Comparing Symplectic and Time-Reversible Methods

Interestingly the time-reversal symmetry provides analogous properties in some respects to the properties of Hamiltonian systems. For example, consider a linear Hamiltonian system dz/dl = JAz, with A a symmetric matrix. If A is an eigenvalue of JA then JAu = Xu for some eigenvector u 0. Because the matrix JA is real, we know that A will also be an eigenvalue. At the same time, we know that since A is an eigenvalue of JA it is also an eigenvalue of its transpose (JAY = A J = —AJ, thus 

Suppose now we have a linear time-reversible map (z) = Tz, then 

Thus A is an eigenvalue of which, in turn, implies that 1 /A is an eigenvalue of T. The matrix being real implies that the conjugates of A and 1 /A are also eigenvalues, thus we have the same eigenvalue quadruplets as for a linear symplectic map. 

Recall that a pair of maps and 9 are said to be conjugate if there is a homomorphism / such that 

In such a case the iterates of the two maps will also be conjugate and, if they are numerical methods, they will have similar stability properties and performance (e.g. the same effective order). It is difficult to separate the relevance of the two properties in cases where symplectic and reversible maps are conjugate. This is however rarely the case and certainly does not hold generically for discrete maps in many dimensions [209]. There is no direct correspondence between reversible and symplectic maps, however each class of maps admits certain theorems of dynamical systems which are in many ways analogous (for example, the Kolmogorov-Arnold-Moser, or KAM, theory for symplectic maps near elliptic fixed points [386] has an analogue for reversible maps [97]). 

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