Time-reversal symmetry in dynamical systems: a survey

Action minimizing orbits in hamiltomian systems

Greene (1979) proposed a relationship between the existence of an invariant circle for an area-preserving map and the 'residues' of periodic orbits with nearby rotation number. It remains without a doubt the most effective practical criterion for calculating the breakup of invariant circles. In this work four results are proved which go a long way towards placing the criterion on a rigorous foundation, and the issues remaining to be resolved in order to formulate a rigorous version of the criterion are discussed.

http://iopscience.iop.org/article/10.1088/0951-7715/5/1/007/pdf

Greene's residue criterion

Let in be an exact, area-preserving, monotone twist diffeomorphism of the cylinder and let 10 be a Liouville number. We will show that arbitrarily close to in the C ∞ topology there exists a C ∞ diffeomorphism with no homotopically non-trivial invariant circle of rotation number ω.

/pdf/destruction-of-invariant-circles-2grs61bfzr.pdf

Destruction of invariant circles

A linear code for the sawtooth and cat maps

We investigate invariant circles for a one-parameter family of piecewise linear twist homeomorphisms of the annulus. We show that invariant circles of all types and rotation numbers occur and we classify them into families. We compute parameter ranges in which there are no invariant circles.

/pdf/invariant-circles-for-the-piecewise-linear-standard-map-b701zjte50.pdf

Shaun Bullett

Papers

Invariant circles for the piecewise linear standard map