Nonergodicity of the motion in three-dimensional steep repelling dispersing potentials

Abstract

It is demonstrated numerically that smooth three degrees of freedom Hamiltonian systems that are arbitrarily close to three-dimensional strictly dispersing billiards (Sinai billiards) have islands of effective stability, and hence are nonergodic. The mechanism for creating the islands is corners of the billiards domain.

Stability in High Dimensional Steep Repelling Potentials

“…The search for the periodic orbit is unnecessary by Lemma 1 (using symmetry and proper parameters) and the need to compute eigenvalues of large matrices is demolished by Lemma 2: for all n we find the solutions of one second-order non-linear equation (2.15) and the monodromy matrix of one second-order linear equation (3.2),(3.3) which depends on n as a parameter. The steep limit is handled as in [20]: we fix ε and increase the size of the billiard domain (r in (2.2)) to get an effectively small ε = ε/r without running into stiffness problems (in the bulk of the domain the motion is essentially inertial and non-stiff).…”

“…To find the stability regions, as shown in Figure 4, we use the continuation scheme which was developed in [20]; first we compute the stability of γ(t) at µ = 0 (the case of a cusp created by n tangent spheres) along the ε-axis (see Figure 4 left). By symmetry (see Lemma 2), Re(|λ n (µ = 0, ε))|) > 1 always corresponds to real eigenvalue (i.e.…”

“…The 3-corner is a point at which three smooth spheres of identical radius intersect in a symmetric fashion, so that only one characteristic parameter µ controls the angle of their intersection (µ = 0 corresponds to a cusp whereas µ = 1 corresponds to a complete overlap of the spheres, see Figure 1). It is demonstrated numerically in [20] that for any value of ε there are intervals of µ values for which the symmetric orbit is elliptic. Here, we generalize this example to the ndimensional case, for arbitrary large n, proving, that for certain classes of smooth repelling potentials (such as the power-law family) the smooth symmetric orbit which enters the vicinity of an n-corner has, for arbitrary small ε, intervals of µ values for which it is elliptic (all its 2n multipliers belong to the unit circle).…”

“…Recently, it was demonstrated numerically that regions of effective stability, hereafter called islands, are created in steep dispersing three-dimensional billiards for what appears to be arbitrarily small ε [20]. Before further describing this construction and its current generalization to the n degrees of freedom case, let us discuss the issue of islands in the multi-dimensional context.…”

“…The islands constructed in [20] are produced by a highly symmetric orbit of the smooth system which visits the vicinity of a symmetric 3-corner. The 3-corner is a point at which three smooth spheres of identical radius intersect in a symmetric fashion, so that only one characteristic parameter µ controls the angle of their intersection (µ = 0 corresponds to a cusp whereas µ = 1 corresponds to a complete overlap of the spheres, see Figure 1).…”