Bifurcations in Hamiltonian Systems with a Reflecting Symmetry

Abstract

A reflecting symmetry q → −q of a Hamiltonian system does not leave the symplectic structure dq ∧ d p invariant and is therefore usually associated with a reversible Hamiltonian system. However, if q → −q leads to H → −H , then the equations of motion are invariant under the reflection. Such a symmetry imposes strong restrictions on equilibria with q = 0. We study the possible bifurcations triggered by a zero eigenvalue and describe the simplest bifurcation triggered by non-zero eigenvalues on the imaginary axis.

Shape curvature effects in viscous streaming

“…We exploit this equivalence to map the transitions seen in our lattice system ( § 4) to well-studied bifurcations in Hamiltonian systems. Once such a bifurcation is identified, we borrow the corresponding reduced Hamiltonian form H(x, y), which mathematically captures topology changes near bifurcating critical points (Bosschaert & Hanßmann 2013;Strogatz 2018). This allows us to predict how the flow evolves upon perturbing shape curvature and/or background flow conditions.…”

“…These flow regions exist only along one of the cylinder symmetry axes. The simplest Hamiltonian form that captures this transition is H(x, y) = axy 2 + bx 3 + βx with ab > 0, which corresponds to a hyperbolic reflecting umbilic bifurcation (Bosschaert & Hanßmann 2013). Here, βx is the unfolding term, that controls the appearance (going from β > 0 to β < 0) of the recirculating region pairs and their size (figure 7e-g).…”

“…After the transition these saddles are located on the horizontal axis, thus recovering Phase VII (figure 7k). The simplest Hamiltonian form that captures this rearrangement is H(x, y) = axy 2 + bx 3 + βx with ab < 0, which corresponds to an elliptic reflecting umbilic bifurcation (Bosschaert & Hanßmann 2013). Here, βx is the unfolding term, that captures whether the saddles are present (β < 0) or absent (β > 0) on the horizontal axis (figure 7l-n), as well as their distance.…”

Equivariant singularity analysis of the 2 : 2 resonance

“…We remark that the Hamiltonian function (1) could also be invariant under other transformations, such as reflections acting on the x and not on the p and viceversa. Our choice to consider reflection symmetries (2) and (3) lies in the Lagrangian description of a reversible system, giving up using all possibilities of the Hamiltonian description [5]. We assume the zero-order term H 0 to be in the positive definite form…”

Periodic Orbits in Hamiltonian Systems with Involutory Symmetries

“…The second problem we discuss is the dynamics near an equilibrium point in an equivariant Hamiltonian system with an involutory (time preserving) symmetry S acting anti-symplectically. Bifurcations of equilibria in Hamiltonian systems with such symmetry have been considered recently by M. Bosschaert and H. Hanßmann [4]. Existence theorems for periodic solutions in symmetric Hamiltonian systems can be found in Montaldi et al [14], [15], but this and related work assumes the symmetry transformation acts symplectically.…”