Outline and Description

1. Synchrony subspaces, balanced colorings, and quotient networks.
2. Classification of regular networks with a small number of identical nodes.
3. Synchrony-breaking bifurcations.
4. Synchrony, phase-shift synchrony and multirhythms in periodic solutions.
5. Feed forward networks.

References

• Overview:  M. Golubitsky and I. Stewart. Nonlinear dynamics of networks: the groupoid formalism. Bull. Amer. Math. Soc. 43 No. 3 (2006) 305–364.
• Technical:  M. Golubitsky, I. Stewart, and A. Torok. Patterns of synchrony in coupled cell networks with multiple arrows. SIAM J. Appl. Dynam. Sys. 4(1) (2005) 78-100.

Outline and Description

1. Normal form theory. Application to elementary bifurcations of reversible systems in small dimensions.
2. Center manifold reduction for infinite dimensional systems. Case of analytical vector fields. Analytic center manifold up to exponentially small term.
3. Case of infinite dimensional reversible systems. Computation of coefficients of the normal form.
4. Travelling waves in the water wave problem. Spatial dynamics formulation. Spectrum. Bifurcations of various types.
5. Travelling waves in infinite lattices. Spatial dynamics formulation. Spectrum. Bifurcations.
6. Remarks on "limiting" cases, where the reduction method does not apply. 

Reference

The following book can be used as a textbook for this course:

Outline and Description

1. Specific focus: Orbital Mechanics of spacecraft about asteroids, comets and planetary satellites. These environments are the most strongly perturbed environments found in nature for artificial satellites. Thus, they provide an excellent forum for the derivation and discussion of spacecraft orbital mechanics in general.
2. Solved Problems of Astrodynamics: the 2-body problem and torque-free rotational motion. At the heart of astrodynamics are the non-trivial integrable problems of orbital motion of two point masses and the rotation of a torque-free rigid body. These solutions also serve as fundamental starting points for the more detailed description and discussion of non-integrable problems in this field.
3. Models of the force environment at small bodies. To adequately discuss the dynamics of spacecraft several specific topics must be discussed, covering aspects of mass distribution models, gravitational potentials, radiation forces, and the like.
4. The equations of motion: Lagrange's, Hamilton's, and Perturbation forms. Fundamentally, all motion in astrodynamics problems can be described using Newton's 2nd Law. However, there are many aspects of these problems that are hidden with such a fundamental approach, thus it is instructive to develop several different approaches to the mathematical description of these problems. A fundamental result is that (nearly) all astrodynamics problems can be reformulated into a Hamiltonian Dynamics form.
5. Solutions and their properties: equilibria, periodic orbits, and general trajectories. Given that all astrodynamics problems can be reformulated into a Hamiltonian structure, there are strong constraints which can be placed on the properties of motion and solutions in these problems. This section provides detailed results on the stability of solutions and other fundamental constraints that exist for these problems.
6. Computation of solutions and constraints: numerical, analytical and topological. Practical issues for the solution of astrodynamics problems are discussed. These include numerical integration, analytical theories of motion, and approximate theories based on veraging.
7. Case studies: Applications of theory and detailed results for orbital motion in extreme environments. To provide insight into the actual implementation of astrodynamics theory, several different canonical problems will be investigated, motivated by actual systems found in the solar system.
   1. Uniformly rotating asteroids: 433 Eros
   2. Tumbling asteroids: 4179 Toutatis
   3. Small asteroids: 25143 Itokawa
   4. d) Binary asteroids: 1999 KW4
   5. Planetary satellites: The Hill 3-Body Problem

References

Any book on Celestial Mechanics would be a good complement to the lectures. Advanced texts include:

• J.M.A. Danby," Fundamentals of Celestial Mechanics"
• A.E. Roy, "Orbital Motion"
• V. Szebehely, "Theory of Orbits"

For a more general introduction to astrodynamics, consider the texts:

• Bate, Mueller and White, "Fundamentals of Astrodynamics"
• Prussing and Conway, "Orbital Mechanics"

A particularly insightful and concise review of Celestial Mechanics is provided in:

• "Mathematical Aspects of Classical and Celestial Mechanics", V.I. Arnold, V.V. Kozlov and A.I. Neishtadt.

I will have proof chapters of my forthcoming book, "Orbital Motion in Strongly Perturbed Environments" to distribute to the class.