the Duffing equation and a forced planar system including the broken line. In the latter system, a grazing bifurcation split the parameter space into the linear and nonlinear response domains. Also, we show that bifurcations of non-resonant periodic orbits appeared in the former system are suppressed in the latter system. Secondly, we obtain bifurcation diagrams by changing a slant parameter of the broken line. We also nd the parameter set that a homoclinic bifurcation arises and the corresponding horseshoe map. It is clari ed that a grazing bifurcation and tangent bifurcations form boundaries between linear and nonlinear responses. Finally, we explore the piecewise linear functions that show the minimum bending angles exhibiting bifurcation and chaos.