Talk:Lin's method

This article describes the fundamentals of so-called Lin's method from an analytical point of view. This method has proved to be a powerful tool for detecting particular orbits staying for all times close to a given heteroclinic cycle -- such as connecting orbits, periodic orbits, or aperiodic orbits which possibly are related to shift dynamics.

The basic idea is to construct discontinuous orbits - sometimes referred to as Lin orbits - near a heteroclinic chain. Note that homoclinic orbits or heteroclinic cycles can be seen as such a chain. Thereby the discontinuities are jumps of size \(G_i\) in a certain direction within cross-sections of the heteroclinic orbits forming the original chain. The core of the method is the proof of existence and uniqueness of Lin orbits, compare Main Result.

To focus on the idea and to avoid technical difficulties the author considers only the easiest case in the context of ordinary differential equations: the heteroclinic orbits connect hyperbolic equilibria which all have the same index (dimension of the unstable manifold). Besides further degeneracies are excluded.

In section 5 several generalizations are itemized. Although it is afore said that this list is by far not complete I want to mention that recently the method has been adapted to cycles involving periodic orbits, see J. Rademacher, Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit, J. Diff. Eqns. 218 (2005) 390-443, or T.Riess, A Lin's method approach to heteroclinic connections involving periodic orbits - analysis and numerics, Ph.D. thesis, TU Ilmenau, 2008.

However, in view of applications one has to find actual orbits among the set of Lin orbits. In other words, one has to solve the bifurcation equations \(G_i=G_i(\{\omega_i\},\mu)=0\), \(i\in\mathbb Z\). Here \(2\omega_i\) are the transition times along the \(i^{th}\) equilibrium and \(\mu\) is the family parameter. For that practicable estimates of the jumps are needed.

Both the proof of the existence result as well as the jump estimates are highly technical in nature. So, wisely the author confines to give an rough idea how to proceed. Thereby the author refers to the original paper (Lin 1990). However, for many applications more delicate jump estimates are necessary. Those can be gained by a strict separation of the influence of the splitting of the stable and unstable manifolds (of the involved equilibria) and the dependence on the transition times \(\omega_i\) -- compare (Sandstede 1993) and the last paragraph of the above section 2.

Summarizing, this article conveys a good idea what Lin's method is about. For applying the method the reader has to revert to original papers - compare the comprehensive list given in section 5.