论文标题
周期性轨道和等速线的数值计算，用于状态依赖的延迟驱动驱动
Numerical computation of periodic orbits and isochrons for state-dependent delay perturbation of an ODE in the plane
论文作者
论文摘要
我们介绍算法及其实现，以计算限制周期及其等级依赖性延迟方程（SDDE），这些方程与具有限制周期的平面微分方程扰动。 请注意，SDDE解决方案的空间是无限的尺寸。我们计算了SDDE解决方案的两个参数家族，该溶液将扰动在限制周期的邻域中收敛到ode的溶液。 我们使用的方法在周期性函数之间（或函数成倍收敛到周期性）。功能方程表示函数求解SDDE。因此，我们考虑使用所需形状的功能空间并要求它们是解决方案，而不是进化初始数据并找到某种形状的解决方案。 这些不变方程的数学理论是在伴侣论文中开发的，该论文开发了“后验”定理。他们表明，如果有足够的近似解决方案（相对于某些明确的条件编号），则有一个接近大约的解决方案。由于数值方法会产生近似解决方案，并提供条件数量的估计值，因此我们可以确保我们考虑近似真实解决方案的数值解决方案。 在本文中，我们选择一种系统的方法来通过有限的数字集（泰勒风格系列）近似功能，并开发一种算法的工具包，以实现运算符（尤其是组合物），这些算法将进入理论。我们还提供了几个实施结果，并介绍了在某些代表性情况下运行算法及其实施的结果。
We present algorithms and their implementation to compute limit cycles and their isochrons for state-dependent delay equations (SDDE's) which are perturbed from a planar differential equation with a limit cycle. Note that the space of solutions of an SDDE is infinite dimensional. We compute a two parameter family of solutions of the SDDE which converge to the solutions of the ODE as the perturbation goes to zero in a neighborhood of the limit cycle. The method we use formulates functional equations among periodic functions (or functions converging exponentially to periodic). The functional equations express that the functions solve the SDDE. Therefore, rather than evolving initial data and finding solutions of a certain shape, we consider spaces of functions with the desired shape and require that they are solutions. The mathematical theory of these invariance equations is developed in a companion paper, which develops "a posteriori" theorems. They show that, if there is a sufficiently approximate solution (with respect to some explicit condition numbers), then there is a true solution close to the approximate one. Since the numerical methods produce an approximate solution, and provide estimates of the condition numbers, we can make sure that the numerical solutions we consider approximate true solutions. In this paper, we choose a systematic way to approximate functions by a finite set of numbers (Taylor-Fourier series) and develop a toolkit of algorithms that implement the operators -- notably composition -- that enter into the theory. We also present several implementation results and present the results of running the algorithms and their implementation in some representative cases.
