论文标题
使用不变歧管和koopman eigenfunctions求解非线性普通微分方程
Solving nonlinear ordinary differential equations using the invariant manifolds and Koopman eigenfunctions
论文作者
论文摘要
非线性普通微分方程很少可以通过分析求解。 Koopman运算符理论提供了一种通过使用特征函数将非线性动力学映射到线性空间来求解非线性系统的方法。不幸的是，很难找到这样的本本函数。我们介绍了一种从非线性颂歌的不变歧管构造本征函数的方法。该方法成功允许我们找到用于恒定系数非线性系统的分析解决方案。以前的数据驱动方法已使用Koopman理论来构建在不同相空间不同区域有效的本地Koopman本征函数近似值。我们的方法找到了精确且在全球有效的分析koopman eigenfunctions。我们演示了我们在一维和2维odes上求解非线性系统的方法。所考虑的非线性示例具有简单的表达式，可为其不变歧管产生可拖动的分析解决方案。因此，我们的方法允许为先前未解决的普通微分方程构建分析解决方案。它还突出了非线性普通微分方程中不变的流形和本征函数之间的联系，并提出了扩展此方法以求解更多非线性系统的途径。
Nonlinear ordinary differential equations can rarely be solved analytically. Koopman operator theory provides a way to solve nonlinear systems by mapping nonlinear dynamics to a linear space using eigenfunctions. Unfortunately, finding such eigenfunctions is difficult. We introduce a method for constructing eigenfunctions from a nonlinear ODE's invariant manifolds. This method, when successful, allows us to find analytical solutions for constant coefficient nonlinear systems. Previous data-driven methods have used Koopman theory to construct local Koopman eigenfunction approximations valid in different regions of phase space; our method finds analytic Koopman eigenfunctions that are exact and globally valid. We demonstrate our Koopman method of solving nonlinear systems on 1-dimensional and 2-dimensional ODEs. The nonlinear examples considered have simple expressions for their invariant manifolds which produce tractable analytical solutions. Thus our method allows for the construction of analytical solutions for previously unsolved ordinary differential equations. It also highlights the connection between invariant manifolds and eigenfunctions in nonlinear ordinary differential equations and presents avenues for extending this method to solve more nonlinear systems.
