Roberts Andrew, Gendinning Paul
Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599-3250, USA.
Department of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom.
Chaos. 2014 Jun;24(2):023138. doi: 10.1063/1.4885502.
We show that a nonlinear, piecewise-smooth, planar dynamical system can exhibit canard phenomena. Canard phenomena in nonlinear, piecewise-smooth systems can be qualitatively more similar to the phenomena in smooth systems than piecewise-linear systems, since the nonlinearity allows for canards to transition from small cycles to canards "with heads." The canards are born of a bifurcation that occurs as the slow-nullcline coincides with the splitting manifold. However, there are conditions under which this bifurcation leads to a phenomenon called super-explosion, the instantaneous transition from a globally attracting periodic orbit to relaxations oscillations. Also, we demonstrate that the bifurcation-whether leading to canards or super-explosion-can be subcritical.
我们表明，一个非线性、分段光滑的平面动力系统能够展现出鸭解现象。非线性分段光滑系统中的鸭解现象在定性上可能比分段线性系统中的现象更类似于光滑系统中的现象，因为非线性允许鸭解从小周期转变为“带头部”的鸭解。鸭解产生于慢零倾线与分裂流形重合时发生的分岔。然而，在某些条件下，这种分岔会导致一种称为超爆炸的现象，即从全局吸引的周期轨道瞬间转变为松弛振荡。此外，我们证明这种分岔——无论是导致鸭解还是超爆炸——都可能是亚临界的。
