Computational Methods for Global Analysis of Connecting Orbits: Development of Algorithms and Applications

弗里德曼 同宿轨道和异宿轨道，也被称为连接轨道，是连接自治常微分方程组的不动点的无限周期轨道。 同宿轨道的出现和消失可以导致周期轨道的产生或破坏。 同宿轨道的余维2分支可以起到有趣的动力学的组织中心的作用。 结构不稳定的连接轨道的存在会导致与奇异吸引子的诞生或死亡相关的分叉。 连接轨道经常出现在抛物型和双曲型偏微分方程的行波解中。 最近Doedel和研究人员开发和分析了一种精确的，鲁棒的，系统的方法来计算分支的连接轨道的情况下，该解决方案接近平衡指数以及在中心流形的情况下，并考虑了一些应用。 研究者在几个方向上扩展了这些结果：1）推广了起始连接轨道的定位算法; 2）改进了分支上分叉的检测和特定分支的选择; 3）定位多峰同宿轨道。4)离散化问题轨道的定性全局误差分析和截尾区间的自适应选择; 5）Banach空间中连通轨道的计算。 应用包括：完成了sine-Gordon方程和Hodgkin-Huxley方程中连接轨分支的研究。 FitzHugh-Nagumo方程行波解的分支及其推广：寻找扭点的异宿环分支。 共振带同宿轨道的计算研究。 电子电路中连接轨道的分叉。 离散时间动力系统中的分岔曲线。 Banach空间中异宿轨道的分叉及其在凝固现象相场模型中的应用。 连接轨道在各种应用中具有特殊意义。 它们已经被证明是流体力学中的非线性现象、数学生物学中的“突发”现象、电路的混沌行为、激光的混沌行为以及光纤应用中的光脉冲、结构力学中的混沌行为和各种化学反应的基础。 因此，迫切需要开发对相互连接的轨道进行全球分析的软件。 研究人员扩展了先前在NSF支持下获得的结果，开发了用于连接轨道全球分析的稳健数值算法，并将所产生的算法纳入了一个众所周知的连续计算机程序，以使更广泛的科学界更容易获得这些结果。 研究人员还考虑了各种独立感兴趣的应用程序，这些应用程序也将用于测试和改进所开发的算法。 最终的目标是创建一个集成的交互系统，包括各种启动程序和延续方法。

Friedman Homoclinic and heteroclinic orbits, also referred as connecting orbits, are orbits of infinite period connecting fixed points of a system of autonomous ordinary differential equations. Appearance and disappearance of a homoclinic orbit can lead to creation or destruction of periodic orbits. Bifurcations of codimension 2 of homoclinic orbits can play the role of organizing centers for the interesting dynamics. Existence of a structurally unstable connecting orbit can lead to bifurcations associated with the birth or death of a strange attractor. Connecting orbits often arise as traveling wave solutions of parabolic and hyperbolic PDEs. Recently Doedel and the investigator developed and analyzed an accurate, robust, and systematic method for computing branches of connecting orbits in the case that the solution approaches the equilibria exponentially as well as in the case of center manifolds, and considered some applications. The investigator extends these results in several directions: 1) Generalizing the algorithm for locating starting connecting orbits, 2) Improving detection of bifurcations on a branch and selecting specific branches, 3) Locating multi-hump homoclinic orbits. 4) Qualitative global error analysis of orbits for the discretized problem and Adaptive choice of the truncating interval using a posteriori error estimates, 5) Computation of connecting orbits in a Banach space. Applications include: The completion of the investigation of the bifurcations of connecting orbits in the sine-Gordon equation and the Hodgkin–Huxley equations. Bifurcations of traveling wave solutions in the FitzHugh-Nagumo equations and its generalization: a heteroclinic loop bifurcation, searching for twisting points. Computational investigation of orbits homoclinic to resonance bands. Bifurcations of connecting orbits in an electronic circuit. Bifurcation curves in dynamical systems with discrete time. Bifurcation of heteroclinic orbits in a Bana ch space, with applications to the phase field model of solidification phenomena. Connecting orbits are of special significance in a variety of applications. They have been shown to underlie the phenomena of intermittency in fluid mechanics, "bursting" phenomena in mathematical biology, chaotic behavior of electrical circuits, chaotic behavior of lasers as well as light pulses in fiber optics applications, chaotic behavior in structural mechanics and in a variety of chemical reactions. The need is therefore paramount for the development of software for global analysis of connecting orbits. The investigator extends the results obtained under prior NSF support on development of robust numerical algorithms for global analysis of connecting orbits and incorporates the resulting algorithms into AUTO, a well known continuation computer program, to make them more accessible to a wider scientific community. The investigator also considers various applications that are of independent interest, and that will also be used to test and refine the developed algorithms. The final goal is to create an integrated interactive system that incorporates various starting procedures and continuation methods.