论文标题
奇异的扰动边界聚焦分叉
Singularly Perturbed Boundary-Focus Bifurcations
论文作者
论文摘要
我们认为将限制的平滑系统限制为具有边界焦点（BF）分叉的分段平滑（PWS）系统的$ε\至0 $。得出合适的局部正常形式后，我们使用几何奇异扰动理论和爆炸的组合研究了$ 0 <ε\ ll 1 $的平滑系统动力学。我们表明，PWS系统中的BF分叉类型确定了$ε-$依赖性域内平滑系统的分叉结构，该结构将其缩小到$ε\ 0 $，从而确定了一个超批评的Andronov-Hopf Bifurcation在一个情况下，以及一个超策略性的Bogdanov-Takanov-Takensbifurcentiation。我们还表明，与BF分叉相关的PWS循环仍然是平滑系统中的松弛周期，并证明存在一个稳定极限循环的家族，该循环将松弛周期与上述$ε-$依赖性领域内的常规周期联系起来。我们的结果应用于牙槽捕食者捕食者相互作用和机械振荡的模型。
We consider smooth systems limiting as $ε\to 0$ to piecewise-smooth (PWS) systems with a boundary-focus (BF) bifurcation. After deriving a suitable local normal form, we study the dynamics for the smooth system with $0 < ε\ll 1$ using a combination of geometric singular perturbation theory and blow-up. We show that the type of BF bifurcation in the PWS system determines the bifurcation structure for the smooth system within an $ε-$dependent domain which shrinks to zero as $ε\to 0$, identifying a supercritical Andronov-Hopf bifurcation in one case, and a supercritical Bogdanov-Takens bifurcation in two other cases. We also show that PWS cycles associated with BF bifurcations persist as relaxation cycles in the smooth system, and prove existence of a family of stable limit cycles which connects the relaxation cycles to regular cycles within the $ε-$dependent domain described above. Our results are applied to models for Gause predator-prey interaction and mechanical oscillation subject to friction.
