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

弗里德曼同斜轨道和异斜轨道，也称为连接轨道，是连接自治常微分方程系统不动点的无限周期轨道。同斜轨道的出现和消失会导致周期轨道的产生或毁灭。同斜轨道的余维2分岔可以为有趣的动力学起组织中心的作用。结构不稳定的连接轨道的存在可能导致与奇异吸引子的诞生或死亡相关的分岔。连接轨道常作为抛物线和双曲偏微分方程的行波解出现。最近，Doedel和研究者开发并分析了一种精确的、鲁棒的、系统的方法，用于计算在解接近指数平衡点的情况下以及在中心流形的情况下连接轨道的分支，并考虑了一些应用。研究者在以下几个方面扩展了这些结果：1)推广了开始连接轨道的定位算法；2)改进了分支上分支的检测和选择特定分支；3)定位了多驼峰同斜轨道。4)对离散化问题的轨道进行定性全局误差分析，并利用后先验误差估计自适应选择截断区间；5)Banach空间中连接轨道的计算。应用包括：完成了sin - gordon方程和hodgkin方程中连接轨道分岔的研究；赫胥黎方程。FitzHugh-Nagumo方程行波解的分岔及其推广：寻找扭转点的异斜环分岔。轨道与共振带同斜的计算研究。电子电路中连接轨道的分岔。离散时间动力系统的分岔曲线。Bana - ch空间中异斜轨道的分岔，并应用于凝固现象的相场模型。连接轨道在各种应用中具有特殊的意义。它们已被证明是流体力学中间歇性现象、数学生物学中的“爆发”现象、电路的混沌行为、光纤应用中激光和光脉冲的混沌行为、结构力学和各种化学反应中的混沌行为的基础。因此，开发对连接轨道进行全球分析的软件的需求是至关重要的。研究人员扩展了先前在NSF支持下获得的结果，用于开发连接轨道的全局分析的鲁棒数值算法，并将结果算法合并到AUTO中，这是一个众所周知的延续计算机程序，使它们更容易被更广泛的科学界使用。研究人员还考虑各种独立的应用程序，这些应用程序也将用于测试和改进开发的算法。最终目标是创建一个整合了各种启动程序和延续方法的综合交互系统。

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.