标量延迟微分方程中双状态依赖延迟的共振现象
Resonance phenomena in a scalar delay differential equation with two state-dependent delays
AI总结：
本文研究了具有两个状态依赖延迟反馈项的标量延迟微分方程，通过双反馈强度平面的分支分析，揭示了状态依赖延迟导致的复杂动力学行为，包括霍普夫-霍普夫分支点、托罗斯分支和共振舌现象。
摘要：
我们研究了一个具有两个延迟反馈项的标量延迟微分方程（DDE），这两个反馈项线性依赖于状态。通过冻结状态依赖性得到的常数延迟DDE是线性的且没有递归动力学。然而，当使用状态依赖的延迟项时，DDE表现出非常复杂的动力学。为此，我们对系统进行了分支分析，并在两个反馈强度的平面上绘制了其分支图。该图由霍普夫-霍普夫分支点组成，这些点导致托罗斯分支曲线和形式为不变托罗斯和共振舌的双频动力学。我们通过计算中心流形上的正常形式确定霍普夫-霍普夫分支点的类型；这需要将定义状态依赖DDE的功能展开为幂级数，其中仅包含常数延迟的项。我们使用Matlab中的符号微分实现了该展开和正常形式系数的计算。托罗斯分支曲线的数值继续验证了我们正常形式计算的正确性。此外，它使我们能够更全局地计算托罗斯分支曲线，并找到与共振舌相关的周期轨道鞍点分支曲线。托罗斯本身在三维投影中计算并可视化，同时还有合适Poincaré截面的平面轨迹。特别是，我们计算了锁定托罗斯上的周期轨道及其相关不稳定流形。这使我们能够研究通过共振舌的过渡和1:4锁定托罗斯的破裂。本文展示了仅状态依赖性就足以产生丰富动力学现象的能力。
英文摘要：
We study a scalar DDE with two delayed feedback terms that depend linearly on the state. The associated constant-delay DDE, obtained by freezing the state dependence, is linear and without recurrent dynamics. With state dependent delay terms, on the other hand, the DDE shows very complicated dynamics. To investigate this, we perform a bifurcation analysis of the system and present its bifurcation diagram in the plane of the two feedback strengths. It is organized by Hopf-Hopf bifurcation points that give rise to curves of torus bifurcation and associated two-frequency dynamics in the form of invariant tori and resonance tongues. We numerically determine the type of the Hopf-Hopf bifurcation points by computing the normal form on the center manifold; this requires the expansion of the functional defining the state-dependent DDE in a power series whose terms up to order three only contain constant delays. We implemented this expansion and the computation of the normal form coefficients in Matlab using symbolic differentiation. Numerical continuation of the torus bifurcation curves confirms the correctness of our normal form calculations. Moreover, it enables us to compute the curves of torus bifurcations more globally, and to find associated curves of saddle-node bifurcations of periodic orbits that bound the resonance tongues. The tori themselves are computed and visualized in a three-dimensional projection, as well as the planar trace of a suitable Poincaré section. In particular, we compute periodic orbits on locked tori and their associated unstable manifolds. This allows us to study transitions through resonance tongues and the breakup of a 1:4 locked torus. The work presented here demonstrates that state dependence alone is capable of generating a wealth of dynamical phenomena.
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