Controlling Hamiltonian chaos via Gaussian curvature

Abstract

We present a method allowing one to partly stabilize some chaotic Hamiltonians which have two degrees of freedom. The purpose of the method is to avoid the regions of V(q(1),q(2)) where its Gaussian curvature becomes negative. We show the stabilization of the Hénon-Heiles system, over a wide area, for the critical energy E=1/6. Total energy of the system varies only by a few percent.

Bailout embeddings, targeting of invariant tori, and the control of Hamiltonian chaos

“…PACS numbers: 05.45.Gg Control of chaos in nonlinear dynamical systems has been achieved by applying small perturbations that effectively change the dynamics of the system around the region -typically a periodic orbit -that one wishes to stabilize [1]. This method has been successful in dissipative systems, but its extensions to control and targeting in Hamiltonian systems [2,3,4,5,6,7,8,9,10] have met various difficulties not present in the dissipative case: the absence of attracting sets, for instance, makes it hard to stabilize anything. In addition, these methods require that one know in advance what one wants to do; in particular, the orbit to be stabilized may have to be known in advance to a fair accuracy.In this Letter we present a novel technique allowing us to control Hamiltonian chaos, in such a way as to keep the original dynamics intact, but which shifts the stability of different kinds of orbits in the dynamics.…”

Pumping a swing revisited: minimal model for parametric resonance via matrix products

“…For example, a particle moving where a static potential in two or more dimensions is concave everywhere (i.e. its principal curvatures-eigenvalues of the Hessian matrix of derivatives-are positive at all points on the orbit) can nevertheless exhibit chaos, notwithstanding occasional suggestions to the contrary [14][15][16], (see also [17][18][19]…”

The potential energy surface and chaos in 2D Hamiltonian systems

“…In this case the system exhibits an exponential instability. It is well known that the Lyapunov exponents depend on the eigenvalues of A, and the criterion is exactly the Lyapunov formula applied to 2D Hamiltonian systems [13].…”

“…The connection between the geometric property of the potential energy surface and the instability [10,13] has been discussed. In this Letter, we consider the TB criterion based on analytic geometry, and give a new insight into the relationship between the geometric property of the potential energy surface and chaotic behavior of 2D Hamiltonian dynamical systems.…”