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.
